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<strong>FRITZ</strong> <strong>NOETHER</strong> (<strong>1884</strong> - <strong>194</strong>?)
The index of a linear operator may be defined as the<br />
difference between the dimension of its kernel and the co-<br />
dimension of its image. The index of an integral operator was<br />
first introduced by Fritz Noether in his important pioneering<br />
paper "Uber eine Klasse singul~rer Integralgleichungen,"<br />
Math. Ann. 82 (1920), 42-63. In the same paper he proposed the<br />
first index formula. During the 65 years which have passed<br />
since then, the topic of the index has become one of the central<br />
topics in mathematics.<br />
This issue of Integral Equations and Operator Theory<br />
is dedicated to the memory of Fritz Noether. It contains a<br />
selection of up-to-date papers on the index theory. Included<br />
is the biography of Fritz Noether with its very tragic end.<br />
This biography is written by his son for this issue. There is<br />
also a list of his publications compiled by A. Dick.<br />
J.A. Dieudonn~ has written for this issue a paper about the<br />
early history of the index theory.<br />
I wish to thank all authors for their contributions<br />
and cooperation.<br />
I. Gohberg
Integral Equations<br />
and Operator Theory<br />
Vol. 8 (1985)<br />
<strong>FRITZ</strong> <strong>NOETHER</strong> (<strong>1884</strong> - <strong>194</strong>?)<br />
Gottfried E. Noether<br />
0378-620X/85/050573-0451.50+0.20/0<br />
9 1985 Birkh~user Verlag, Basel<br />
Fritz Noether was born on October 7, <strong>1884</strong>, in the<br />
university town of Erlangen in southern Germany, where his father,<br />
Max Noether, was University Professor of Mathematics. Though<br />
Erlangen was the smallest of the three universities in the State<br />
of Bavaria, it had a high reputation in the field of mathematics.<br />
In 1872, in his inaugural lecture, Felix Klein had developed<br />
what became known as the "Erlanger Programme." Two years later,<br />
Paul Gordan, who was to become known as the "King of Invariants,"<br />
came to Erlangen. He was joined one year later by Max Noether,<br />
at first as ausserordentlicher (associate) professor, and from<br />
1888 until his retirement in 1918 as Ordinarius (full professor).<br />
The NSther ancestors had been well-to-do Jewish<br />
tradesmen in the Black Forest area of Germany. (The spelling of<br />
the name was changed to the present spelling by Max.) In 1838,<br />
two brothers, Joseph and Hermann, moved to Mannheim to start a<br />
wholesale iron business that remained in NSther hands for i00<br />
years until its "aryanization" in 1937 by the Nazi regime. Of<br />
the three sons of Hermann NSther, two continued in the family<br />
business, but the third son, Max, handicapped since the age of<br />
fourteen from the effects of infantile paralysis~ turned to the<br />
study of mathematics, an interest that seems to have come from<br />
his mother's family, Jewish merchants in Mannheim. In 1868,<br />
I am greatly indebted to Dr. August Dick in Vienna, author<br />
of Emmy Noether (1885 - 1935), Boston: Birkhauser, 1981,<br />
for compiling detailed information about the life of<br />
Fritz Noether, partfedlarly his early life.
574 Noether<br />
after study in Mannheim, Heidelberg, Giessen, and G~ttingen,<br />
Max Noether was awarded the Ph.D. in Mathematics from the<br />
University of Heidelberg. After five years as Privatdozent in<br />
Heidelberg during which he began to establish his reputation as<br />
an algebraic geometer, he was called to Erlangen, where he would<br />
stay until his death in 1921.<br />
In 1880, Max Noether married Ida Kaufmann, who came<br />
from a well-to-do Jewish family in the Cologne area. They had<br />
four children. Emmy, the oldest, born in 1882, was to become<br />
Paul Gordan's only doctoral student. Of her, Edmund Landau<br />
would say that "she was the coordinate origin of the Noether<br />
family."<br />
The third child of Max and Ida Noether was Fritz.<br />
Fritz received the customary education for boys of the bourgeois<br />
middle class, elementary school from age six and humanistic<br />
Gymnasium from age ten. In 1903, he passed the Abitur entitling<br />
him, after fulfilling his military duty, to admission to the<br />
university. Starting with the winter semester 1904/5, he<br />
studied mathematics and related subjects for nine semesters,<br />
first at the University of Erlangen and then in Munich. In<br />
March 1909, he defended his doctoral dissertation Uber rollende<br />
Bewegung einer Kugel auf Rotationsfl~chen. His Ph.D. adviser<br />
at the University in Munich was Professor A. Voss. But in the<br />
dissertation, the candidate gave equal thanks to Professor<br />
A. Sommerfeld for valuable inspiration. A close working relation-<br />
ship between Fritz Noether and A. Sommerfeld is indicated by the<br />
fact that Part 4 (Die technischen Anwendungen der Kreisel-<br />
theorie) in F. Klein's and A. Sommerfeld's Uber die Theorie des<br />
Kreisels, published in 1910 by Teubner in Leipzig, bears the<br />
notation: Prepared for publication and supplemented by Fritz<br />
Noether. Indeed, Sommerfeld's introductory remarks refer to<br />
Fritz Noether as collaborator for this fourth and final part of<br />
the Klein and Sommerfeld work on gyroscopes.<br />
While Max and Emmy Noether were primarily interested<br />
in pure mathematics, Fritz Noether embarked on a career in
Noether 575<br />
applied mathematics and mechanics. After postdoctoral studies in<br />
G~ttingen, he became assistant to Karl Heun at the Technische<br />
Hochschule in Kar!sruhe. In the summer of 1911, on presentation<br />
of Uber den G~Itigkeitsbereich der Stokesschen Widerstandsformel,<br />
he obtained the venia legendi (permission to lecture) and was<br />
appointed Privatdozent for Theoretical Mechanics and Mathematics.<br />
In December 1911, he married Regina W~rth of Randegg on the<br />
Bodensee, daughter of a customs official. There were two sons,<br />
Hermann, born September 21, 1912, and Gottfried, born January 7,<br />
1915.<br />
During World War I, Fritz Noether saw active service on<br />
the German-French front and, after being wounded, was assigned to<br />
doing research in ballistics. At the end of the war, he returned<br />
to teaching at the Teehnische Hochsehule in Karlsruhe, where in<br />
the meantime he had been promoted to associate professor. In<br />
1921 he took a leave of absence to do industrial research for<br />
the Siemens-Konzern in Berlin. In the fall of 1922 he accepted<br />
a full professorship at the Technische Hochschule in Breslau,<br />
which he held until 1934, when, though not yet 50, he was forced<br />
into retirement for "racial reasons." He thus shared Emmy<br />
Noether's fate, whose permission to teach - along with that of<br />
several other Gottingen mathematicians - had been withdrawn one<br />
year earlier. While Emmy Noether accepted a guest professorship<br />
at Bryn Mawr College in the United States, Fritz Noether decided<br />
to accept the offer of a professorship at the Institute of<br />
Mathematics and Mechanics at the University of Tomsk in the USSR.<br />
In the summer of 1934, brother and sister met for a last time<br />
before departing in opposite directions. Less than a year later<br />
Emmy Noether died very suddenly in Bryn Mawr following a tumor<br />
operation. Fritz Noether was the honored guest at the Emmy<br />
Noether Memorial Session held September 5, 1935, in Moscow with<br />
P.S. Alexandrov, then President of the Moscow Mathematical<br />
Society, delivering the eulogy. Barely two years later, in<br />
November 1937, at the height of the Stalinist purges, Fritz<br />
Noether was arrested by the NKVD in his home in Tomsk.
576 Noether<br />
Informa about Fritz Noether after his arrest is<br />
fragmentary. At some time, most likely before the end of 1939,<br />
he was in Orel. In December 1939 and January <strong>194</strong>0, he was seen<br />
in Butyrka Prison in Moscow. There is a report that he was seen<br />
in the center of Moscow toward the end of <strong>194</strong>1 or the beginning<br />
of <strong>194</strong>2. But numerous attempts, official and unofficial, to<br />
find out more about his fate remain unsuccessful.<br />
At the time of his arrest, Fritz Noether had completed<br />
the manuscript of a book on Bessel functions. The manuscript<br />
had been translated into Russian and was scheduled for publica-<br />
tion in the Soviet Union. It is not known whether the book has<br />
ever been published.*<br />
The University of Connecticut<br />
Storrs<br />
CT 06268<br />
U.S.A.<br />
The author would appreciate receiving any relevant<br />
information
Integral Equations<br />
and Operator Theory<br />
Vol. 8 (1985)<br />
i.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
i0.<br />
Ii.<br />
12.<br />
13.<br />
14.<br />
15.<br />
LIST OF PUBLICATIONS OF <strong>FRITZ</strong> <strong>NOETHER</strong><br />
0378-620X/85/050577-0351.50+0.20/0<br />
9 1985 Birkh~user Verlag, Basel<br />
(Assembled by Dr. Auguste Dick, Vienna)<br />
Uber rollende Bewegung einer Kugel auf Rotationsfl[chen,<br />
Leipzig, Teubner, 1909. Inaugural Dissertation<br />
Zur Kinematik des starren K~rpers in der Relativtheorie,<br />
Annalen der Physik (4) 31 (1910~ 919-944<br />
Uber den G[itigkeitsbereich der Stokesschen Widerstands-<br />
formel, Leipzig, Teubner, 1912. Habilitationsschrift<br />
Z.f. Mathematik und Physik, 62 (1914) 1-39<br />
Uber die Entstehung einer turbulenten Fl[ssigkeits-<br />
bewegung, Sber. Bayerische Akad. Wiss 1913, 309-329<br />
Zur Theorie der Turbulenz, Jber. DMV 23 (1914),<br />
138-144<br />
Zur Theorie der Turbulenz, Nachr. Ges. Wiss. G@ttingen,<br />
1917, 199-212<br />
Uber analytische Berechnung der Geschosspendelungen,<br />
Nachr. Ges. Wiss. G6ttingen, 1919,373-391<br />
(Uber Geschosspendelungen) ~<br />
Artilleristisehe Monatshefte, Mai/Juni, 1919, 170-204<br />
Bemerkung ~ber die L@sungszahl zueinander adjungierter<br />
Randwertaufgaben bei linearen Differentialgleichungen,<br />
Sber. Heidelb. Akad. Wiss., 1920, 1-12<br />
Uber eine Klasse singul~rer Integralgleichungen,<br />
Math. Ann. 82 (1920), 42-63<br />
Das Turbulenzproblem, ZAMM i (1921), 125-138; 218-219<br />
Uber Stromaufnahme in Metallrohrleitungen und<br />
verwandte Erdungsfragen, Wiss. Ver@ff. aus dem<br />
Siemenskonzern, i (1921), 35-60<br />
Uber die Abstimmung der L6sehdrosseln I, Elektroteehn.<br />
Zeitsehr. 1921, 1478-1482<br />
Uber die Abstimmung der L6sehdrosseln II, Elektrotechn.<br />
Zeitschr. 1922, 385-388<br />
Uber eine Aufgabe der Kapazit[tsberechnung, Wiss.<br />
Ver@ff. aus dem Siemenskonzern, 2 (1922), 198-202<br />
ohne Titel erw~hnt in (7), S. 17<br />
Titel wie (7)[12.3.82 A.D.]
578 Dick<br />
16.<br />
17.<br />
18.<br />
19.<br />
20.<br />
21.<br />
22.<br />
23.<br />
24.<br />
25.<br />
26.<br />
27.<br />
28.<br />
29.<br />
30.<br />
Uber ein Problem der Strom!eitung , ZAMM 2 (1922),<br />
274-278<br />
Zur asymptotischen Behandlung der station[ren LSsungen<br />
im Turbulenzproblem, ZAMM 6 (1926), 232-243~ 497-498<br />
Uber Stromverdr[ngung in zylindrischen Leitern von<br />
allgemeiner Querschnittsform, Annalen der Physik (4)<br />
84 (1927), 775-778<br />
Bestimmung der Stromverteilung in zylindrischen<br />
Leitern von allgemeiner Querschnittsform, ZAMM 7<br />
(1927), 452-454<br />
Uber den "Strahl" begriff in der Fresnelschen und der<br />
Maxwellschen Krystalloptik, Probleme der modernen<br />
Physik, Hsg. P. Debye, Leipzig, Hirzel, 1928, 58-64<br />
Berechnung von elektrisehen StrSmungsfeldern,<br />
Handbuch der Physik, 13 (1928) 76-102<br />
Kerl Heun T (Naehruf), ZAMM 9 (1929) 167-171<br />
Bemerkungen zur 0seenschen Hydrodynamik, ZAMM 9 (1929)<br />
5O9<br />
Zur statistischen Deutung der K~rm~nschen Ahnlich-<br />
keitshypothese in der Turbulenztheorie, ZAMM, ii<br />
(1931) 224-231<br />
Uber die Verteilung des Energiestroms bei der Total-<br />
reflexion, Annalen der Physik (5) ii (1931), 141-146<br />
Ausbreitung elektrischer Wellen 6ber der Erde<br />
Funktionentheorie und ihre Anwendung in der Technik,<br />
Hsg. R. Rothe u.a. Berlin, Springer, 1931, 154-170<br />
Integrationsprobleme der Navier-Stokesschen Differen-<br />
tialgleichungen, Handbuch der Physikalischen und<br />
techn. Mechanik, Bd. 5, 3. Leiferung, Leipzig, Barth,<br />
(1931), 719-796<br />
Anwendung der Hillschen Differentialgleichung auf<br />
die Wellenfortpflanzung in elektrischen oder<br />
akustischen Kettenleitern, Verh. des 3. Internat.<br />
Kongr. f. Techn. Mechanik, Stockholm, 3 (1931), 143-149<br />
mit E. Waetzmann: Uber akustische Filter, Annalen<br />
der Physik (5) 13 (1932), 212-228<br />
Dynamische Gesichtspunkte zu einem statistischen<br />
Turbulenztheorie, ZAMM 13 (1933), 115-120
Dick 579<br />
31.<br />
32.<br />
33.<br />
34.<br />
35.<br />
36.<br />
Bemerkungen 6ber das Ausbreitungsgesetz f6r lange<br />
elektrische Wellen und die Wirkung der Heavisides-<br />
chieht, Elektr. Nachrichtentechnik, i0 (1933), 160-172<br />
Das station[re (und quasistation[re) elektromagnetische<br />
Feld Differential- und Integralgleichungen der<br />
Mechanik und Physik, hsgg. von Phillip Frank und<br />
Richard von Mises. 2., vermehrte Auflage, zugleich 8.<br />
Aufl. von Riemann-Webers partiellen Differential-<br />
gleichungen der mathematischen Physik. Vieweg,<br />
Braunschweig, 1935; im 2. (physikal.) Teil 627-755,<br />
Unver[nderter Nachdruck, New York, Dover Pub.,<br />
Braunschweig, Vieweg, 1961"<br />
Elektromagnetische Wellen an einem Draht bei konzen-<br />
trierter Energiequelle, Physikal. Zeitschr. der<br />
Sowjetunion, 8 (1935) (1-24)**<br />
Uber die Rekursionsformeln der Besselschen und<br />
Hermiteschen Funktionen. (In russischer Sprache mit<br />
deutscher Kurzfassung), Mitt. des Forschungsinstituts<br />
f~r Mathematik und Mechanik an der Kujbyschew-<br />
Universit~t Tomsk, i (1935), 120-125<br />
Asymptotische Darstellungund geometrische Optik<br />
(In deutscher Sprache mit russischer Kruzfassung)<br />
Mitt. des Forschungsinstituts fir Mathematik und<br />
Mechanik an der Kujbyschew-Universit~t Tomsk, i<br />
(1935), 175-189"**<br />
Uber elektrische Drahtwellen, C.R. Congr. Internat.<br />
Math., 0slo, 2 (1937), 234-235<br />
Zweiter/physikalischer/Teil, unter Mitarbeit von<br />
G. Beck-Kansas (USA), R. F6rth-Prag, R. v. Mises-<br />
Istanbul, F. Noether-Tomsk, G. Schulz-Berlin,<br />
A. Sommerfeld-M~nchen, E. Trefftz-Dresden<br />
Paginierung nach dem Separatum<br />
gezeichnet ist diese Arbeit mit: Tomsk, 25. 2. 1936<br />
und enth[it einen Hinweis des Autors auf ein demn[chst<br />
in russischer Sprache erscheinendes Buch 6ber<br />
Besse!sche Funktionen.
Integral Equations<br />
and Operator Theory<br />
Vol. 8 (1985)<br />
0378-620X/85/050580-I051.50+0.20/0<br />
9 1985 Birkh~user Verlag, Basel<br />
THE INDEX OF OPERATORS IN BANACH SPACES<br />
J.A. Dieudonn@<br />
The notion of index of an operator is a central one in<br />
the theory which leads to the famous Atiyah-Singer theorem, one<br />
of the highlights of modern Global Analysis. In this paper I<br />
want to give an historical sketch of the development of that<br />
notion during the first half of the 20th century, providing the<br />
tools of "Functional Analysis" used in the Atiyah-Singer proof.<br />
I. The Fredholm Alternative<br />
The fundamental result proved by I. Fredholm in 1900 is<br />
called the "Fredholm alternative." Let I = [0,i] and suppose<br />
for simplicity that we start with a continuous "kernel" K(s,t)<br />
in I x I. Consider the linear integral operator K in the<br />
space C(~) of continuous functions, defined by<br />
(i) (K.~)(t) = flK(s,t)~(s)ds ;<br />
then the kernel N = Ker(l + K) of the operator I + K in<br />
C(I) has finite dimension, and the image F = Im(l + K) is the<br />
space of functions ~[C(I) satisfying the "orthogonality<br />
relations"<br />
(2) f1~(t)ki(t)dt = 0<br />
for the functions k. forming a basis of the kernel N' of the<br />
i<br />
"transposed" integral operator I + K' , where<br />
(3) (K'.~)(t) = fiK(t,s)~(s)ds 9<br />
Furthermore the dimensions of N and N' are equaZ, which<br />
implies that F is a closed subspace of C(1), of finite<br />
codimension given by<br />
(4) eodim F = dim N<br />
This result did not seem at all surprising, since relation (4)<br />
was well known for endomorphisms of finite dimensional spaces,<br />
and I + K appeared as a kind of "limit" of such operators.
Dieudonn4 581<br />
II. The Riesz-Fredholm theory of compact operators.<br />
Already Fredholm had seen that his results would still<br />
be valid if it was only assumed that Ix-yleK(x,y) is continu-<br />
ous for some e < i After the introduction of the Hilbert<br />
space L 2, it was easily seen that the "Fredholm alternative"<br />
also held when Ks X I) and the operator K in L2(1) is<br />
defined by (i) for ~6L2(I).<br />
The big step forward in the theory of these operators<br />
was taken by F. Riesz. He was inspired by Hilbert's theory of<br />
"bounded operators" in Z2, and in particular by his concept of<br />
"completely continuous" operators, which included the integral<br />
operators K defined by (i) for a symmetric kernel, but were<br />
more general. Hilbert's definition was given in the language of<br />
bilinear forms~ F. Riesz translated it in terms of operators:<br />
at first, for him, a completely continuous operator on L 2 (or<br />
in L p for i < p < + ~) was defined by the condition that it<br />
transforms aweakly convergent sequence into a strongly conver-<br />
gent one. Then in 1915, he realized that this definition is<br />
equivalent to the following one: the operator transforms any<br />
bounded set into a relatively compact one. But then that defini-<br />
tion obviously can be extended to a linear map u: E -> F when<br />
E and F are arbitrary normed spaces, and this justifies the<br />
name compact operator, later adopted for such maps. In a beauti-<br />
ful paper published in 1918, F. Riesz showed that the Fredholm<br />
theorems were valid for any endomorphism v = i + u of the space<br />
C(1), where u is a compact endomorphism; at that time<br />
general normed spaces had not yet been defined, but Riesz's<br />
arguments apply without change to an arbitary normed space.<br />
For future reference, it is convenient to rapidly<br />
describe Riesz's procedure (which, even for the integral opera-<br />
tor (i), gives more information that Fredholm's). Initiating<br />
in Analysis, a method first used in algebra by E. Weyr in 1890<br />
to obtain the Jordan normal form of a complex matrix, he<br />
considers, for the iterated operators v m, the subspaces<br />
F m = Im(v m) and N m = Ker (vm). By an elementary (but clever)
582 Dieudonn4<br />
use of the definition of compactness, he shows that the increas-<br />
ing sequence<br />
(5) 0cNICN2 c'''cNmc''"<br />
consists of finite dimensional subspaces, and the decreasing<br />
sequence<br />
(6) E~FI~F2 ~'''~Fm~''"<br />
consists of closed subepaces of finite codimension.<br />
Furthermore, both sequences are stationary: there is<br />
a smallest integer k for which Nk+ j = N k and Fk+ j = F k for<br />
all j > 0 ; E is the topological direct sum F k ~ N k ; the<br />
restriction of v to F k is an automorphism of F k , and the<br />
restriction of v to N k is a nilpotent endomorphism of that<br />
finite dimensional space, such that v m is not identically 0<br />
k<br />
in N k for m < k, but v = 0.<br />
Several authors later found more general endomorphisms<br />
of Banach spaces for which the preceding decomposition holds;<br />
for these operators v, one therefore always has the relation<br />
(7) dim Ker (v) = codim Im(v) (= dim Coker(v))<br />
The most interesting ones were described by Ruston in 1954; in<br />
the Banach space End(E), let K be the closed subspace con-<br />
sisting of compact operators; Ruston's operators u are<br />
"close to K ', in the fol]owing sense: if d is the distance<br />
in End(E),<br />
(8) lim (d(un,K)) I/n = 0.<br />
Then the Riesz theory applies without change to these operators:<br />
the spectrum of u is at most countable, each point h # 0 in<br />
the spectrum is an isolated eigenvalue of finite multiplicity,<br />
and there is a decomposition in a topological direct sum<br />
E = F(h) ~ N(h), such that the restriction of u -X.I E to<br />
F(1) is an automorphism of that subspace, N(~) has finite<br />
dimension and the restriction of u-X.l E to N(X) is a nil-<br />
potent endomorphism of N(~).
Dieudonn4 583<br />
III. The F. Noether paper of 1921<br />
The first paper in which appeared in a natural way an<br />
operator v : E + F for which Ker(v) and Coker(v) are both<br />
finite dimensional, but have different dimensions, was published<br />
in 1921 (of course, it was later easy to concoct ad hoe examples,<br />
such as the shift in Z2). Its author was Fritz Noether, Emmy<br />
Noether's brother, who later was a professQr in a Russian uni-<br />
versity and disappeared in Siberia before World War II.<br />
In his paper he considers a "singular integral equa-<br />
tion" which had first been studied by Hilbert, in his address to<br />
the International Congress of Mathematicians at Heidelberg in<br />
1904. He wanted to solve the "Riemann problem" for holomorphic<br />
functions: given a bounded plane domain A, limited by a<br />
smooth curve F, find a function ~ + i~ continuous in A O F,<br />
holomorphie in A and such that on F<br />
(9) a(s)~(s) + b(s)~(s) : f(s)<br />
where s is a parameter varying in [-~, + z] and proportional<br />
to arc length in F , and a, b, f are continuous periodic<br />
functions of period 2z. Using classical properties of the<br />
Green function in A, Hilbert showed that, if<br />
Cl0)<br />
k(s,t) : ~b(s).cot~-e + A(s,t)<br />
where A<br />
equation<br />
is continuous, then ~ must be a solution of the<br />
r (K.~)(s) ~ a(s)~(s) + v.p.J~K(s,t)~(t)dt- = f(s)<br />
where v.p. means the Cauchy "principal value" of the integral<br />
(since the integrand has a pole of order I at the point t = s).<br />
Hilbert's solution of (i0) consists in applying again the<br />
operator K to both sides, and proving by direct computation<br />
that the equation<br />
r K.(K.~) = K.f<br />
is a Fredholm integral equation (i) with continuous kernel.<br />
This may look like a trick, but in the light of modern Function-<br />
al Analysis it is quite natural: K is a pseudo-differential<br />
operator of order -i on the circle, hence I K.X is a pseudo-<br />
differential operator of order -2, and it is well known that
584 Dieudonn4<br />
such operators are of type (i).<br />
Hilbert probably thought that the "Fredholm alter-<br />
native" still held for the operator K, for when he wanted to<br />
prove uniqueness of a solution of (Ii), he only checked that<br />
~.~ = 0 had no nontrivial solution, whereas he should have con-<br />
sidered instead the "transposed" equation tK.~ = 0, K(s,t)<br />
being replaced by K(t,s).<br />
F. Noether's discovery was that Hilbert's assumption<br />
was not always true. He observed that if ~ is a nontrivial<br />
solution of K.~ = 0, the corresponding holomorphic function<br />
+ i~ should satisfy<br />
(13) fFd(log(~+i~)) = fFd(log(a-ib)) = 2~n<br />
where n is a positive or negative integer. If n < 0, this<br />
would mean that ~ + i ~ has poles, in 4, so there is no<br />
solution to K.~ = 0; on the contrary, if n Z 0, ~.~ = 0 has<br />
2n linearly independent solutions, so that the difference<br />
between both sides of (7) is 2n, given by the integral along F<br />
~fFd(log(a-ib))<br />
(first example of expression of the index by topological means).<br />
IV. Small Perturbations<br />
It is probably an old idea to study an equation which<br />
"differs slightly" from another one by assuming that the solu-<br />
tions of both equations also "differ slightly" from one another:<br />
it is for instance the principle which has been used in celestial<br />
mechanics since the beginning of the 18th century. In spectral<br />
theory, very soon the same idea was found useful, for instance<br />
in Hilbert's and E. Sehmidt's work on linear operators in<br />
Hilbert space, which they "approximate" by operators of finite<br />
rank; or in the investigations of G.D. Birkhoff-Kellogg and<br />
Schauder on fixed point theorems for operators in function<br />
spaces, with the help of Brouwer's fixed point theorems in<br />
finite dimensional spaces.<br />
A more general viewpoint may be found in the series<br />
of papers which Rellich, in 193Y-<strong>194</strong>2, devoted to operators in
Dieudonn4 585<br />
Hilbert space which depend "continuously" or "analytically" on a<br />
complex parameter; they were essentially concerned with the way<br />
in which the spectrum of the operator varies with the parameter.<br />
In <strong>194</strong>3, in connection with Bourbaki's program of study<br />
of topological vector spaces, I was led to investigate what<br />
happens in general when a continuous linear map u : E + F<br />
between normed spaces is "perturbed" by the addition of a "small"<br />
map w : E § F . I naturally gave to the word "small" the mean-<br />
ing that the norm llwll would be small; the investigation was<br />
chiefly directed to finding, for v = u + w, relations between<br />
Ker(u) and Ker(v) , and between Im(u) and !m(v)<br />
An elementary case had been known and used for a long<br />
time: if u is an isomorphism of E onto a subspace of F ,<br />
then the same is true for v if I I w I I < ]I u-ll I 9 In<br />
addition, if u(E) is closed in F , and F is the topological<br />
direct sum of u(E) and a closed subspace G , then F is also<br />
the topological direct sum of G and v(E) (which is therefore<br />
closed in F).<br />
The theory of normed spaces had introduced a generali-<br />
zation of the notion of isomorphism onto a subspace: any<br />
continuous linear map u : E + F factorizes in<br />
E P > E/(Ker(u)) v_~_> F<br />
and it was natural to consider the case in which v is an iso-<br />
morphism of E/Ker(u) onto a subspace of F ; such maps u are<br />
now called strict homomorphisms. Even when E and F are<br />
complete (i.e. Banach spaces), it is trivial to see that there<br />
exist strict homomorphisms u such that v = u + w is not a<br />
strict homomorphism for arbitrary small llwll (for instance<br />
u = 0). In this ease Ker(u) has infinite dimension and Im(u)<br />
infinite codimension; I realized that only by strong assump-<br />
tions on Ker(u) or Im(u), results similar to the elementary<br />
theorem on isomorphisms could be ohtained for strict homomorphisms.<br />
For Banach spaces E, F, Banach had proved two deep<br />
results on strict homomorphisms:<br />
i) A continuous linear map u : E + F is a strict<br />
homomorphism if and only if u(E) = Im(u) is closed in F.
586 Dieudonn4<br />
2) If E ~ and F* are the Banach spaces dual to E<br />
and F (with the strong topology), then in order that<br />
Im(u) = F, it is necessary and sufficient that the transposed<br />
t<br />
map u : F* § E* be an isomorphism of F* onto a (necessarily<br />
closed) Subspace of E*<br />
For Banach spaces E,F, these theorems and the<br />
elementary isomorphism theorem imply that if u : E § F is such<br />
that Im(u) = F, then, for llwll small enough, one has also<br />
Im(v) = F for v = u + w ; if in addition E is the topologi-<br />
cal direct sum of Ker(u) and a closed subspace G, then it is<br />
also the topological direct sum of Ker(v) and G.<br />
None of these properties holds for non complete normed<br />
spaces E,F, even if u is a strict homomorphism.<br />
The main special cases of applications of these resultS<br />
to a continuous linear map u : E § F and its perturbed map<br />
vmu+w are:<br />
(i) If E and F are Banach spaces, u a strict<br />
homomorphism and dim Coker (u) < + ~ then, for I lwl I small<br />
enough, v is a strict homomorphism and dim Coker (v)<br />
dim Coker (u). This is not true if E and F are not complete.<br />
(ii) If E and F are Banach spaces, u a strict<br />
homomorphism and dim Ker (u) < + ~ then, for Ilwll small<br />
enough, v is a strict homomorphism and dim Ker (v) !<br />
dim Ker (u) This is still true if E and F are not<br />
complete.<br />
(iii) If E and F are Banach spaces and both<br />
dim Ker (u) and dim Coker (u) are finite, this already implies<br />
that u is a strict homomorphism; if<br />
C14) ind(u) : dim Ker(u) - dim Coker(u) (the index<br />
of u ) then ind(u + w) = ind(u) for I lwl I small enough (I<br />
did not explicitly introduce the number ind(u) ).<br />
V. Compact perturbations<br />
Published in occupied France during the war, my paper<br />
remained unnoticed until 1951; in that year three papers on<br />
the index of strict homomorphisms of Banach spaces were published,<br />
one in the United States and two in the USSR.
Dieudonn4 587<br />
(a) The Yood paper. B. Yood first added some<br />
interesting complements to my results (which he knew) on "small"<br />
perturbations. He separately studied the strict homomorphisms<br />
u : E + F between Banach spaces for which dim Ker(u) < +<br />
(which he called "property A") and those for which dim Coker(u)<br />
< + ~ (which he called "property B"). He gave an important<br />
equivalent characterization of property A: for each compact<br />
-i<br />
set K c F , all bounded sets in u (K) are relatively<br />
compact. To the fact that<br />
property B for u property A for<br />
t<br />
u<br />
Which I had proved in my paper), he added the similar result<br />
t<br />
property A for u property B for u<br />
(the two results are not equivalent when E and F are not<br />
reflexive).<br />
Furthermore, he showed that if two strict homomorphisms<br />
between Banaeh spaces u : E + F , v : F + G have property A<br />
(resp. B), then v Ou has property A (resp. B). Conversely, if<br />
v ~ u has property A (resp. B), then u has property A (resp. v<br />
has property B).<br />
The main novelty in Yood's paper is his idea of con-<br />
sidering as a "perturbation",instead of a "small" map, a compact<br />
map. Using his criterion for property A, he showed that if u<br />
has property A (resp. B) and if w : E + F is compact, then<br />
u + w has property A (resp. B)~ if u has both properties A<br />
and B, then ind (u + w) = ind (u)<br />
Strict homomorphisms of Banach spaces which possess<br />
both properties A and B are now usually called Fredholm opera-<br />
tors. Yood showed that Fredholm operators are characterized by<br />
the following condition: there exist continuous maps v : F + E<br />
and w : F § E such that i X - wo u and iy - uo v have<br />
finite rank.<br />
Finally, he observed that the technique of Weyr-F.Riesz,<br />
which succeeds for maps v = 1 X + u , where u is a compact<br />
endomorphism of E , cannot always be applied when 1 X is<br />
replaced by an arbitrary a~tomorphism f of E ; he gave an<br />
example in which the kernels of (f - u) n have dimensions
588 Dieudonn4<br />
which increase indefinitely with n.<br />
(b) The Atkinson and Gohberg papers. In <strong>194</strong>8, the<br />
British mathematician F. Atkinson submitted a paper to Mat.<br />
Sbornik, but it was published only in 1951. He could not of<br />
course mention Yood's paper, but he was not aware of mine either.<br />
He independently reproved the results of Yood and mine, and added<br />
a property which had not been proved by Yood: if both u : E § F<br />
and v : F + G are Fredholm operators, then v c u : E + G is<br />
also a Fredholm operator, and<br />
(15) ind (v o u) = ind (v) + ind (u)<br />
In the same year, 1951, I. Gohberg independently obtained the<br />
same results, and announced them in a Doklady Note.<br />
VI. Further results<br />
The theorems published in the three preceding papers<br />
are the only ones needed for the proof of the Atiyah-Singer index<br />
formula, where they are applied to Sobolev spaces. But many<br />
papers Were published after 1951, extending these theorems in<br />
various ways, in particular to unbounded operators and to more<br />
general topological vector spaces, such as Fr6chet spaces. We<br />
mention some of them in the Bibliography.<br />
BIBLIOGRAPHY<br />
i. Atkinson, F.V.: The normal solubility of linear<br />
operators in normed spaces, Mat. Sbornik, N.S.28 (70),<br />
pp. 3-14 (Russian)<br />
2. Dieudonn6, J.: Sur les homomorphismes d'espaces<br />
norm6s, Bull. Sci. Math. (2), 67 (<strong>194</strong>3), pp. 72-84.<br />
3. Gohberg, I.C.: On linear equations in normed spaces,<br />
Dokl. Akad. Nauk, SSR (N.S.) 76 (1951), pp. 477-480<br />
(Russian)<br />
4. Gohberg, l.C. and Krein, M.: The basic propositions on<br />
defect numbers, root numbers and indices of linear<br />
operators, A.M.S. Transl. (2), 13 (1960), pp. 185-264<br />
(Uspehi Mat. Nauk (N.S.), 12 (1957), pp. 43-118).<br />
5. Kato, T. Perturbation theory for nullity, deficiency<br />
and other quantities of linear operators, Journ. Anal.<br />
Math., 6 (1958), pp. 281-322.
Dieudonn4 589<br />
6.<br />
7.<br />
8.<br />
9.<br />
i0.<br />
Noether, F.: Uber eine Klasse singul~rer Integral-<br />
gleichungen, Math. Ann., 82 (1921), pp. 42-63.<br />
Rellich, F.: St~rungstheorie der Spektralzerlegung, I,<br />
Math. Ann., 113 (1937) pp. 600-619.<br />
Ruston, A.F.: Operators with a Fredholm theory,<br />
Journ. London Math. Soc., 29 (1954), pp. 318-326.<br />
Schwartz, L.: Homomorphismes et applications eompl~tement<br />
continues,<br />
pp. 2472-2473.<br />
C.R. Aead. Sci. Paris, 236 (1953),<br />
Yood, B.: Properties of linear transformations<br />
preserved under addition of a completely continuous<br />
transformation, Duke Math. Journ., 18 (1951),<br />
pp. 599-612.<br />
10 Rue du G~n~ral Camou<br />
75007 Paris<br />
France<br />
Submitted: January 7, 1985
Integral Equations<br />
and Operator Theory<br />
Vol. 8 (1985)<br />
0378-620X/85/050590-2451.50+0.20/0<br />
9 1985 Birkh~user Verlag, Basel<br />
FREDHOLM THEORY OF WIENER-HOPF EQUATIONS<br />
IN TERMS OF REALIZATION OF THEIR SYMBOLS<br />
H. Bart, I. Gohberg and M.A. Kaashoek<br />
The Fredholm properties (index, kernel, image, etc.) of<br />
Wiener-Hopf integral operators are described in terms of realiza-<br />
tion of the symbol for a class of matrix symbols that are analy-<br />
tic on the real line but not at infinity. The realizations are<br />
given in terms of exponentially dichotomous operators. The re-<br />
sults obtained give a complete analogue of the earlier results<br />
for rational symbols.<br />
0. INTRODUCTION<br />
This paper concerns the Fredholm theory of Wiener-Hopf<br />
integral equations. Consider<br />
(0.i) r - f k(t - s)r s : f(t), 0 _< t < ~,<br />
0<br />
where k(.) is an m m matrix function of which the entries are<br />
in LI(-~,~). For 1 ~ p ~ ~, p fixed, let I - K be the operator on<br />
Lm[0,~) defined by the left hand side of (0.I). It is well-known<br />
p<br />
(see [7]) that the operator ! - K is a Fredholm operator if and<br />
only if the symbol of the equation (0.1) has no zeros on the real<br />
line, i.e.,<br />
(0.2) det(l m - k(l)) ~ O, -~ < I < ~.<br />
Here k(-) is the Fourier transform of k(-). Furthermore, in that<br />
case the Noether index of I - K is equal to the winding number<br />
relative to the origin of the curve parametrized by<br />
det(l m - k(~)), -~ < ~ < ~.<br />
In the framework of the general theory additional in-<br />
formation about I - K, its kernel, its image, etc. is available
Bart, Gohberg and Kaashoek 591<br />
only in terms of a Wiener-Hopf factorization and the partial in-<br />
dices of the symbol I m - k('). However, as is well-known, for<br />
non-scalar symbols, the factorization and the indices cannot be<br />
constructed explicitly except for some rare cases, and even then<br />
the final answers are depending on complicated operations like<br />
the Fourier transform and its inverse.<br />
For the case of rational matrix symbols the difficul-<br />
ties referred to above were overcome in [1,2], where the analysis<br />
of the Wiener-Hopf equation (0.&) is based on a representation of<br />
the symbol in the following form:<br />
(0.3) ! m - k(1) : ! m + C(I - A)-IB, -~ < I < ~.<br />
Here A is a square matrix with no eigenvalues on the real line of<br />
which the order n may be different from m. Further, B and C are<br />
matrices of sizes n x m and m x n, respectively. In systems theo-<br />
ry ([i0,11], also [9,1]) the right hand side of (0.3) is called a<br />
state space realization. By analyzing systematically the operator<br />
I - K in terms of a realization of its symbol the following re-<br />
sults have been obtained (see [1,2,3,6]). First of all let us<br />
mention that condition (0.2) is equivalent to the requirement<br />
that the matrix A x :: A - BC has no eigenvalues on the real line;<br />
in other words, I - K is Fredholm if and only if A x has no real<br />
eigenvalues. Assuming this to be the case, then<br />
Ker(l - K) = {~ ~ Lm[0,~) I ~(t) Ce -itAx<br />
= X<br />
p<br />
x ~ Im P n Ker pX},<br />
9 X<br />
L m 0,~) I f pXelSA Bf(s] d s<br />
Im(l - K) : {f ~ P 0<br />
c Im P + Ker pX},<br />
Ind(l - K) = dim(Im P n Ker P - codim(!m P + Ker pX),<br />
where P and pX are the Riesz projections corresponding to the<br />
eigenvalues in the upper half plane of A and A x, respectively.<br />
Furthermore (see [3,6]), a generalized inverse of the Fredholm
592 Bart, Gohberg and Kaashoek<br />
operator I - K is given by the integral operator<br />
with<br />
((I - K)+~)(t) = ~(t) + f y+(t,s)~(s)ds,<br />
0<br />
9 x + 9 x<br />
f iCe -ltA (I - ~ )e lsA B,<br />
y+(t,s) = . . . .<br />
-iCe -itA H+e IsA B,<br />
s < t,<br />
s > t,<br />
0 _< t < ~,<br />
where H+ = I - pX _ (I - Px)(p P)+PX and (pxllm P)+ is a<br />
generalized inverse of the finite dimensional operator<br />
pXllm P: im P + !m P.<br />
When the symbol of (0.1) is analytic on the real line<br />
and at infinity the above results remain true ([2,3,6]). The main<br />
change is that for A one has to take a bounded linear operator<br />
acting on a possibly infinite dimensional Banach space which has<br />
no spectrum on the real line.<br />
The main aim of the present paper is to extend the con-<br />
structive approach described above to a more general class of<br />
symbols, namely to matrix symbols that are analytic on a strip<br />
containing the real line but not at infinity. This requires that<br />
in the realization (0.3) one allows the operator A to be an un-<br />
bounded linear operator. But then the following questions arise.<br />
How to build the projections P and pX and how to give a meaning<br />
to the exponential expressions appearing in the description of<br />
the kernel, the image and the generalized inverse? To answer<br />
these questions we have developed in [5] (see also [4]) a little<br />
theory of exponentially dichotomous operators. In the present<br />
paper we show that with the machinery of exponentially dichoto-<br />
mous operators also a constructive Fredholm theory can be devel-<br />
oped for Wiener-Hopf equations with non-rational symbols. In fact<br />
in this way we get a complete analogue of our results in the<br />
rational case for matrix symbols of Wiener-Hopf equations of<br />
which the kernel k(.) is continuous one\ {0}, exponentially de-<br />
caying and has a jump discontinuity at the origin.
Bart, Gohberg and Kaashoek 593<br />
The paper is organized as follows. In the first section<br />
we review the results of [4,5] about exponentially dichotomous<br />
operators and realization as far as they are needed in the pres-<br />
ent paper. Section 2 contains the main theorems. In Section 3 the<br />
Wiener-Hopf integral operator is reduced to an operator whose<br />
Fredholm properties can be analyzed easily and which appears as a<br />
natural generalization of the indicator (see [3,6]) of the ratio-<br />
nal case. In Section 4 the proofs of the main theorems are given.<br />
The final section contains an application to the Riemann-Hilbert<br />
boundary value problem.<br />
Canonical factorization of symbols of the type con-<br />
sidered here and the corresponding inverse formulas for Wiener-<br />
Hopf equations on the half line and the full line appear in [5].<br />
The study of the partial indices and non-canonical factorization<br />
are planned for another publication.<br />
We shall use the symbol ! to denote an identity opera-<br />
tor. The symbol I with a subscript X stands for the identity<br />
operator on the space X. Instead of I m we write I m.<br />
operator<br />
(1.1)<br />
i. REALIZATION AND SPECTRAL EXPONENTIAL REPRESENTATION<br />
In this paper we deal with the Wiener-Hopf integral<br />
(I - K)~(t) : ~(t) - S k(t - s)~(s)d s, 0 s t < ~,<br />
0<br />
assuming its symbol I - k(-) admits a realization of the type<br />
m<br />
(1.2) I m - k(1) = I m + C(l! X - A)-IB, -~ < I < ~,<br />
where X is a (possibly infinite dimensional) complex Banach space,<br />
B: ~m § X and C: X § ~m are bounded linear operators and -iA<br />
(X § X) is exponentially dichotomous in the sense of [4,5]. So,<br />
by definition, A is a densely defined closed linear operator and<br />
the space X in which A acts admits a topological direct sum de-<br />
composition
594 Bart, Gohberg and Kaashoek<br />
(1.3) X : x_ ~ x+<br />
with the following properties: the decomposition reduces A, the<br />
restriction of iA to X is the infinitesimal generator of an ex-<br />
ponentially decaying strongly continuous semigroup and the same<br />
is true for the restriction -iA to X+. These requirements deter-<br />
mine the decomposition (1.3) uniquely and the projection of X<br />
onto X_ along X+ is called the separating projection for -iA. In<br />
case A is bounded, -iA is exponentially dichotomous if and only<br />
if the spectrum o(A) of A does not meet the real line and then<br />
the separating projection is just the Riesz projection correspon-<br />
ding to the part of o(A) lying in the upper half plane. In gen-<br />
eral, the condition that -iA is exponentially dichotomous in-<br />
volves a more complicated spectral splitting of the (possibly<br />
connected) extended spectrum of A. The details (including a char-<br />
acterization of exponentially dichotomous operators in terms of<br />
two-sided Laplace transforms) may be found in [5] or [4].<br />
For what follows it is essential that the assumption<br />
concerning the symbol made above can be rephrased in terms of the<br />
kernel k appearing in (1.1). What it amounts to is that k has the<br />
spectral exponential representation<br />
k(t) : iCE(t;-iA)B, t # 0,<br />
where E(.;-iA) is the bisemigroup generated by -iA, i.e.,<br />
(1.4)<br />
E(t ;-iA)x :<br />
[<br />
mieWit ~AI Xm~px~<br />
{ ie -it(AI X+) (I - P)x,<br />
t < O,<br />
t > O.<br />
Note that the function E(.;-iA) takes its values in L(X), the<br />
space of all bounded linear operators on X.<br />
As was proved in [5] (also [4], Section 7) the (matrix)<br />
function k admits a spectral exponential representation if and<br />
only if k is continuous one\ {0}, exponentially decaying (in<br />
both directions) and has a jump discontinuity at the origin in
Bart, Gohberg and Kaashoek 595<br />
the sense tha~ the left and right limits exist but need not be<br />
equal. Given a function k subject to these conditions, a spectral<br />
exponential representation for k can be constructed in the<br />
following expliclt way. Let X denote the complex Banach of all<br />
bounded [m-valued functions on ~\ {0} having uniformly continuous<br />
restrictions to the half lines (-~,0) and (0,~). The algebraic<br />
operations in X are defined pointwise and the norm is the usual<br />
supremum norm. Take -q < ~ < 0, where q is a positive constant<br />
suchthat<br />
sup e qltl llk(t)kl < ~,<br />
t~0<br />
and define A (X § X), B: ~m § X and C: X § ~m by<br />
D(A) : {f c X I f is differentiable and f' E X}.<br />
[Af](t) = I -i~f(t) + if'(t), t < O,<br />
[ i~f(t) + if'(t), t > 0,<br />
[By](t) = e-Wltlk(t)y, t # 0,<br />
Cf = i(lim f(t) - lim f(t)).<br />
~t§ t+0+<br />
Here, of course, D(A) stands for the domain of A. A straighfor-<br />
ward argument shows that -iA is exponentially dichotomous, B and<br />
C bounded and k(t) : iCE(t;-iA)B, t # 0. For details, see [5]<br />
(also [4], Section 7).<br />
The following lemma, taken from [5] (also [4], Section<br />
8), plays an important role in this paper.<br />
BASIC LEMMA. Suppose W(1) has the realization<br />
W(1) : I m + C(hI X - A)-IB,<br />
where (as above) B and C are bounded and -iA is exponentially
596 Bart, Gohberg and Kaashoek<br />
dichotomous. Then the following three statements are equivalent:<br />
(i) det W(h) # O, h { ~,<br />
(ii) A X :: A - BC has no spectrum on the real line,<br />
(iii) -iA := -i(A - BC) is exponentially dichotomous.<br />
The non-trivial part of the lemma is the fact that (i)<br />
implies (iii). The proof given in [4,5] is based on Wiener's<br />
theorem and involves the spectral splitting of a possibly connec-<br />
ted (extended) spectrum.<br />
For later use we recall a few simple facts about bi-<br />
semigroups. Suppose -iA (X § X) is exponentially dichotomous, and<br />
let E(.;-iA) be the corresponding bisemigroup. Take x e X. The<br />
function E(.;-iA)x is continuous on ~\ {0} and exponentially de-<br />
caying (in both directions). It also has a jump (discontinuity)<br />
at the origin and in fact<br />
(1.6) lim E(t;-iA)x = -Px, lim E(t;-iA)x = (I - P)x,<br />
t§ ~ t+0+<br />
where P is the separating projection for -iA. If x belongs to the<br />
domain D(A) of A, then E(.;-iA)x is differentiable one\ {0} and<br />
(1.7) d E(t;-iA)x = -iE(t;-iA)Ax = -iAE(t;-iA)x,<br />
dt<br />
t # 0.<br />
Obviously, the derivative of E(';-iA)x is continuous one\ {0},<br />
exponentially decaying (in both directions) and has a jump at the<br />
origin. From (1.4) it is clear that<br />
(1.8) E(t;-iA)P = PE(t;-iA) = E(t;-iA), t < 0,<br />
(1.9) E(t;-iA)(l - P) = (I - P)E(t;-iA) = E(t;-iA), t > 0.<br />
Moreover, the following semigroup properties hold:<br />
(1.10) E(t+s;-iA) = -E(t;-iA)E(s;-iA), t,s < 0,
Bart, Gohberg and Kaashoek 597<br />
(1.11) E(t+s;-iA) = E(t;-iA)E(s;-iA), t,s > O.<br />
2. MAIN THEOREMS<br />
First we fix some notation. As before K denotes the<br />
integral operator on L~[0,~) given by (1.1), i.e.,<br />
[K~](t) = S k(t - s)~(s) d s, t ~ 0.<br />
0<br />
Here p is given, 1 s p s ~, and the kernel k is an integrable<br />
m x m matrix function. It will be assumed that k has the spectral<br />
exponential representation<br />
(2.1) k(t) = iCE(t;-iA)B, t # 0.<br />
Recall that A x stands for A - BC. The space in which A and A act<br />
is denoted by X.<br />
THEOREM 2.1. Assume the kernel k of the integral<br />
operator K has the spectral exponential representation (2.1).<br />
Then I - K is a Fredholm operator if and only if -iAx<br />
(:= -i(A - BC)) is exponentially dichotomous. In that case,<br />
letting P and px denote the separating projections for -iA and<br />
-iA x, respectively, the following statements hold true:<br />
(i) Im P n Ker pX is finite dimensional, Im P+ Ker pX<br />
is closed with finite codimension (in X) and<br />
ind(l- K) = dim(Im P n Ker pX) _ codim(Im P+ Ker pX),<br />
(ii) a function ~ belongs to Ker(l - K) if and only if<br />
there exists a (unique) x ~ Im P n Ker pX such<br />
that<br />
~(t) = CE(t ~-mA . x )x, t a 0,<br />
(iii) dim Ker(l - K) = dim(Im P n Ker pX),
598 Bart, Gohberg and Kaashoek<br />
(iv)<br />
(v)<br />
a function ~ in L~[0,~) belongs to Im(l - K) if<br />
and only if<br />
oo<br />
r j ~ v "<br />
0<br />
;-zA )B~(t)d t ~ Im P + Ker P ,<br />
codim Im(l - K) = codim(Im P + Ker pX).<br />
An operator T + is called a generalized resolvent of an<br />
operator T if T = TT + T. As soon as such an inverse is known,<br />
+<br />
then T y is a solution of Tx = y whenever this equation is solv-<br />
able. Each Fredholm operator has a generalized inverse.<br />
THEOREM 2.2. Assume the kernel k of the integral<br />
operator K has the spectral exponential representation (2.1), and<br />
let -iA x (:= -i(A - BC)) be exponentially dichotomous with<br />
separating projection pX. Then a generalized inverse of the<br />
Fredholm operator ! - K is given by<br />
[(I - K)+~](t) = ~(t) + f y+(t,s)~(s)ds,<br />
0<br />
oo<br />
t a O,<br />
+<br />
u (t,s) = iCE(t-s;-iAX)B - iCE(t;-iAX)H+E(-s;-iAX)B<br />
= iCE(t-s;-iAX)B + iCE(t;-iAX)(l - H+)E(-s;-iAX)B,<br />
where H + = I - pX _ (I - Px)(pXllm p)+pX and (pXllm P) is a<br />
generalized inverse of the Fredholm operator pXllm P: Im P § ImpX.<br />
+ .<br />
The operator H zs easily seen to be a projection of X<br />
onto Ker pX. Theorem 2.2 remains true if "generalized inverse" is<br />
replaced by "left inverse", "right inverse" or "(two-sided) in-<br />
verse". In fact, left invertibility occurs if and only if<br />
Ker H + c Im P, right invertibility if and only if Im P c Ker H+<br />
and (two-sided) invertibility if and only if Ker H+ = Im P. So<br />
I - K is invertible if and only if X = Im P r Ker pX and in that<br />
case H + is the projection of X onto Ker pX along Im P. Via<br />
factorization the "if part" of the latter result was also proved
Bart, Gohberg and Kaashoek 599<br />
in [5]. In general these inversion results look very similar to<br />
those for the case of rational symbols (see [i], Section 4.5,<br />
[2], Theorem 1.3.4, [6], Section 2).<br />
3. COUPLING AND RELATED RESULTS<br />
In this section we establish two lemmas that will play<br />
an important role in the proofs of the main results. The first is<br />
a coupling lemma (cf. [3]). Notation is as in Section 2. In<br />
particular A x stands for A - BC.<br />
LEMMA 3.1. Assume the kernel k of the integral<br />
m ~ ~<br />
operator K on L_[O,~) has the spectral exponenttal representation<br />
P x<br />
(2.1), and let -iA be exponentially dichotomous. Denote the<br />
separating projection for -iA and -iA x by P and pX, respectively,<br />
and introduce<br />
U: Im pX § L [0,~),<br />
uX: im P § L~[0,~),<br />
R: L~[0,~) + Im P,<br />
R x ~ pX<br />
: L [0,~) § !m ,<br />
J: Im pX § Im P,<br />
[Ux](t) : iCE(t;-iA)x,<br />
[U : iCE(t ;-IA " )x,<br />
R~ = # E(-t;-iA)B~(t) d t,<br />
0<br />
X . X<br />
R ~ = - I E(-t;-iA )B~(t) d t,<br />
0<br />
Jx = Px,<br />
jx x x<br />
: Tm P + Im P , jXx : P x,<br />
K X : L [0,~) § L [Q,~), [KX~o](t) = ~ kX(t - s)~o(s)d s,<br />
where k x is given by the spectral exponential representation<br />
kX(t) = -iCE(t;-iAX)B, t # 0.<br />
Then all these operators are well-defined, linear and bounded.<br />
Moreover<br />
: L [0,~) ~ Im pX + Lm[O ~) $ Im P<br />
P<br />
0
600 Bart, Gohberg and Kaashoek<br />
is invertible with inverse<br />
K x ]<br />
I U Lm[0,~) ~ Im P § Lm[0,~) ~ !m<br />
R x jx p p<br />
In terms of [3], the lemma says that the operators<br />
I K and jx<br />
- are matricially coupled with coupling relation<br />
II- I<br />
The operator J is also called an indicator for I - K. Note that<br />
jx = pXll m P: im P § Im P already appeared in Theorem 2.2.<br />
PROOF. All operators appearing in Lemma 3.1 are well-<br />
defined, linear, bounded and acting between the indicated spaces.<br />
In this context two observations should be made. The first is<br />
that since ~m is finite dimensional, the operator functions<br />
E(.;-A)B and E(';-iA are continuous one\ {0} with a jump at<br />
the origin, all with respect to the norm topology. The second is<br />
that in view of (1.8), we have R = PR and R x = pxRX. In particu-<br />
lar R maps into Im P and R x maps into Im pX.<br />
Checking the coupling relation (3.1) amounts to verify-<br />
ing eight identities. The computations are long and tedious,<br />
which is the reason why we restrict ourselves to proving only<br />
that RX(l - K) + jXR = 0. The proof of the other seven identities<br />
follow the same pattern.<br />
Take ~ ~ L~[0,~). Then<br />
RXK~ = -i 55 E(-t;-iA -iA)B~(s) d s d t.<br />
00<br />
Observe that it is permitted to change the order of integration.<br />
Doing this, we get<br />
: f [-i f E(-t iA S.<br />
0 L 0 J<br />
For the moment, let x c D(A). In view of (1.7) and the definition<br />
pX
Bart, Gohberg and Kaashoek 601<br />
of A x, we have<br />
d (E(-t ;-iA " )E(t-s;-iA)x)<br />
dt<br />
: -iE(-t;-iA " )BCE(t-s;-iA)x<br />
where t # 0,s. Taking advantage of (1.6), it follows that<br />
co<br />
-i J E(-t . x : _<br />
;-iA )BCE(t-s;-iA)x dt P E(-s;-iA<br />
0<br />
Since D(A) is dense in X, this identity actually holds for all<br />
x c X. But then<br />
co co<br />
RXK~ = f pxE(-s;-iA)B~(s) d s - f E(-s;-iAX)B~0(s) d s<br />
0 0<br />
= P + R<br />
This can also be written as RX(l - K)~ + jXR~ = 0. D<br />
The second lemma is concerned with the difference of<br />
the separating projections for -iA and -iA x where it is assumed<br />
that both these operators are exponentially dichotomous.<br />
LEMMA 3.2. Let P and pX be the separating projections<br />
for -iA and -iA x, respectively, and define F: Lm[O,~) § X and<br />
P<br />
AX: X + Lm[0,~) by<br />
P<br />
co<br />
r~ = I E(-s;-ia)B~(s) d s,<br />
[AXx](t) iCE(t . x<br />
= ;-iA )x.<br />
Then F and A x are well-defined bounded linear operators and<br />
FA x = p _ pX. Moreover F and P - pX are compact.<br />
PROOF. It is easy to see that s and A x are well-<br />
defined bounded linear operators (cf. the first paragraph of the<br />
proof of Lemma 3.1). Take x ~ D(A). In view of (1.7) and the<br />
definition of A x, we have
602 Bart, Gohberg and Kaashoek<br />
and so<br />
Hence, by (1.6),<br />
d (E(-t;-iA)E(t;-iAX)x) = iE(-t;-iA)BCE(t;-IA" X)x, t ~ 0,<br />
dt<br />
X<br />
tAx : i ~ E(-t;-iA)BCE(t;-iAX)xdt<br />
--oo<br />
9 ;1.<br />
FAXx : -(I - P)pXx + P(I - P : Px - P<br />
Since D(A) is dense in X, we may conclude that FA x = P - P<br />
It remains to prove that F is compact. Write<br />
We shall establish the compactness of F+. For F_ the argument is<br />
of course analogous.<br />
Put h(0) = -PB and h(s) = E(-s;-iA)B, s > 0. Then h is<br />
an exponentially decaying continuous function from [0,~) into<br />
L(~m,x), the Banach space of all bounded operators from {m into<br />
X. Note that<br />
oo<br />
F+~ : # h(s)~(s)d s, ~ 9 Lp[O,~).<br />
0<br />
For 0 < T < ~, let H T be the bounded linear operator from L~[0,~)<br />
into X given by<br />
T<br />
HT~ = # h(t)o(t) d t.<br />
0<br />
Then ILE+ - HTll does not exceed the Lq-norm of h on the interval<br />
[T,~), where, as usual, p-i + q-1 : 1. Since h is exponentially<br />
decaying, it follows that H T § F+ when T § ~. Hence it suffices<br />
to show that the operators H are compact.<br />
T<br />
Fix T > 0. For notational convenience, we consider H T
Bart, Gohberg and Kaashoek 603<br />
as an operator on Lm[0,T). Let n be a positive integer and define<br />
P<br />
the function h n on [0,T) by giving it the constant value h(kn-IT)<br />
on the interval [kn-&T,(k+l)n-IT). As h is uniformly continuous<br />
on bounded intervals, we have that h n<br />
+ h uniformly on [0,T) But<br />
then the bounded linear operator H : Lm[0,x) § X given by<br />
~,n p<br />
T<br />
HT,n~ : f h (t)~(t) d t<br />
0 n<br />
tends to H when n § ~. Observe that<br />
T<br />
n-1 (k+l)n-lT<br />
= ~ h(kn-l~) f-1 ~(t) dt.<br />
HT'n~ k=0 kn<br />
Now all the values of h are of finite rank. Thus HT, being the<br />
limit of a sequence of operators of finite rank is compact.<br />
4. PROOFS OF MAIN THEOREMS<br />
In view of the Basic Le~ma of Section 1, the first part<br />
of Theorem 2.1 can be rephrased as follows. The operator I - K is<br />
Fredholm if and only if the determinant of the symbol ! - K does<br />
not vanish on the real line. This fact is known from [7] for<br />
arbitrary integrable matrix valued kernels. Here, we present a<br />
more direct argument based on the special form (2.1) of k, there-<br />
by aZso proving the rather explicit second part of Theorem 2.1.<br />
PROOF OF THEOREM 2.1 (FIRST PART). Suppose -iAx is<br />
exponentially dichotomous, and let P and P be the separating<br />
projections for -iA and -iA x, respectively. Then P - pX is com-<br />
pact by Le~na 5.2. It follows that I - P + pX and I - pX + p are<br />
Fredholm operators. In particular ~er(l - P + pX) is finite<br />
dimensional and Im(I - pX + p) is closed with finite codimension<br />
(in X). Since Im P n Ker pX c Ker(l - P + pX) and<br />
Im(l - pX + p) c Im P + Ker pX, the same conclusions hold for<br />
Im P n Ker F x and Im P + Ker pX, respectively.<br />
By Lemma 3.1, the operator jx = pXllm P: Im P § Im pX<br />
is an indicator for I - K. Taking into account the special form<br />
of the operators appearing in the coupling relation (3.1) and the
604 Bart, Gohberg and Kaashoek<br />
results obtained in Section 1.2 of [3], we see that (ii) and<br />
(iii) are satisfied. For (iv) and (v) the situation is slightly<br />
more involved. In first instance we obtain that codim Im(! - K) =<br />
= dim Im pXllm J and<br />
Im(I - K) = {~ c Lm[o,oo) I RX9 E !m jx}<br />
where R x is as in Lemma 3.1. From this one obtains the assertions<br />
(iv) and (v) by a straightforward argument based on the fact<br />
that Im J = !m P and Im R c Im pX. Note that (iii) and (v)<br />
imply that I - K is Fredholm with index as indicated in (i). This<br />
completes the first part of the proof.<br />
For the second part of the proof of Theorem 2.1 we need<br />
one more lemma which is concerned with the behaviour of separa-<br />
ting projections under simple perturbations. It is convenient to<br />
introduce the following notation. Let P and Q be projections of a<br />
given Banach space X. We write P ~ Q if Im P c Im Q and<br />
Ker P ~ Ker Q. Observe that P ~ Q if and only if P = PQ = QP. The<br />
relation < defines a partial ordering on the set of all projec-<br />
tions of X.<br />
LEMMA 4.1. Let S be a linear operator acting in a<br />
complex Banach space X, and let a,6 ~ ~, ~ < 6. Suppcse a - S and<br />
B - S are exponentially dichotomous with separating projections<br />
P(a) and P(6), respectively. Then P(a) ~ P(~). ALso P(a) = P(6)<br />
if and only if p - S is exponentially dichotomous for all p in<br />
the closed interval [a,B].<br />
The result is practically obvious when S happens to be<br />
a bounded operator on X. Indeed, in that case P(a) and P(B) are<br />
the spectral projections corresponding to the parts a (S) and<br />
aB(S) of d(S) lying in t~e open half planes Re h < a and Re h < B,<br />
respectively. Hence P(a) = P(6) + P0' where P0 is the spectral<br />
projection associated with the part ~ ,B(S) of ~(S) contained in<br />
the open strip ~ < Re h < B.
Bart, Gohberg and Kaashoek 605<br />
In the general, not necessarily bounded case, the situ-<br />
ation is considerably more complicated due to the (possible)<br />
presence of ~ in the extended spectrum of S. Again the (finite)<br />
spectrum ~(S) of S splits into the three disjoint closed subsets<br />
~ (S), ~(S) and ~ ,B(S) described above. But even in the seem-<br />
ingly innocent case when ~ ,B(S~ is a compact subset of ~ (hence<br />
a spectral set of S) it need not be true that P(~) is the sum of<br />
P(~) and the spectral projection corresponding to ~ ,~(S). As an<br />
example, let S be the infinitesimal generator of a strongly con-<br />
tinuous group having the property that ~(S) = r (cf. [8], Section<br />
23.16). For a a sufficiently large negative real number and ~ a<br />
sufficiently large positive real number, we have that ~ - S and<br />
S - ~ both generate exponentially decaying strongly continuous<br />
semigroups. Hence ~ - S and B - S are exponentially dichotomous<br />
with separating projections P(~) = 0 and P(B) = I, respectively.<br />
Since ~(S) is empty in this case, it certainly is a compact<br />
subset of the complex plane. Note that this example exhibits the<br />
following interesting phenomenon: For the operator ~ - S the in-<br />
finite spectral point ~ has to be counted as lying completely at<br />
the left hand side of the imaginary axis. By shifting ~ - S to<br />
B - S it changes its position (of course without actually moving<br />
as finite spectral points would do) and goes completely to the<br />
right hand side of the imaginary axis.<br />
PROOF OF LEMMA 4.1. The projection P(B) is a bounded<br />
linear operator commuting with B - S, hence with a - S. From the<br />
integral representation for separating projections (cf. [5];[4],<br />
Theorem 2.2) it is clear that P(B) also commutes with P(~). Put<br />
P0 = P(~) - P(~)P(B). Then P0 is a projection commuting with S<br />
and Im P0 = Im P(~) n Ker P(B). Let S O be the restriction of S<br />
to X 0 = Im PO" Thus Sox = Sx for all x in the domain D(So) =<br />
= D(S) n X 0 of S O . Since PO and S commute, S O is a linear opera-<br />
tor acting in X 0.<br />
Let S i be the restriction of S to Im P(~) considered as<br />
a linear operator acting in X i = Im P(a). Then S I - ~ is the<br />
infinitesimal generator of an exponentially decaying strongly
606 Bart, Gohberg and Kaashoek<br />
continuous semigroup of bounded linear operators acting on X I.<br />
The restriction P1 of P0 to X 1 is a projection of X 1 commuting<br />
with S I. Also Im P1 = Im P0 = X0 and the restriction of S 1 to<br />
Im PI coincides with S O . It follows that S O - ~ generates an<br />
exponentially decaying strongly continuous semigroup too. But<br />
then ~(S 0) lies in the open half plane Re ~ < ~ and (~ - S0 )-I is<br />
bounded on the closed half plane Re ~ a ~.<br />
In exactly the same way (using that X 0 c Ker P(B)) one<br />
shows that ~(S 0) lies in the open half plane Re ~ > B and<br />
(~ - S0 )-I is bounded on the closed half plane Re ~ ~ B. We con-<br />
11 ~ , ~<br />
clude that o(S0~ = 0 and (~ - S O ) is a bounded entlre functlon.<br />
Hence (~ - S O ) is constant by Liouville's theorem. This implies<br />
X 0 = (0), that is, P(a)= P(~)P(B) = P(B)P(~) or, equivalently,<br />
P(a) ~ P(~).<br />
Assume now that P(a) = P(B). With respect to the decom-<br />
position X = Im P(a) ~ Ker P(a), we write<br />
s 2<br />
Then S I - ~ and ~ - S 2 both generate strongly continuous semi-<br />
groups of negative exponential type. Taking ~ ~ p ~ 6, the same<br />
is obviously true for S I - P and 0 - S 2. Thus P - S is exponen-<br />
tially dichotomous for all P in [~,B].<br />
In order to complete the proof we make the following<br />
general observation. Suppose T is an exponentially dichotomous<br />
operator of negative exponential type ~ (cf. [4] and [5]) and<br />
with separating projection Q. Then for all complex E with<br />
IRe ~I < -~, the operator T + a is again exponentially dichoto-<br />
mous with separating projection Q. This is immediate from the<br />
definitions. So, if for all p in [~,B], the operator P - S is<br />
exponentially dichotomous, say with separating projection P(O),<br />
then P(O) is locally constant on [~,~]. Since [~,B] is connected,<br />
it follows that P(O) is constant on [~,B] and, in particular,<br />
P(~) = P(~).
Bart, Gohberg and Kaashoek 607<br />
PROOF OF THEOREM 2.1 (SECOND PART). It remains to<br />
prove that if I - K is a Fredholm operator, then -iA x is exponen-<br />
tially dichtomous. For this we shall use a perturbation argument.<br />
Assume -iA is of exponential type ~ < 0 (cf. [4] and<br />
[5]) and take ~ < s < -~. Then the operator c - iA is exponen-<br />
tially dichotomous, its separating projection is the same as that<br />
for -iA and<br />
Putting<br />
E(t;c - iA) = eStE(t;-iA), t ~ 0.<br />
ka(t) = iCE(t;s - iA)B, t ~ 0,<br />
we have ks(t) : eStk(t),- where as in the theorem k is given by<br />
(2.1). One checks without difficulty that k s converges to k in<br />
Li(~,L(~m)) when s tends to zero. Define K s on L~[O,~) by<br />
Ks~ : f ks(t - s)~(s) d s, t a 0.<br />
0<br />
Then K s converges to K in L~[0,~) when s tends to zero. It<br />
follows that for Isl sufficiently small the operator I - K is<br />
Fredholm with the same index as I - K.<br />
The Fourier transform k of k is analytic in the strip<br />
llm 11 < -~. Fix 0 < h < -~. Then<br />
lim ~(I) : 0.<br />
IIm llsh<br />
Hence the analytic function det(I - k(1)) has only a finite<br />
m<br />
number of zeros in the closed strip IIm 11 s h. So taking Isl<br />
non-zero and sufficiently small, we have that det(I m - k(1)) ~ 0<br />
for all real I.<br />
Let s be a sufficiently small positive real number.<br />
Then det(I m - ks(1)) = det(I m - k(1 - ie)) ~ 0 for all real I. By<br />
the Basic Lemma of Section i, this implies that e - iA z is expo-
608 Bart, Gohberg and Kaashoek<br />
nentially dichotomous. Its separating projection will be denoted<br />
by PX(s). From the first part of the proof of Theorem 2.1 we know<br />
that I - K is Fredholm and<br />
S<br />
dim Ker(i - K s ) = dim(Im P n Ker PX(s)) < ~,<br />
codim Im(l - K s ) = codim(Im P + Ker Px(s)) < ~.<br />
Since s was taken sufficiently small, the operator -s - iAx is<br />
exponentially dichotomous too, while<br />
dim Ker(l - K ) = dim(Im P n Ker P < ~,<br />
-a<br />
codim Im(! - K ) = codim(Im P + Ker PX(-s)) < ~.<br />
-S<br />
Here PX(-s) denotes the separating projection for -e - iA x<br />
pX<br />
By Lemma 4.1, we have P ~ (s). In particular<br />
Ker PX(-s) n Ker PX(s), and so<br />
dim Ker(l - K_a) ~ dim Ker(l - Ks) ,<br />
codim Im(l - K_s) ~ codim Ker(l - Ks).<br />
We may assume that ind(l - K_s) = ind(l - K) = ind(l - Ks). It<br />
follows that the above inequalities are in fact equalities. But<br />
then<br />
Im P n Ker PX(-s) = Im P n Ker PX(a),<br />
Im P + Ker PX(-s) = Im P + Ker PX(a).<br />
Take x ~ Ker PX(-a). Then x can be written in the form x = Pu + v<br />
with v ~ Ker PX(s) c Ker PX(-s). Hence 0 = PX(-a)x = PX(-s)Pu,<br />
and so Pu ~ Im P n Ker PX(-a) = Im P n Ker PX(s). We conclude<br />
that PX(s)x = 0. This proves that Ker PX(-s) = Ker PX(s). Also<br />
pX(_s) ~ P Thus PX(-e) = PX(s). Applying (the second part
Bart, Gohberg and Kaashoek 609<br />
of) Lemma 4.1, we see that 0 - iA x is exponentially dichotomous<br />
for all P in the interval [-E,a]. In particular -iA x is exponen-<br />
tially dichotomous. D<br />
PROOF OF THEOREM 2.2. From Lemma 3.1 we know that the<br />
operators I - K and J = P P: Im P § Im pX are matricially<br />
coupled with coupling relation (3.&). Combining this with Theorem<br />
2.& and [3], Theorem 2.1, we see that ! - K and jx are both<br />
Fredholm. Also, given a generalized inverse (jx)+ of jx, one for<br />
I - K can be defined by<br />
(I - K) + = I - K x - uX(jX)+R x,<br />
where K x, U x and R x are defined as in Lemma 3.1.<br />
Let @ ~ Lm[0,~). Taking advantage of the special form<br />
x P x<br />
of the operators K , U and R , we obtain<br />
[(T - K)+~](t) = ~(t) + f ~+(t,s)~(s~ds<br />
f . ,<br />
0<br />
+<br />
where the generalized resolvent y is given by<br />
co<br />
y+(t,s) = iOE(t-s;-iAX)B+ iCE(t;-iAX)(jX)+E(-s;-iAX)B.<br />
In view of the identities (1.8) and (1.9), the second term in<br />
the right hand side of this expression can be rewritten as<br />
-iCE(t;-iAX)H+E(-s;-iAX)B or iCE(t;-iAX)(l - ~+)E(-s;-iAX)B,<br />
where the projection ~+ of X onto Ker pX is defined by<br />
~+ = z - P - (z - P 215215<br />
5. THE RIEMANN-HILBERT BOUNDARY VALUE PROBLEM<br />
In this section we deal with the Riemann-Hilbert boun-<br />
dary value problem (on the real line)<br />
(5.1) w(z)r = r -~ < z < ~,<br />
the precise formulation of which reads as follows: Given an m x m
610 Bart, Gohberg and Kaashoek<br />
matrix function W(h), -~ < I < ~, with continuous entries,<br />
describe all pairs 9+, 9_ of ~m-valued functions such that (5.1)<br />
is satisfied while, in addition, 9+ and 9_ are the Fourier trans-<br />
forms of integrable Tm-valued functions with support in [0,~) and<br />
(-~,0], respectively. For such a pair of functions, We have that<br />
9+ (resp. 9_) is continuous on the closed upper (resp. lower)<br />
half plane, analytic in the open upper (resp. lower) half plane<br />
and vanishes at infinity.<br />
The functions W that we consider are of the form<br />
W(/) = I - k(1), where k admits a spectral exponential represen-<br />
m<br />
tation. In other words W(1) admits a realization as in (1.2).<br />
This implies that W(1) is analytic in a strip around the real<br />
axis.<br />
As before, A stands for A - BC.<br />
THEOREM 5.1. Consider the function W(I) : I m - k(1),<br />
where k has the spectral exponential representation (2.1) and<br />
(consequently)<br />
(5.2) W(h) : I + C(l - A)-IB, -~ < I < ~.<br />
m<br />
Assume the operator -iA is exponentially dichotomous (or,<br />
equivalently, det W(1) ~ 0 for all I e ~), and let P and P be<br />
9 X<br />
the separating projections for -iA and -IA , respectively. Then<br />
the pair of functions 9+, 9_ is a solution of the Riemann-Hilbert<br />
boundary value problem (5.1) if and only if there exists a<br />
(unique) vector x in Im P n Ker P such that<br />
5.3) r : C(l - A : S eilt(-iCE(t;-iA x) d t,<br />
0<br />
0<br />
5.4) g_(h) : C(I - A)-lx : S eilt(-iCE(t;-iA)x)dt, h c ~.<br />
Here E(';-iA) and E(-;-iA denote the bisemigroups generated by<br />
-iA and -iA respectively.<br />
h e JR,
Bart, Gohberg and Kaashoek 611<br />
by<br />
PROOF.<br />
Take x in Im P n Ker P , and define ~+ and ~0_<br />
~+(t) : -iCE(t;-iA<br />
~_(t) : iCE(t;-iA)x.<br />
Then ~+ and ~_ are integrable functions. Since x ~ Ker pX, it<br />
follows from (1.8) that ~+ has its support in [0,~). In a similar<br />
way one infers from x ~ Im P and (1.9) that ~_ has its support in<br />
(-~,0]. We denote Fourier transforms of ~+ and ~_ by @+ and ~_,<br />
respectively. The identities (5.3) and (5.4) are now clear from<br />
the fact that for an exponentially dichotomous operator S the<br />
resolvent has the following representation (see [4,5]):<br />
(k -<br />
S)-Ix<br />
co<br />
: ~ e-ktE(t;S)xdt,<br />
--co<br />
Re k = O.<br />
Also, a straightforward computation using (5.2) shows that (5.1)<br />
is statisfied. This settles the "if part" of the theorem.<br />
In order to establish the "only if part", assume that<br />
the pair r r is a solution of the Riemann-Hilbert boundary<br />
value problem 5.1). Write ~+ and r in the form<br />
r = ~ eilt~+(t) d t,<br />
0<br />
0<br />
r = f eilt~_(t)dt,<br />
where ~+ ~ L~[0,~) and ~_ ~ L~(-~,0]. A routine argument,<br />
based on (5.1) and the fact that W(~) = I m - k(~), yields<br />
~+(t) - f k(t - s)~+(S) d s : O, t a O.<br />
0<br />
Since k has the form (2.1), it follows from Theorem 2.1 that<br />
there exists a unique x ~ Im P n Ker P such that<br />
~+(t) = -iCE(t ;-IA " )x .<br />
But then (5.3) is satisfied As
612 Bart, Gohberg and Kaashoek<br />
~_(~) = W(h)r = C(h - A)-ix, the identity (5.4) holds too. D<br />
REFERENCES<br />
i. Bart, H., Gohberg, I. and Kaashoek, M.A.: Minimal<br />
Factorization of Matrix and Operator Functions. Opera-<br />
tor Theory: Advances and Applications, Vol.1,<br />
Birkh~user Verlag, Basel etc., 1979.<br />
2. Bart, H., Gohberg, I. and Kaashoek, M.A.: Wiener-Hopf<br />
integral equations, Toeplitz matrices and linear sys-<br />
tems, in: Toeplitz Centennial (ed. I. Gohberg), Opera-<br />
tor Theory: Advances and Applications, Vol.4,<br />
Birkh~user Verlag, Basel etc., 1982, p. 85-135.<br />
. Bart, H., Gohberg, I. and Kaashoek, M.A.: The Coupling<br />
method for solving integral equations, in: Topics in<br />
operator Theory, Systems and Networks, The Rehovot<br />
Workshop (ed. H. Dym, I. Gohberg), Operator Theory:<br />
Advances and Applications, Vol.12, Birkh~user Verlag,<br />
Basel etc., 1984, p, 39-73.<br />
. Bart, H., Gohberg, I. and Kaashoek, M.A.: Exponential<br />
dichotomous operators and inverse Fourier transforms,<br />
Report 8511/ M, Econometric Institute, Erasmus Univer-<br />
sity, Rotterdam, The Netherlands, 1985.<br />
. Bart, H., Gohberg, I. and Xaashoek, M.A.: Wiener-Hopf<br />
factorization, inverse Fourier transforms and exponen-<br />
tially dichotomous operators, J. Functional Analysis,<br />
to appear.<br />
. Bart, H. and Kroon, L.G.: An indicator for Wiener-Hopf<br />
integral equations with invertible analytic symbol.<br />
Integral Equations and Operator Theory 6(1983), 1-20.<br />
See also the addendum to this paper: Integral Equations<br />
and Operator Theory 6(1983), 903-904.<br />
. Gohberg, I. and Krein, M.G.: Systems of integral equa-<br />
tions on a half line with kernels depending on the<br />
difference of argument, Uspehi Mat. Nauk 13 (1958), no.<br />
2 (80), p.3-72 [Russian]. Translated as: Amer. Math.<br />
Soc. Trans. (2) 14 (1960), 217-287.<br />
. Hille, E. and Phillips, R.S.: Functional analysis and<br />
semigroups. Amer. Math. Soc., Providence R.I., 1957.<br />
. Kailath, T.: Linear Systems. Prentice Hall Inc.,<br />
Englewood Cliffs N.J., 1980.<br />
10. Kalman, R.E.: Mathematical description of linear dynam-<br />
ical systems, SIAM J. Control i (1963), 152-192.
Bart, Gohberg and Kaashoek 613<br />
ii. Kalman, R.E., Falb, P. and Arbib, M.A.: Topics in<br />
Mathematical System Theory. McGraw-Hill, New York etc.,<br />
1960.<br />
H. Bart,<br />
Econometrisch Instituut<br />
Erasmus Universiteit<br />
Postbus 1738<br />
3000 DR Rotterdam<br />
The Netherlands<br />
M.A. Kaashoek<br />
Subfaculteit Wiskunde en<br />
Informatica<br />
Vrije Universiteit<br />
Postbus 7161<br />
1007 MC Amsterdam<br />
The Netherlands<br />
Submitted: May 6, 1985.<br />
I. Gohberg<br />
Dept. of Mathematical Sciences<br />
The Raymond and Beverly Sackler<br />
Faculty of Exact Sciences<br />
Tel-Aviv University<br />
Ramat-Aviv<br />
Israel
Integral Equations<br />
and Operator Theory<br />
Vol. 8 (1985)<br />
PRINCIPAL CURRENTS<br />
Richard W. Carey and Joel D. Pincus<br />
0378-620X/85/050614-2751.50+0.20/0<br />
9 1985 Birkh~user Verlag, Basel<br />
An operator theoretic analog of a local Chern class is<br />
discussed for certain finitely generated operator algebras. This<br />
gives rise to an index theory for the maximal ideals of the<br />
algebra. The algebraic receiver for this index is the Grothendieck<br />
group of the algebra completed at a maximal ideal.<br />
INTRODUCTION<br />
In [ 6 ] we reported the construction of a new index<br />
theory for the maximal ideals of a finitely generated commutative<br />
Banach algebra of operators on Hilbert space. The index was the<br />
local degree of a holomorphic chain supported on the Taylor<br />
spectrum of the generators. The specification of the chain<br />
corresponds to the evaluation of certain weights determined by<br />
composition series associated with the module action of the algebra.<br />
The boundary of the holomorphic chain is a distinguished cycle<br />
concentrated on the joint essential spectrum of the generators.<br />
The index which we introduced via this association of<br />
holomorphic chains to operator algebras differs from others<br />
in that it does not vanish when the dimension of the Taylor<br />
spectrum is less then the number of the given generators,and ill<br />
that it is not constant on the Fredholm components. Our index<br />
has jumps on the singular locus of the underlying spectrum. But,<br />
because of this, has a natural transformation law under morphisms.<br />
Here we give a synoptic account of the ramifications of<br />
our idea of using analytic continuation to construct a local index<br />
theory in terms of the multiplicity theory of ideals.<br />
82 The analytic index<br />
We begin by recalling the main ideas associated with<br />
the so-called principal function of an operator. It has stability
Carey and Pincus 615<br />
under perturbation by trace class operators, and has geometric<br />
meaning. This object is associated with pairs of possibly non-<br />
closed subspaces of Hilbert space which are operator ranges.<br />
Recall that an operator range is a linear subspace that is the<br />
range of some linear operator. The collection of all operator<br />
ranges is the lattice L generated by the closed subspaces. An<br />
ideal in L is a subset I of L such that if D'aL and D E I with<br />
D' < D , then D'EI; furthermore, If both D and D' are in I then<br />
so is D+D'. An ideal I is said to be invariant if whenver D is in<br />
I then every D' unitarily equivalent to D is also in I.<br />
If F is the ideal of finite dimensional subspaces of L<br />
then two elements R and S of L are said to be congruent modulo F<br />
provided that R+d = S+d' for some elements d and d' in F.<br />
Suppose that R~S(modF). Then there are positive operators<br />
(A,B,C,D) with R(/A) =R, R(/B)=S, R(/A+-C)=R(~-D-) where the ranges<br />
R(/C) and R(/D) are in F. See[15] for references to the study Of<br />
range spaces,<br />
Consider now the perturbation problems<br />
A~A+C and B~B+D.<br />
The functions displayed above the arrows are the phase<br />
shifts corresponding to the perturbations. Thus,<br />
and<br />
for smooth f.<br />
trace [f(A+C)-f(A)] = /f'(1)~(1)dl<br />
trace [f(B+D)-f(B)] = ff'(1):(~)dl<br />
It is quite easy to see that if R =R(T) and S=R(T*) for<br />
a Fredholm operator T, then Index T = T(O) -:(0).<br />
Suppose that f, g and h are in L m , with h> 0 and all<br />
=<br />
the functions having compact support in [0,~]. The function f is<br />
said to be h related to g at the origin provided that<br />
If f(x)-g(x) dx I< /h(x) dx for s > O.<br />
x + s = x+~<br />
Recall that v is an approximate limit of the function f<br />
at x provided that for every ~ > O, { y:]v-f(y)l
616 Carey and Pincus<br />
approximate limits of h at g consists only of 0 and f and g are h<br />
related, we will write f=g. This is an equivalence relation and<br />
the set of equivalence classes [f] forms an abelian group K(L)<br />
under the natural identification [f+f'] = [f] + [f'].<br />
We have proved in [15] that if g and g' are represent-<br />
atives of [f] with {g(O)} = w+(0,g) and {g'(0)}=m+(O,g'), then<br />
g(O) = g'(O). If there is a g in [f] which is right approximately<br />
continuous at the origin, then it follows that the value g(O)<br />
does not depend upon the choices of g and h. We will denote this<br />
value by f~(O), and say that 0 is in the extended Lebesgue set of<br />
f.<br />
Suppose that R and S are elements of the lattice L, and<br />
that RES(modF). We proved the following result in ~5].<br />
THEOREM I.i The equivalence class[~-T] is independent<br />
of the implementing operators {A,B,C,D}. It depends only upon the<br />
subspaces R and S.<br />
Thus if {W,V,X,Y}is another choice of positive operators<br />
n P<br />
with R(~-) = R, and R(~ = S, R(/~ = R(/V-$-Y), V+V+X, W§<br />
then [~-T ]=[D-p].<br />
We will say that two subspaces R and S in L are F<br />
comparable if Rs and the origin belongs to the extended<br />
Lebesgue set of the class [~-~9<br />
In this case the value (T~)~(O) will be denoted by<br />
ind(R,S). We note that if T is a Fredholm operator then the<br />
subspaces TH and T~H are F-comparable, and ind(TH,T~H) is the<br />
usual Noether index.<br />
stability result.<br />
We have also been able to establish [15] the following<br />
THEOREM 1.2 Suppose that T and K are bounded operators<br />
on the Hilbert space H with R(K) in F. Suppose that TH and T~H are<br />
F-comparable. Then [T+K]H and [T+K]~H are F-comparable and<br />
ind([T+K]H,[T+K]~H) = ind(TH,T~H)<br />
Now let Tz= T-zI and form the self adjoint operators<br />
T ~T and T T with spectral resolutions E~ and F~ (respectively).<br />
Z Z Z Z<br />
If z is not in the essential spectrum of T, it is clear<br />
that both spectral projections are constant finite rank operators
Carey and Pincus 617<br />
for sufficiently small positive %. Thus, if p(T) is a polynomial<br />
in T, we have<br />
z z<br />
J(p)(z) = trace [p(T)(E%-F%)] = p(z) Index(T-z).<br />
Since we are interested in the case where z is in the<br />
essential spectrum of T, we form another functional. Let<br />
J (p)(z) = 2--~rtrace p(T) f (E%-F4) z z<br />
~(1%1~) d%<br />
where ~ is a real smooth compactly supported function normalized<br />
2<br />
so that ~ s@ (s) ds = I.<br />
Of course J e(p)(z ) = J(p)(z) for e sufficiently small<br />
if z is not in the ess~tial spectrum of T. But it is basic to<br />
Our analysis that the hypothesis T~T-TT ~ is trace class implies<br />
that there exists an Ll,real-valued, compactly supported function<br />
gT,called the principal function of T, so that<br />
1<br />
Jg(p)(z) = -g2f P(~)~(I~-z[/~)gT(
618 Carey and Pincus<br />
But there are circumstances in which it is important to<br />
refine the definition so that we obtain a function which is defined<br />
on a larger set. Thus, when the essential spectrum is curve like<br />
it will be possible to define the principal function ~ the curve<br />
as an average of the values on both sides of the curve. This even<br />
makes sense when the principal function is weakly differentiable<br />
i.e. when the distributional derivatives Sg /~x and 8g/~y are<br />
measures. It is known ~4] that the principal function of an operator<br />
T satisfies this condition provided that the spectral multiplicity<br />
functions of Re T and Im T are Lebesgue summable.<br />
In this case a theorem of Federer [i~ asserts that for<br />
H I almost all b in R 2, and f a weakly differentiable function<br />
lim ff(b+Ez)~(Izl)dL2z = 89 lim inf f(z) + (L 2) ap iim supf~,<br />
E+0 + z~b z§<br />
whenever ~ satisfies the conditions above. Thus, we see that if the<br />
subspaces T H and T~H are F-comparable then 6 ~ (0) is equal to the<br />
z z z<br />
average of the lower and upper approximate limits of gT at z.<br />
It therefore is reasonable to take this as a redefinition<br />
of the principal function on the set of points z where the rotation<br />
-ally symmetric regularizaNons of the principal function converge<br />
Hausdorff one measure almost everywhere. Then ind(T~H,TzH ) = gT(z)<br />
H 1 almost everywhere~<br />
For an example, consider the unilateral shift. If T is<br />
the shift and w is a complex number with modulus one, then<br />
R(T-w)~/R(T-w) is a one dimensional vector space. Both ranges are<br />
dense subspaces of H, but 89 =-gT(0) - by Theorem 1.1 - describes a<br />
stable feature of their geometry, a new relative dimension.<br />
Other examples of a simple nature are furnishad by<br />
singular integral operators on smooth contours in the plane with<br />
rational symbols that vanish. These operators actually have N(T)<br />
and N(T ~ finite dimensional, but the difference of these dimensions<br />
is not stable against finite rank perturbations. See [17].<br />
82 Fundamental properties<br />
The principal function of a single operator T is used to<br />
define a current, called the principal current of the unital C*-<br />
algebra corresponding to T. By definition the principal current
Carey and Pincus 619<br />
for the C ~ algebra which corresponds to a T such that [T,T ~] is<br />
in trace class is T 1 E21 gT<br />
- 2~i -<br />
basic properties:<br />
The two-current defined by this relation has certain<br />
With a suitable functional calculus<br />
(i) ~T(~) = index f(T), when f ~ 0 on Oess(T).<br />
(ii) ~T(fdh) = trace [h(T~,T),f(T~,T)]<br />
(iii) Mass(T) < 89 II[T~,T]II trace norm<br />
(iv) T~= ~#(T), where ~(T,T ~) is a "smooth" function of T and T ~<br />
and T~ is the current formed from the operator ~(T#T~).<br />
(v) TT=TT+ k for k in trace class.<br />
The first of these relations remains the motivation for<br />
the entire theory, ic led originally to the conjecture by the second<br />
author that the principal function coincides with index T when z<br />
z<br />
is a Fredholm point. This conjecture was first established in full<br />
generality in [12] .<br />
It is fundamental to our approach that we consider<br />
the functionals which appear in (ii) (or those that appear in higher<br />
dimensional versions) not only as being represented by a global object<br />
like a homology class, or some measure [13],but by an object defined<br />
pointwise on the spectrum of the algebra.<br />
Indeed, a variety of conditions are known which guarantee<br />
even that the principal function of a single operator takes on only<br />
integer values. This is so, for example, if T is subnormal. Integer<br />
values are also assumed if T+T ~ and(T-T~)/2i have integrable<br />
spectral multiplicity functions, [14].(We speak here of a determina-<br />
tion up to a set of zero with respect to two dimensional Lebesgue<br />
measure.)Thus, there is geometry associated with these point values.<br />
Desideratu<br />
We can rewrite (i) as an intersection of currents<br />
index f(T) = [div (f) liT](1)<br />
provided that f is analytic and non-vanishing on the essential<br />
spectrum of T.<br />
We put aside for the moment the general question of the<br />
meaning of intersections with principal currents in the non-Fredholm<br />
situation, and ask what the natural multidimensional extension of<br />
this intersection result may be in the multidimensional Fredholm case.
620 Carey and Pincus<br />
Suppose that we are given s commuting operators<br />
=(TI,...,Ts) with joint essential spectrum X a closed 2k-1<br />
dimensional smooth submanifold of CS.Then for holomorphic functions<br />
= (fl ..... fk ) and h, the operators ~(~) and h(~) make sense<br />
if they are understood in the sense of the Taylor functional<br />
calculus, and we can ask if there are hypothesis which guarantee<br />
the existence of a holomorphic (k,k) chain T in cS/x so that<br />
x[h,E~(s = [div(i)(l T ](h)<br />
where E~(~) denotes the Koszul complex of the tuple ~(T);<br />
x[h,Em(~)] is the Lefshetz number of the endomorphism induced on<br />
E~(~) by h(T), and div(~) is the current of integration which<br />
corresponds to the zero set of ! counted according to multiplicity.<br />
Thus, div(! ) = d(d c logll~N2a(ddClogIl~II2) k-l, where dC=i/4~(~-~),<br />
see [ 4 ].<br />
The Brown-Douglas-Fillmore-Kasparov theory is associated<br />
with a 2k-i current ~ with support in X so that<br />
x[h,E~(!) ] = ~(h 3log II!ll2A[dd c log ]lill2] k-1<br />
The restriction of ~ to X defines an element of KI(X ).<br />
But the C ~ algebra structure does not determine solutions Tof<br />
dT = ~. Rather, it is our fundamental idea that the C ~ algebra<br />
index theory is determined by analytic structure associated to<br />
Commuting subalgebras generating the C ~ algebra. Now we define<br />
our concept of local index theory. Recall that dual to the Chern<br />
map ch~: K~(X)§ there is ch~:K~(X)+H~(X;Q) and an index<br />
theorem is obtained by identifying ch~(~) = [~]. Thus~ if Fg~I(X)<br />
then index(F)= ch~(~)ch~(~), corresponding to the pairing<br />
H~(X;Q) OH~(X;Q) +Q under the Chern maps. Note that the boundary<br />
-i<br />
map~ takes ch~(~) to a class in H~(cS,x;Q). When ~ is holomorphic<br />
with Bott symbol [B(!)] (see [ 3 ]) in ~I(X), ch~[B(!)] maps to<br />
a bivariant class [div(~)] in H~(cS/x,(cS/x)/zero(~); Q]<br />
We get<br />
x(E~(~))=ch~(~)ch~[B(~)]=~-Ich~(~)'[div(~)]<br />
:e([T]- [div(s<br />
Here the e applied to a class of dimension zero of the<br />
form ~gj hj[pj] , where [pj] is a zero cycle,i.e, a point, gives<br />
~jhj gj.
Carey and Pincus 621<br />
Note here that since we are treating [div(~)]as a<br />
bivariant class it maps H~(cS/x) to H~(zero ~) and we think of [T]<br />
as being in H~(cS/x), so that [T]'[div(~)] is a class of dimension<br />
zero in H~(zero ~) having the form [gjhj[pj].<br />
Thus we want to realize the restriction ofthe Ext class<br />
ch~(~) to Bott symbols as an intersection with a holomorphic (k,k)<br />
chain T in cS/x. If T and div(~) are holomorphic chains of<br />
complementary dimensions k and s-k which meet in a finite number<br />
of points {m} of the Taylor spectrum of ~, we write<br />
e([T].[div(!)]) = ~ (T,div(~)) m<br />
where (T,[div(~)) m depends only onT and div(~) in a neighborhood<br />
of m.<br />
The establishment of a local index theory then means<br />
that we must find a way to calculate the pairing (T,(div(~)) m.<br />
In this note we explain our results from [6] in these<br />
terms rather completely for the case when the Taylor spectrum<br />
has one dimensional structure, and we indicate how the general case<br />
may be treated.<br />
For maximal ideals which are 1-analytic (a concept which<br />
we introduce below), we will define the operator theoretic local<br />
Chern class ch~(~,m) by forming the unital Banach algebra generated<br />
by T. Let A denote this algebra, and let A ^ denote the completion<br />
-- m<br />
of A with respect to the filtration{mn}, where bar denotes closure.<br />
Under the generating condition it will turn out that<br />
A^m is Noetherian~d we form the Grothendieck group K(A~) of<br />
isomorphism classes of all finitely generated A ^ modules.<br />
m<br />
We view the Hilbert space H in two different ways:<br />
For x in H and r in A, we consider the two actions (r,x)§<br />
and (r,x)§ We let H~ denote the adj0int module associated<br />
with the second action.Then we complete the Hilbert space regarded<br />
as an A module by taking the filtration{mnH}. The resu]ting<br />
completion is denoted by H^'m Similarly, we form H~, the completion<br />
formed by taking the filtration{mnH~}<br />
The local chern class ch~(~,m) is defined as the element<br />
of K(A~) given by the difference [Him ] - [H ^] when m is 1-analytic.<br />
m
622 Carey and Pincus<br />
We note here that the condition of one-analyticity implies that<br />
H ^ and H ^ are Noetherian A ^ modules.<br />
m *m m<br />
If a is an ideal of a ring R and E is an R-module,<br />
denote by ~R[a,E] the Hilbert-Samuel multiplicity of a with<br />
respect to E over R whenever it is defined. Recall [7] that<br />
this is the coefficient in the so-called X-polynomial of E.<br />
The multiplicity of the two modules above can be<br />
^<br />
written as limits. Take m to denote the closure of mA ^ in A ^. Then<br />
m m<br />
and<br />
~A m^ [~,H~]= lim dimcmnH/mn+iH<br />
UA~[~,H~%] : lim dimcmnH~/mn+iH ~<br />
n §<br />
If a is an ideal in A~, we will define its index by<br />
setting I(a) = ~A~[a,H~] - ~A~[a,H~] , whenever the multiplicities<br />
exist. Thus, the indexof a one-analytic maximal ideal m is obtained<br />
by applying the multiplicity homomorphism ~Am[~,.] : K(A ) + Z,<br />
to ch~(2,m) which gives<br />
I(m) =UA^[~,H~ ] -~A^[~,H~]<br />
m m<br />
82 k-analyticity<br />
We now give a definition which makes it possible to carry<br />
out the program which we have outlined for the construction of the<br />
local index. The definition which follows is merely a sufficient<br />
condition which can certainly be weakened.<br />
Let A be the Banach algebra generated by ~ = ~i ..... Ts}'<br />
a commuting s-tuple of operators. Let (z) be the ideal in A generated<br />
by T-z. We will say that (z) is k-analytic if k is the smallest<br />
integer for which there is a surjective linear map ~ ={~i,~2 H<br />
from C s onto C k such that the image of T-z in the Calkin algebra<br />
is an invertible k-tuple in the sense of Taylor.<br />
In other words, (z) is k-analytic if there are k<br />
linear combinations of the generators T-z which form a Fredholm<br />
k-tuple, but fewer than k such combinations are not Fredholm.<br />
A condition such as this one guarantees that the<br />
Taylor spectrum in a neighborhood of (z) consists of k -dimensional<br />
analytic variety.
Carey and Pincus 623<br />
result.<br />
Results from [16 ] can be used to establish the following<br />
THEOREM 4.1 Let (z) be !k-analytic maximal ideal of<br />
C s<br />
A. Then there is a neighborhood of (z) in which the Taylor<br />
spectrum of T is an analytic subvariety of dimension at most k.<br />
It is a consequence that the Taylor spectrum in a<br />
neighborhood of a non-isolated one-analytic (z) consists of a finite<br />
union of irreducible complex one-dimensional analytic varieties<br />
{vj}.<br />
THEOREM 4.2 ~(z) i__s on,analytic, then for all w<br />
sufficiently near z, the module completions H2w). and H$(w) are<br />
No etherian A~ . modules; fur____!therm______qor ~ th___!e multiplcities<br />
~i~ dimc (w)~H)(w)n+lWl and ~i~ dimc (w)nH~/(w)n+iH~ exist.<br />
This result seems to be the first realization that<br />
Noetherian ring theory and index theory are related. We found it<br />
by considering the standard definition of the Noether index of<br />
a Fredholm operator, and rewriting it in the following way:<br />
PROPOSITION 4.3 Let T be a Fredholm operator. Then<br />
Index T = lim dim T~nH/T~n+IH - lim dim TnH/Tn+IH<br />
Because the proof depends on a fundamental insight due<br />
to Hilbert, we will give it here.<br />
For n a positive integer,<br />
Index T = Index T n+l - Index T n = (dim ker T n+l- dim ker T n)<br />
- ( dim ker T ~n+l - dim ker T~n).<br />
But Index T = Index TITn H because Index is invariant<br />
1<br />
under finite rank perturbations. Thus, with P the orthogonal<br />
n<br />
projection onto TnH, we have<br />
Index T = dim ker T TnH - dim ker P T ~<br />
n TnH<br />
= (dim ker T n+l - dim ker T n) - dim TnH/Tn+IH.<br />
as well as<br />
Combining the two expressions for Index T, we get that<br />
dim ker T ~n+l - dim ker T ~n = dim Tn/Tn+IH<br />
dim ker T n+l - dim ker T n = dim T~nH/T~n+IH<br />
Thus, Index T = dim T~nH/T~n+IH - dim TnH/Tn+IH for n=l'2 ....
624 Carey and Pincus<br />
A classical theorem of Hilbert which underlies all of the<br />
multiplicity theory of rings will now imply that both terms on<br />
the right stabilize. Let X be a finitely generated graded module,<br />
X = ~ ~ X k over the polynomial ring C[x 0 ..... Xr]. Then there is<br />
a polynomial Px(t) of degree at most r with rational coefficients,<br />
called the Hilbert polynomial of the module X, so that dim C X k<br />
= Px(k) when k is sufficiently large.<br />
To apply this result, we take X k = TkH/Tk+IH and define<br />
the action of C[x0] as x 0 : h + Tk+IH § + Tk+2H , with a similar<br />
construction for the adjoint term. Since r=0, the Hilbert polynomial<br />
result implies that the dimensions stabilize.<br />
It is remarkable that this result seems not to have been<br />
previously observed; but we also have the following result which<br />
shows that index theory is closely related to Noetherian ring<br />
theory, and in particular to the multiplicity theory of ideals.<br />
COROLLARY 4.4 Let T be a Fredholm operator, then<br />
Z<br />
H(z ) and H~(z) are Noetherian C[x O] modules and<br />
Index %= length C[[x0~(H~(z) 8 C[[xo]] ) - length C[[XoI]](H(~ ) 8 C[[x0]]) "<br />
For the case of the commuting s-tuple T we have not yet<br />
introduced the analog of the principal function. Although the<br />
local index of a single Fredholm operator coincides with the<br />
classical Noether index, and thus with gT' this is a feature of<br />
the geometry of the plane which does not carry to higher dimensions<br />
where the spectrum can have singular loci imbedded in it.<br />
We have defined Ch~(T,m) as a difference element in<br />
the Grothendieck K(AI). Since this group is generated by the<br />
^<br />
quotients A~/p , where p is a prime ideal of Am, the local index<br />
can be thought of as being the ~ement in K(Am) defined by the<br />
difference [H ~m ^ ] - [H ~ ] , with the decomposition<br />
^<br />
[H.m] - [H i] = ~gp[A:/p]<br />
The sum here is taken over the distinct primes. These will all<br />
have depth zero or one, and the integer coefficients gj which<br />
occur as the coefficients of primes of depth one will correspond<br />
to the principal function.
Carey and Pincus 625<br />
Under the assumption of one-analyticity, we can give a<br />
separate formula for these "principal numbers".<br />
Let Z denote the Taylor spectrum of ~. Let B = C(ZN U),<br />
the continuous complex-valued function in a CS-neighborhood, U,<br />
of a one-analytic maximal ideal m. Let P:A§ be the Gelfand-Mazur<br />
map composed with restriction to U. We can take the closures of<br />
A<br />
the images p(m n) to form the algebra completion B ^ and a P which<br />
m<br />
extends P , and maps A^ to BA.<br />
m m<br />
Let E (resp. E~) be the annihilator in ^Am of H ^m (resp. H~)<br />
the localization (A~)p~ of Am at the minimal prime<br />
and form Pj<br />
of E which corresponds to the i~reducible variety V. introduced<br />
J<br />
after theorem 4.1 above. In other words, pj is the minimal prime^<br />
ideal of E whose image under the surjective map (induced by p)<br />
^ ^<br />
given by Am/E§ ) yields the prime ideal corresponding to<br />
the sheet V and maximal ideal m. We note that this makes sense<br />
J<br />
only when the image is a minimal prime, i.e. V. may not appear<br />
^ J<br />
in the prime spectrum of B /P(Z).<br />
m<br />
Now we can use the associativity formula for the<br />
multiplicity of ideals. See,for example,Nagata [7 ].<br />
THEOREM 4.5 If (z) is one-analytic, then the principal<br />
numbers are given by<br />
gJ = lengthA(z)p~j[H~(z) ^ ^ 8 A(z)^ (A i z)p~j ]<br />
- lengthA~z)pj[H(z)8 ^ A(z ^ ) (A)iz)p j ]"<br />
The lengths which occur here are of course defined only<br />
when the indicated primes exist. The lengths are set equal to<br />
zero when the ideals are not well defined.<br />
This theorem is the natural extension of corollary 4.4.<br />
The proof, of course, is more than just an application of the<br />
associativity law from algebra. The main point of the proof is<br />
that we are able tO get rid of the quasi-nilpotent elements.<br />
But the perhaps unexpected result is that the principal<br />
numbers are unchanged under trace class perturbations of the<br />
generators T which preserve commutativity.
626 Carey and Pincus<br />
THEOREM 4.6 If L. = T.+K., where K. is in the trace class<br />
-- 1 I 1 I<br />
and if the commutator [Li,Lj] = O, then all of the numbers gi<br />
remain unchanged when they are computed in terms of the perturbed<br />
ideals.<br />
We will see below that more can besaid if it is assumed<br />
that the essential spectrum of the s-tuple ~ has some smoothness.<br />
It is very important that we can now state transformation<br />
lawsfor the local index under morphisms in terms of these principal<br />
numbers.<br />
For the simplest of these relations, consider a morphism<br />
= (fl .... fr)- By the spectral mapping theorem the Taylor spectrum<br />
of the r-tuple of operators ~(~) is the image set ~(~) in C r.<br />
Suppose that the r-tuple ~(~) is one-analytic at y in ~(Z).<br />
-I<br />
Suppose also that ! (y) consists of one-analytic ideals.(The<br />
analyticity assumption implies that this set is finite.)Suppose<br />
also that ~(Z) is topologically unibranch at y.<br />
THEOREM 4.7 The index of the maximal ideal which<br />
corresponds to the new generators ~(~)- y is given by<br />
I(!-y) =[{~mult f gj(x) }<br />
J x:!(x~=y j<br />
xEV. ]<br />
where mult f.is the degree of the unramified covering defined by<br />
x--3~--<br />
restricting f to a small neighborhood o~f x in V..<br />
- - ]<br />
The point of such theorems as this is that there is no<br />
restriction on the relation between r and s, and we are not merely<br />
dealing with a global transformation law far homology classes-<br />
which we could also construct- but with a transformation law for<br />
the local index in terms of the principal numbers. The module<br />
action of the algebra A on the Hilbert space has been broken up<br />
into a purely geometric part corresponding to the multiplicities<br />
associated to the map ~ on the varieties V. and lengths of the<br />
]<br />
composition series given by the gi"<br />
For a very simple example, consider the pair ($2,S 3)<br />
where S is the unilateral shift. The Taylor spectrum is the<br />
patch {(Zl'Z2):<br />
Z 3<br />
1 -<br />
2<br />
z2 = O, z 3. < i, j=l,2}. Since the principal<br />
function of the shift (its index) is -I inside the unit disc, we
Carey and Pincus 627<br />
see that the theorem then implies that the local index of the<br />
maximal ideal generated by (S2-zl,S3-z2) on this patch is -i<br />
except at the origin where it takes the value -2. Indeed, if we<br />
consider the map ! : C+C 2 given by z+(zkP,z kq) where p and q are<br />
relatively prime and p < q,then it is clear that mult0~ = kp.<br />
We will return to this example below, where we will again<br />
compute the local index but in terms of a system of parameters<br />
for the maximal ideal.<br />
Now we turn to the computation of the local index<br />
in terms of the transformation law. This is the analog of (i) in<br />
82 we expressed that formula in terms of the residue theorem).<br />
THEOREM 4.8 Suppose that f is a complex-valued analytic<br />
function which does not vanish on Oess(~) , Then<br />
Index f(T) = X{Xmult~ fj gj<br />
~(x)=8<br />
x in V.<br />
]<br />
82 System of parameters<br />
An alternate description of the local index can be<br />
given in terms of a system of parameters for the one-analytic<br />
maximal ideal. We let (z) be a one-analytic ideal and let ~ belong<br />
to the Grassmanian Gc(s-l,s ) so that:<br />
a) the hyperplane corresponding to N intersects the<br />
tangent cone to the Taylor spectrum at (z) only at the vertex point<br />
z.<br />
b) The operator ~1(~-z) is Fredholm.<br />
The notation ~ refers here both to an element of the<br />
Grassmanian and to the s-i linear combinations of the generators<br />
with the indicated zero space. We prove that the ambiguity involved<br />
here is harmless in the sense that different linear combinations of<br />
the generators which vanish on the same subspace give the same<br />
result in the next theorem. N(~-z) is the s-1 tuple of operators<br />
formed by choosing a dual basis{Nl,-..,Ds_ I} for ~, and then forming<br />
the operators Dl(~-z ) ..... Ds_l(~-z). The notation ~i(~-z) refers<br />
J<br />
to the operator D(T-z) obtained from any dual vector N to 9-, the<br />
orthogonal complement to K relative to the standard inner product<br />
in C s .
628 Carey and Pincus<br />
THEOREM 5.1 If (z) is one-analytic, there is an open<br />
set ~ in the Grassmanian Gc(S-l,s ) such that a) and b) above are<br />
satisfied and such that if H is in ~ , then the local index of<br />
the maximal ideal (z) is ~y_e_E by<br />
. L<br />
l(z) dim ker ~ (~-z)(l root space H (~-z)<br />
I<br />
- dim ker ~(~-z) ~ r~ root space H (~-z) ~<br />
In this statement, we have used root space ~(~-z) to mean the<br />
set of vectors in Hilbert space which are annihilated by some<br />
positive power of all of the s-I operators ~(~-z). The first term<br />
in I(z) above is the dimension of the space of these vectors that<br />
I<br />
lie also in the kernel of the single operator ~(~-z). Because of<br />
b) this last space is finite dimensional.<br />
We turn now to the example (S 2,S 3) treated in 82 above,<br />
but now we wish to illustrate how we get I(O) for all ~ that do<br />
not pass through the tangent cone.<br />
We have seen that the Taylor spectrum is {(Zl,Z2):<br />
3 2 = O, Izjl~ i}. This has a cusp singularity at the origin.<br />
Z 1 ~ Z 2<br />
The tangent cone at the origin is the zero set of the<br />
2<br />
polynomial z 2 , and this is the plane C z 1<br />
Thus, to calculate the index by the formula in theorem<br />
5.1, we can use ~($2,S 3) = aiS2+agS 3 as long as a2~ O. Then we<br />
must have ~I($2 ~3. . ~2 . ~S 3<br />
,~ ) = Dlb +D 2 w ere albl+a2b2 = 0 and O~Ib I ~Ib21.<br />
The last condition is a result of the Fredholm assumption b).<br />
Because ker(blS2+b2S3)is the zero vector, we have<br />
i 2 3~<br />
I(0) = - dim ker ~ (S ,S ) r] root space ~ ($2,$3) ~.<br />
But, with % = - bl/b 2 , we have<br />
i r span {l,e i0 } for Ibll > [b21<br />
ker ~ ($2,$3)~ = I span {l,e i0 ~ %neinO } for 0 la21 . This<br />
inequality is equivalent to ]bll < Ib21 . Thus~ine in0 is not in<br />
the intersection. Hence, I(0) = -2.<br />
The theorem above is the form in which we originally<br />
announced our results in [6].
Carey and Pincus 629<br />
There is yet another way in which we can express the<br />
local index. It is the Euler characteristic of a Koszul complex<br />
given in terms of the local parameters.<br />
We regard the Hilbert space as an A module, and we form<br />
the vector space H z, which is defined to be the localization of H<br />
at the maximal ideal corresponding to z. The operator ~<br />
!<br />
induces an endomorphism [~'(~-z)] on Hz, and it is easy to give<br />
another form to the last theorem:<br />
THEOREM 5.2 I(z) is the Euler-Poincare characteristic<br />
I<br />
of the complex [H-(~-z)]<br />
82 Holomorphic chains<br />
O+ Hz +H z +0<br />
One of the characterizations of a holomorphic one-chain<br />
is as a locally finite sum ~ gj[Vj], where gj is an integer and<br />
[Vj] indicates integration of two-forms over the regular points<br />
of the irreducible variety V . It is a theorem of Lelong that<br />
3<br />
this defines a current.<br />
The local degree of such a chain is defined to be<br />
gjO(ll[Vj!i[ ; z), where 0(I [ [Vj] II; z) is the melong number of<br />
[Vj] at z.<br />
Recall that by definition the density of a current A<br />
having measure coefficients is defined in terms of its associated<br />
variation measure. For an open set Q in some fixed compact set<br />
containing the support of A , this is defined to be<br />
IIA(~)II ~ sup { I&(~) : ~Dk(~) with II~(x) II < I, all x ~ Q}<br />
The density or Lelong number 0(I I ~ l];z) at the point<br />
z in ~ , is lim @ II & II;r,z), the limit of the average mass of<br />
r§<br />
in a ball of radius r about z. When & = [Vj] it is known that<br />
O(II[Vj]II;z ) = ~j(z), the multiplicity of the local ring of the<br />
variety V. at z.<br />
3 We have already seen that the Taylor spectrum of T in<br />
a neighborhood of a non-isolated one-analytic (z) consists of a<br />
finite union of irreducible complex one dimensional analytic<br />
varieties V..<br />
3<br />
By the local principal current we shall mean a pair
630 Carey and Pincus<br />
C s<br />
{Uz,[Vz]}Consisting of a neighborhood U z of z and a holomorphic<br />
one-chain [Vz] defined in U z so that UznZ consists of one-analytic<br />
ideals, and [Vz] has the form ~ gj(z)[Vj]. Here gj(z) is the<br />
principal number at z, and [Vj] indicates the current of integra-<br />
tion of two forms over the regular points of V. with the induced<br />
3<br />
orientation.<br />
The local principal current has local degree equal to<br />
the maximal ideal index I(w) for w in U , and the local currents<br />
z<br />
piece together to give the globally defined principal current<br />
THEOREM 7.1 If (z) and (w) are one analytic ideals<br />
then the local currents [Vz] and [Vw] agree on the intersection<br />
U :~ U .<br />
z w<br />
Using theorem 5.8 above we can we can sometimes evaluate<br />
Index f(~) when f is a complex valued analytic function without a<br />
zero on the joint essential spectrum of ~ and [Ti,Tj~] is compact.<br />
But there are hypothesis on the joint essential spectrum which allow<br />
us to conclude more.<br />
According to a definition in Harvey [9], suppose that M<br />
is a compact subset of a Riemannian manifold N. Suppose that there<br />
is a closed subset S (the scar set ) of M with Hausdorff 2k-I<br />
measure zero such that the set of points in M but not is S, M/S, is<br />
a real 2k-I dimensional submanifold of N/S of class Clhaving finite<br />
volume. Such sets are called scarred (2k-l) manifolds.<br />
The assumption that the joint essential spectrum of<br />
is contained in a scarred one-manifold has strong consequences.<br />
The following theorem implies that all the maximal ideals of the<br />
algebra A which are not in the joint essential spectrum are one-<br />
analytic, and shows how they assemble globally.<br />
THEOREM 7.2 Suppose that T is a commuting s-tuple o__~f<br />
operators with [Ti,Tj~ ] compact for all i and j, and suppose that<br />
th 9 .~oint essential spectrum of T is contained in a scarred one<br />
manifold M. If z is not in the joint essential spectrum of ~,<br />
then (z) i__~s one-analytic, and there is a holomorphic one-chain $<br />
i__~n CS/oess(~) whose local degree a__tt z i__~s th e maximal ideal index<br />
I(z). This chain $ has a unique compactly supported current extension<br />
T to c s with finite mass. T is the unique compactly supported
Carey and Pincus 631<br />
A~<br />
rectifiable (i,i) current so that aT(y) = Index f(~) when f i__ss<br />
analytic and not vanishing on ~ (~).<br />
- - ess<br />
Under the additional assumption that the commutators of<br />
the C ~ algebra generated by ~ are in trace class, we can evaluate<br />
the boundary of T.<br />
THEOREM 6.3 I__f_f [Ti,Tj~] is in the trace class for all<br />
i and j, then ~T(fdh) = trace [ h(~,~), f(T,~)] where f(~,T~)<br />
and h(~,~ ~) are operators defined for test functions f and h by<br />
the Weyl functional calculus.<br />
Because we are assuming that the two dimensional Hausdorff<br />
measure of the joint essential spectrum of ~ is zero, we can use<br />
the Hartogs-Rosenthal theorem to conclude that the continuous<br />
functions on the joint essential spectrum of ~ can be approximated<br />
by rational functions having poles in the complement of Gess(~).<br />
It follows then from the last part of Theorem 6.2, that<br />
~T determines the Kl(Oess(~homology class which corresponds to the<br />
tuple T.<br />
The boundary current here is a real MC cycle, where we<br />
recall that a one dimensional real MC cycle on a Stein manifold<br />
is a closed one-current ~ for which ~(~) = 0 whenever ~ is a<br />
holomorphic one-form.<br />
In [8] Harvey and Lawson eonslder the construction of<br />
holomorphic chains which span real MC cycles supported on real<br />
2k-i dimensional submanifolds of C s. This work was extended by<br />
Harvey to the scarred case in [9].<br />
The crux of the cited construction is the introduction<br />
of a certain line bundle defined in terms of a function which is<br />
shown (after some difficulty) to be rational using results of<br />
Hadamard. We proceed directly to construct this function in terms<br />
of Lefshetz numbers associated with the z-parameterized Koszul<br />
complexes introduced above. In the one chain case we identify the<br />
cocycles which determine the line bundle as a quotient of character-<br />
istic polynomials of endomorphisms which act on the Koszul complex<br />
of Theorem 5.2 above. Our local construction contrasts with the<br />
Harvey-Lawson use of global data associated with the boundary cycle.
632 Carey and Pincus<br />
A version of this construction holds in all dimensions.<br />
While we do not know the invariance properties of the<br />
local index beyond those described in Theorem 4.6 in general, the<br />
additional assumptions of Theorem 6.2 imply stability under compact<br />
perturbations for the principal current T.<br />
THEOREM 6.4 Under the assumptions of Theorem 6.2, the<br />
local index I(z) is stable under compact perturbations of the<br />
generators which preserve commutativit[. The chain T is obtained<br />
from the collection of operators TI+KI,...,Ts+K s if K. is compact<br />
for all j, and these new generators commute.<br />
- - J - -<br />
Although the results above may seem somewhat special,<br />
they are in fact just the relevant ones for certain geometric<br />
applications. There is a canonical relationship between algebraic<br />
geometry and operator theory which emerges from our considerations<br />
and which may be fruitful for the further study of the singular<br />
locus of analytic varieties.<br />
THEOREM 6.5 Let W be a complex one-dimensional analytic<br />
variety supported in an open se____~t of C s Let x be a point of W,<br />
and let B~(x) be the ball of radius 6 about x __in C s. Let ds denote<br />
arc length measure on F= W n ~B6(x), and let H2(ds) be the closure<br />
i__n_n L2(F,ds) of the polynomials i__n_n (z I .... Zs). Then the subnormal<br />
operators T defined to be multiplication by the coordinate<br />
J<br />
functions zj on th____ee Hardy space H2(ds), have commutators [Tj,Ti~ ]<br />
in trace class. There is a principal current for the algebra<br />
generated by these T. and the local index I(~) i__~s equal to the<br />
J<br />
negative of the multiplicity of the local ring of W for ~ in the<br />
intersection of W and the ball.<br />
We pose the question of connecting the multidimensional<br />
versions of this theorem with the study of more subtle invariants<br />
of singularities.<br />
82 Remarks<br />
We will not give a full development of the multidimensional<br />
theory here. Instead we will merely indicate how the local index can<br />
be obtained. A full description will be given elsewhere9<br />
First, let us remark that the theory which we have
Carey and Pincus 633<br />
introduced has the study of pseudo-differential operators which are<br />
microlocally like Toeplitz operators as one of its applications.<br />
Accordingly, in our limited space here, we will treat<br />
only one case, that of the so-called crypto-integral algebras.<br />
These algebras are generated by pseudo-differential operators of<br />
order zero with scalar symbols which are asymptotically homogeneous<br />
and which have compact support See [13].<br />
The local index theory is most easily developed in terms<br />
of the Koszul complex associated with a system of parameters,and<br />
we will discuss the case where assumptions are made on ~ss(T).<br />
We willassume that: T is a commuting s-tuple<br />
of operators which generate a crypto-integral C~-algebra of<br />
dimension two, and suppose that the joint essential spectrum i__ss<br />
contained in a C 1 smooth scarred three-manifold<br />
Form the real MC cycle defined by setting<br />
~(fldf2~ df 3 ~ df4) = -2~ 2 trace [fl(~),f2(~),f3(~),f4(~) ]<br />
where [',',','] denotes the completely antisymmetric form with<br />
four entries.<br />
Assume that ~ is a flat current. Of course this needs<br />
further discussion, and will be analyzed in a forthcoming account.<br />
Suppose also that z is a Fredholm point for T. Then the set of<br />
projections ~ for which ~$(z) is not in HI(M ) -- is an open dense set<br />
in the complex Gc(s-2,s). For such a ~ , let the pair<br />
of operators ~ (~-z),obtained as before through the choice of a<br />
dual basis, be denoted by (V1(z),V2(z)).<br />
The Koszul complex defined by this commuting pair of<br />
operators is<br />
where the boundary maps are<br />
and<br />
d 2 d 1<br />
0 + H § H @ H § H § 0<br />
d2(x) = -V1 (z)x ~ V2 (z)x<br />
d2(xSy ) = Vl(z)x + VZ(z)y<br />
We will let H k = ker dk/range dk+l, and suppose for<br />
the following theorem that H satisfies the conditions above.
634 Carey and Pincus<br />
THEOREM 7.1 Consider the endomorphisms induced on the homology of<br />
the Koszul complex by the s-2 operators ~(~-~). There is a holomorphic<br />
two chain $ in cS/o ,~. whose local degree is equal to<br />
- - ess~i ) -- __<br />
I(z) = dim fl root space H(~-~)i H 0 + dim N root space~(~-~)IH 1<br />
- dim l~ root space H (~-~) H<br />
This chain has a unique compactly 2 supported current<br />
extension T to C s with finite mass. T is the unique compactly<br />
supported rectifiable (2,2) current so that ~T = ~. ibove'<br />
the range hyperplane associated to ~ intersects the tangent cone<br />
to the Taylor spectrum only at the vertex z.<br />
Theorem 5.2 above was obtained by writing the dimensions<br />
of the intersection of the corresponding root spaces as dimensions<br />
of homology modules of a localized Koszul complex. We can rewrite<br />
theorem 7.1 in the same way. We again view the Hilbert space as a<br />
module which we localize at the maximal ideal which corresponds to<br />
z. Then the pair of operators {V1(z),V2(z)} induces commuting<br />
endomorphisms {[V1(z)] [V2(z)]} on H . The Koszul complex E(Hz,[V])<br />
' Z<br />
is defined to be the localization of the complex of the preceeding<br />
page obtained by replacing H by H z, and by replacing the boundary<br />
maps by the ones formed from the commuting endomorphisms of H just<br />
z<br />
described.<br />
9 s<br />
THEOREM 7.2 The Euler-Polncare characteristic of the Koszul complex<br />
E(H [V]) is I(z).<br />
When the Taylor spectrum has higher dimensionality<br />
we proceed similarly. If we have a k-analytic maximal ideal, we<br />
form parameters which correspond to projections in the Grassmanian<br />
Gc(s-k,s), and the local index can be defined to be the Euler<br />
characteristic of the corresponding multistep localized Koszul<br />
complex. There is a corresponding Grothendieck group construction.<br />
Just as in Theorem 4.8 above, we can express the<br />
Noether index of operators formed from the generators in terms of<br />
the Functional calculus by taking the intersections of divisors<br />
and principal currents.<br />
Suppose,for example, that F = (FI,F2):Cz+C2~ , 0 ~ F(M)
Carey and P 635<br />
for holomorphic F . Then we can consider the Bott operator<br />
]<br />
I<br />
-F2(2)" FI(I)~ ]<br />
on H@H. The Noether index of this Fredholm operator is equal to<br />
the current intersection T {] [div([)](1), where [div(~)] is the<br />
so-called Poincare-Martinelli divisor of F. See [2].<br />
Our main idea from [6] was that there is a holomorphic<br />
chain representation of the Ext class with operator theoretic<br />
meaning for the local degree. The study of holomorphic chains can<br />
be expressed sheaf theoretically, and we will exploit this language<br />
elsewhere. But we note here that our attention has been called, at<br />
a recent conference organized by INCREST, to a preprint by M. Putinar<br />
[18], and an announcement by R. Levi [19] where sheaf theoretic<br />
language is used to study a generalized index theory associated<br />
with the Euler-PQincare characteristics of Koszul complexes<br />
constructed from s-tuples of operators. There are interesting<br />
relations between this work and ours; but neither of these<br />
cited authors introduces the concept of a local index, and they<br />
do not make contact with the multiplicity theory of ideals nor with<br />
the relationships between traces of commutators and boundary<br />
currents. Yet their work has connections with ours and may be<br />
looked upon (from our viewpoint) as being a step towards the<br />
kind of localization which we have described above as the con-<br />
struction of operator-theoretic local Chern classes.<br />
Our interpretation of local Chern classes- and especially<br />
the relationships between principal numbers and the local Euler<br />
obstruction in higher dimensional situations will be treated elsewhere<br />
in terms of the Grothendieck local cohomology. See [20].(The local<br />
Euler obstruction was introduced by MacPherson in [21 ]as a numerical<br />
invariant associated to singular points of algebraic spaces, and<br />
used to construct Chern classes in homology for algebraic spaces.)<br />
See [22] and [23]. We envisage the fully developed local index theory<br />
of our operator algebras as providing relationships between ~ and<br />
m<br />
these geometric objects as well as relationships between integrals<br />
of curvature forms in the neighborhood of singularities and appropriate<br />
traces involving multicommutators.
636 Carey and Pincus<br />
Of course the restriction of the theory to the situation<br />
where the Taylor spectrum has analytic structure involves too much<br />
loss of generality. There is the beginning of a more general<br />
formulation for situations in which there are no underlying analytic<br />
varieties.<br />
We describe such an example now. Suppose that N is a<br />
bounded normal operator and that Y is a bounded self-adjoint<br />
operator such that the commutator [N,Y] is in the trace class. There<br />
is no fully satisfactory notion in the literature,of which we are<br />
aware, for the spectrum of the non-commutative algebra generated<br />
by such a pair of operators. The joint essential spectrum is some<br />
subset of R 3, and when this is a smooth curve one can talk in terms<br />
of finding a spanning surface. But there are many ways to span-<br />
even minimally.<br />
Despite this we have a result which is the protQtype<br />
for more general ones involving greater numbers of operators.<br />
Using the Weyl calculus, it is possible to define a real closed<br />
one-current A concentrated on the joint essential spectrum of<br />
the pair of operators N and Y (without any smoothness assumptions<br />
on the joint essential spectrum) by setting<br />
A(RdS) = 2~i trace [R(N,Y),S(N,Y)].<br />
THEOREM 7.2 A is a flat current having the form d[G] for [G]<br />
rectifiable current having the form [G] = (H2 I_ E^ g~) where ~ i__ss<br />
an orientation function, ~: Z§ (unit simple two vectors in R 3<br />
tangent t__oo E, a bounded two dimensional rectifiable set.<br />
The role of the principal function here is played by<br />
the function g, which need now not be an integer, and the<br />
rectifiable set E plays the role of a new spectrum.<br />
It seems quite remarkable that the trace assumption on<br />
the commutator [N,Y] implies so much geometric structure. But it<br />
is also the case that the current [G] is constructed from local<br />
data expressed in terms of symbols.<br />
In [I0] this theorem is used to study self-adjoint<br />
(and symmetric) Toeplitz operators on multiply connected plane<br />
domains. The symbol of these operators has no smoothness beyond<br />
essential boundedness, and the joint essential spectrum can be
Carey and Pincus 637<br />
dense in Z. Nevertheless, the principal function is integer valued<br />
H 2 almost everywhere on Z. A certain multiplication operator plays<br />
the role of N in this example; and the joint essential spectrum of<br />
the Toeplitz operator and N is computed as the H 2 I_ Z essential<br />
discontinuity set of the principal function. We have thus construced<br />
an integral varifold which spans the essential spectrum and whose<br />
support is understood as a spectrum for the algebra.<br />
It should be noted that the Euler characteristic of<br />
a Koszul complex built directly from the operators N and Y in the<br />
preceeding theorem - by passing to the Calkin algebra- v&nishes.<br />
See [ 27] or [28]. This is also the case in the situation of<br />
Theorem 7.2 above, where the number of generators is greater than<br />
the "intrinsic" dimension of the Taylor spectrum. This circumstance<br />
may perhaps make the distinction between the underlying ideas of<br />
the original principal function approach as contrasted with its<br />
variant in [13] more pointed. In the single operator case the<br />
integrability of the principal function was perhaps the main fact.<br />
Now rectifiability of the principal function seems to be of cardinal<br />
importance in relation to an associated slicing and intersection<br />
theory. We hope for a theory in which geometric singularities<br />
are related to the local structure of the underlying operator algebras.<br />
The trace can be associated with a type II factor.<br />
For, in fundamental work, M. Breuer [25],[26] has created an index<br />
theory in this context. We will close our remarks by giving an<br />
example of a principal current in the yon Neumann algebra setting.<br />
It is known that the C e algebra generated by the translations on<br />
L 2 (R +) can be represented faithfully in a type II factor. In<br />
[24] the authors showed how the principal function of a Toeplitz<br />
operator whose symbol is a finite sum of exponentials coincides<br />
with the Lagrange mean motion of the symbol. Now we wish to study<br />
the more special Fredholm version of this result in which the<br />
frequencies in the exponential are rational and we have Bohr mean<br />
motion.<br />
Let { Tj } be the commutative s-tuple of Wiener-Hopf<br />
operators with the symbols exp(-2~iljt) _ where I. is a rational<br />
J
638 Carey and Pincus<br />
number expressed in lowest terms as %j = pj/qj, and where we take<br />
0 < %1 < %2 < "'"
Carey and Pincus 639<br />
[9]<br />
[10]<br />
[11]<br />
[12]<br />
[13]<br />
[14]<br />
[15]<br />
[16]<br />
[17]<br />
[18]<br />
[19]<br />
[20]<br />
[21]<br />
[22]<br />
[23]<br />
[24]<br />
............ On boundaries of complex analytic varietiies<br />
II, Annals of Math. 106 (1977) 213-238<br />
Harvey, F.R., Holomorphic chains and their boundaries,<br />
Proc. Sympos. Pure Math., 3 0:I (1977) 309-382, Amer.<br />
Math. Soc., Providence, Rhode Island<br />
Pincus, J.D. and Xia, J. Self-adjoint and symmetric<br />
Toeplitz operators on multiply connected plane domains,<br />
Jour. Functional Analysis 59 (1984) 397-444<br />
Federer, H., Geometric Measure Theory, Grundlehren<br />
Math. Wiss. 153, Springer Verlag (1969) Berlin and<br />
New York<br />
Carey, R.W. and Pincus, J.D., An invariant for certain<br />
operator algebras, Proc. National Acad. of Sciences<br />
71 (1974) 1952-1956<br />
Helton, J.W. and Howe, R.e., Integrql operators,<br />
commutators, traces, index and homology, Lecture Notes<br />
in Math. 345 (1973) 141-209<br />
Carey, R.W. and Pincus, J.D., Pincipal functions, index<br />
theory, geometric measure theory, and function algebras,<br />
Integral Eq. and Operator Theory 2 (1979) 441-483<br />
Carey, R.W., and Pincus, J.D., Index theory for operator<br />
ranges, preprint (1984)<br />
Markoe, A., Analytic families of differential complexes,<br />
Jour. Functional Analysis 9 (1972) 181-188<br />
Prossdorf, S., On the stability of the index of a one<br />
dimensional singular integral operator with vanishing<br />
symbol, Prob. Math. Analysis, Leningrad State Univ.<br />
(1966) 70-79 Translation by Consultants Bureau, New York<br />
Putinar, M., Base change and the Fredholm index,<br />
Increst preprint No. 33 (1984)<br />
Levi, R.N., Cohomology of collections of essentially<br />
commuting operators, Functional Analysis and its<br />
applications, 17 (1984) 229-230, Translation by Plenum<br />
Press, New York<br />
Hartshorne, R., Local cohomo~gy , Lect. Notes in Math.,<br />
41 (1967) Springer Verlag, Berlin and New York<br />
MacPherson, R., Chern classes for singular varieties,<br />
Annals of Math i00 (1974) 423-432<br />
Gonzalez-Sprinberg, G., L'obstruction locale d'Euler<br />
et le theoreme de Mac-Pherson, Seminare E.N.S. (1978-79)<br />
7-32 in ASTERISQUE 82-83 , Soc. Math. de France<br />
Langevin, R., Courbures au voisanage d'une singularitie<br />
algebraique isolee, Seminare E.N.S. (1978-79)33-43<br />
in ASTERISQUE 82-83, Soc. Math. de France<br />
Carey, R.W., and Pincus , J.D., Mean motion, Principal<br />
functions, and the zeros of Dirichlet series, Integral<br />
Equations and Operator Theory 2 (1979) 484-502
640 Carey and Pincus<br />
[25]<br />
[26]<br />
[27]<br />
[28]<br />
Breuer, M., Fredholm theories in von Neumann algebras,l<br />
Mth. Annalen 178 (1968) 243-254<br />
Breuer, M., Fredholm theoried in yon Neumann algebras,II<br />
Math. Annalen 180 (1969) 313-325<br />
Curto, R. Fredholm and invertible n-tuples of operators.<br />
The deformation problem, Trans. Amer. Math. Soc. 266<br />
(1981) 129-159; and Fredholm and invertible tuples of<br />
linear operators, Dissertation ,SUNY, Stony Brook,1978<br />
Vasilescu,F.H.,On pairs of commutifig operators,Studia<br />
Math. , 62 (1978) 203-207<br />
Mathematics Department Mathematics Department<br />
University of Kentucky StateUniversity of New York<br />
Lexington, Kentucky 40506 Stony Brook, New York 11794<br />
This research was partially supported by grants from the<br />
National Science Foundation.<br />
Submitted: January 7, 1985
Integral Equations<br />
and Operator Theory<br />
Vol. 8 (1985)<br />
ON THE SYMBOL HOMOMORPHISM<br />
OF A CERTAIN FRECHET ALGEBRA<br />
OF SINGULAR INTEGRAL OPERATORS<br />
0378-620X/85/050641-0951.50+0.20/0<br />
9 1985 Birkh~user Verlag, Basel<br />
Heinz-0tto Cordes and Elmar Sehrohe<br />
We prove the surjeotivity of the symbol map of the<br />
Frechet algebra obtained by completing an algebra of convolution<br />
a~d multiplication operators in the topology generated by all<br />
z . n . ,<br />
L -Sobolev norms. The proof is based on an R -verslon of Egorov s<br />
theorem valid for non-homogeneous principal symbols, discussed in<br />
[5] , [6] . We use the hyperbolic equation ~u/~T:ilDl~u , 0
642 Cordes and Schrohe<br />
rem on commutative C -algebras that the symbol map is surjective:<br />
im ~ = A ker ~ = K(H ) In [4],[8] also a Frechet algebra A<br />
s ~ s "<br />
was investigated, obtained by completing A 0 above in the locally<br />
convex topology generated by all Sobolev norms {II. II : s E N }<br />
s<br />
This algebra A also possesses a symbol homomorphism o:A + C(~) ,<br />
governing the Fredholm properties of its operators.Moreover, A<br />
may be remarkable by its property of being a ~ -subalgebra of<br />
i(H ) , for any fixed s It shares one of the most important pro-<br />
s ,<br />
perties of C -subalgebras :If A 6 A is invertible in [(Hs) , then<br />
the inverse is in A (ef.Gramsch [I0] ).<br />
oo<br />
On the other hand it seems an open problem whether the<br />
symbol homomorphism o still is surjective,as for A s ,sE~. In that<br />
respect a first result was given in [4] : ira o at least contains<br />
a space of functions in C(~) which are infinitely differentiable<br />
in every x-direction at points (x,~) with finite x .<br />
In the present paper we will show that no condition<br />
at all is required. In other words, also the symbol of the algebra<br />
A maps onto C(~) The proof consists of a refinement of the<br />
methods developed in [ 4] and [ 8] , replacing the translation ope-<br />
iDz i< D) ~t<br />
rators e used there by the one-parameter family e ,<br />
with some fixed 0 < D < I . The important point is that the cha-<br />
racteristic flow corresponding to this family vanishes at infinity<br />
so that no differentiability of the symbol at all is required.<br />
A very similar Freehet algebra on a compact manifold without<br />
boundary was investigated by Atiyah and Singer [ I] Our present<br />
method is also applicable for that algebra,as will be discussed in<br />
a separate publication [II]<br />
wing notations.<br />
i. DEFINITIONS AND PRELIMINARY RESULTS.<br />
1 9 1 . DEFINITION: For x E Nn , s @ 9 we use the follo-<br />
(a) (x) : (l+IxI2) I/2 , ~x = (2~)-n/2dXl...dXn<br />
(b) f^= Ff denotes the Fourier transform of a<br />
tribution of H = {fES' : IIfll = ~ If^($)(~)Si2d $ < ~} denotes the<br />
s s ~ n
Cordes and Schrohe 643<br />
L2-Sobolev space of order s over ~n (a Hilbert space)<br />
(c) L =L(H ) is the algebra of bounded operators<br />
S S<br />
H s§ ' Ks=Ks(Hs ) is the ideal of compact operators in Ls<br />
(d) For f 6 L~(2 n) define the (Fourier-)multipliers<br />
f(M) and f(D) by f(M)u(x) = f(x)u(x) , f(D) = F-If(M)F ,on H 0<br />
Here f(D) 6 L ,for all s ,and llf(D) ll = sup{llf(D)ull :IIutT
644 Cordes and Schrohe<br />
have A-SAA s - A E K t , for all s,t E ~ .<br />
,<br />
(e) A /K is a commutative C -algebra with unit and<br />
8 S<br />
thus isometrically isomorphic to some space C(~) of continuous<br />
functions over a compact Hausdorff space ~, by the Gelfand-Naimark<br />
theorem. We call the function a 6 C(~) corresponding to A+K the<br />
S<br />
the symbol of A , denoted by o(A) or OA By (b) all the A s<br />
are isometrically isomorphic, and the 'symbol spaces' 9 for diffe-<br />
rent s may be identified. By (d) this can be done such that the<br />
'symbol map' A + ~ for the generators is independent of s<br />
(f) (Calderon interpolation theorem) If s
Cordes and Schrohe 645<br />
PROOF: Use prop.l.6 and induction.<br />
An important tool in the theory of ~do's is the result<br />
below,which is an immediate consequence of the Calderon-Vaillan-<br />
court theorem (cf.[2]),and calculus of ~do's ([3],[6],for example)<br />
1.9.THEOREM: The pseudo-differential operator A=Op a<br />
with symbol in CB~(N n) is in L for every s. Moreover, for every<br />
s<br />
fixed s we have an estimate<br />
I'AII < c sup {ID~D~a(x,~)I : x,~ E ~n IeI IBI~<br />
S -- S ' ' ~S } '<br />
with constants c s ' Ys depending on s and n , but not on a .<br />
2.RESULTS AND PROOFS.<br />
The following theorem gives the main result of this article.<br />
2.1. THEOREM: If a E C(N) , then there exists an<br />
operator A ~ A with ~ = a .<br />
We will break up the proof of thm.2.1 into a series of<br />
lemmata. In the remainder of this section we keep a real number<br />
, 0 < ~ < i given fixed.<br />
2.2. DEFINITION: For A E A 0 ,t 6 N define the operator<br />
A t = e-iA-~tA eiA-nt ,<br />
with the unitary operators e iA-qt = Op (e i(~)nt) of every H s .<br />
2.3. LEMMA: Let A be an operator of the algebra A 0<br />
(a) We have A t E A , for all real t .<br />
(b) The map t + A t is differentiable from N to As '<br />
in the operator norm of L s , for every s.<br />
(c) The commutators, below, are in every L , and<br />
s<br />
d/dt A t = i[At,A-n] = i[A,A-n] t .<br />
(d) We have<br />
OAt(X,~) = OA(X,~) , x,~ e N .<br />
PROOF: Clearly it is sufficient to give the proof for<br />
the generators of A 0. We get At= A for A = ~(D) , ~ = ~ , sj ,<br />
so that the statement is trivial for the Fourier multipliers. On<br />
n<br />
the other hand, for the multipliers we must invoke the N -vers!on
646 Cordes and Schrohe<br />
of Egorov's theorem discussed in [5] (for a discussion in full<br />
detail we refer to [6] (to appear)). We Only require the result<br />
for the simplest case of the l system du/dt = int<br />
with the solution operator e This equation satisfies all<br />
assumptions of thm.4.1 in [5] (or better,thms.12.1 and 12.2)<br />
In particular it is important that we do not require the hamilto-<br />
nian system defining the characteristic flow to be homogeneneous<br />
in $ ,as most other authors do. That system here is of the form<br />
x" = (
Cordes and Schrohe 647<br />
radicand R we get the estimate (2+t2) -I _< R ~ (2+t 2) = nt 2<br />
First of all, R!l+2ulvIIwI+u21wI2 qt 2 . Furthermore, if IvI>ultI, then we<br />
get R>u2+(Ivl-ultl)2=l-2ulvIItI+u2t2>l+t2-~(l-u2)-~-lt2u2=(l-E)_<br />
+u2(E-(e-l-l)t 2) , with arbitrary 0
648 Cordes and Schrohe<br />
(b) The commutator [A-n,B A] (first considered as an<br />
operator in L(Hq,H_n) ) is (extends to) an operator in A 0. We have<br />
[A-q,BA] = -iA+iB A , hence [A-q,B A] has symbol zero .<br />
PROOF: (a) is an immediate consequence of lemma 2.5.<br />
For (b) we first assume A E A 0 again. One gets<br />
t ~ t<br />
-A = I ~ d/dt(e-tAt)dt =- I e- Atdt_i I e- [A-I,At]dt =_B+[A-I,B] ,<br />
0 0 0<br />
where the fact may be used that A and B are @do's in ~c 0 (ef.[5]).<br />
Then, for a general A E A 0 we again do the approximation in the<br />
proof of lemma 2.5, observing that -iA+iB A converges in A 0 . Hence<br />
[A-q,BA]U converges for u E H q , and the limit is (-iA+iBA)U .<br />
Hence BA(dOm A -q) C dom A -n, so that [A-I,BA ] : dom A -q + H 0 is<br />
well defined as a commutator of a bounded and an unbounded opera-<br />
tor. Moreover, that commutator even is bounded and equal to<br />
-iA+iB A ,as stated. Since we have OA=OB it then is trivial that<br />
the symbol of the commutator vanishes.<br />
2.8.LEMMA: Let a E C(N), and let k=l,2,...<br />
be arbitrary. Pick any operator P E A 0 with symbol a , and define<br />
A = Bk(p) . Then o A = a . Moreover, for j=0,...,k-i we have<br />
Aj = ad]A-n(A) = i](I-B-I)JBk(p) E A 0 ,and OA. = 0 ,<br />
3<br />
with the operator B=BA=B(A) of 2.6, and B-l(Br(Q))written for<br />
Br-I(Q)<br />
The proof is an evident consequence of Lemma 2.7.<br />
2.9.LEMMA: Given any a C C(N) It is possible<br />
to find an operator A E A 0 ,j=0,1,..., with symbol a such that<br />
Aj = ad]A-q(A) 6 A 0 , and OA.= 0 , j=l,2, ....<br />
]<br />
PROOF: By lemma 2.8, for k=l,2,..., we get a set<br />
of operators A k , Alk ,..., Akk corresponding to the operators<br />
A , Aj , of the lemma. Thus the proof of 2.9 is reduced to the<br />
construction of an operator A from the operators A k , all having<br />
symbol a , in a way that extends the properties of lemma 2.8 to<br />
all j=8,i,2, .... The way to achieve this is described in detail<br />
in [4] ,p.162-164.
Cordes and Schrohe 649<br />
PROOF OF THEOREM 2.1. Let {Aj} be the sequence<br />
of 2.9. We have ~ = a ,and a(adkA-~(A))=0 . Hence all operators<br />
adkA-l(A)A k and Akad~-l(A) have their symbol vanishing over 9 .<br />
This means that they are compact. Now lemma 1.8 implies that<br />
A E A<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
I0.<br />
II.<br />
REFERENCES<br />
Atiyah, M., and Singer, I.: The index of elliptic ope-<br />
rators, Ann.of Math.66 (1968) 484-530.<br />
Calderon, A.P., and Vaillancourt, R.: On the bounded-<br />
hess of pseudodifferential operators, JJ ~th. Soe. Japan<br />
23 (1971), 274-378.<br />
Cordes, H.O.: A global parametrix for pseudodifferen-<br />
tial operators over Nn, preprint SFB 72 Math.lnst. Uni-<br />
versitaet Bonn (1976).<br />
Cordes, H.0.: Elliptic pseudodifferential operators,<br />
an abstract theory, Springer Lecture Notes in Math.<br />
voi.756, Berlin Heidelberg New York 1979.<br />
Cordes, H.0.: A version of Egorov's theorem for<br />
systems of hyperbolic pseudodifferential equations,<br />
J.of Functional Analysis 48 (1982) 285-300.<br />
Cordes, H.0.:Techniques of pseudodifferential opera-<br />
tors, to appear.<br />
Cordes, H.O. and Herman, E.: Gelfand theory of pseudo-<br />
differential operators, Amer. J.Math.90 (1968) 681-717,<br />
Cordes, H.0. and Williams, D.: An algebra of pseudo-<br />
differential operators with nonsmooth symbol, Pac. J.<br />
Math. 70 (1977) 101-116.<br />
Gohberg, I.C.: On the theory of multi-dimensional<br />
singular integrals, Soviet Math.l ~1960) 960.<br />
Gramsch, B.: Relative Inversion in der Stoerungstheo-<br />
rie von Operatoren und ~-algebren, Math. Annalen 269<br />
(1984) 27-71.<br />
Schrohe, E.: The symbol of an algebra of pseudodiffe-<br />
rential operators, to appear.<br />
H. 0. Cordes,<br />
Department of Mathematics,<br />
University of California,<br />
Berkeley,Ca. 94720<br />
U.S.A.<br />
Submitted: May 13, 1985<br />
Elmar Schrohe<br />
PB Mathematik<br />
Johannes-Gutenberg Univ.<br />
Saarstr.21<br />
6500 Mainz<br />
West Germany
Integral Equations<br />
and Operator Theory<br />
Vol. 8 (1985)<br />
0378-620X/85/050650-2451.50+0.20/0<br />
9 1985 Birkh~user Verlag, Basel<br />
WIENER-HOPF OPERATORS AND GENERALIZED ANALYTIC FUNCTIONS<br />
Raul E. Curto*, Paul S. Muhly*, and Jingbo Xia<br />
An almost periodic generalization of H +C is defined<br />
and analyzed9 Applications of this analysis are made to the type<br />
II index theory of Wiener-Hopf operators with almost periodic<br />
symbols9<br />
w INTRODUCTION<br />
In [5], Coburn, Douglas, Schaeffer, and Singer developed<br />
an index theory9 relative to a certain II~ factor, for Wiener-<br />
Hopf or Toeplitz operators with continuous almost periodic sym-<br />
bols. Their results generalize the well-known theory of Toeplitz<br />
operators on the circle whose symbols lie in C(~). Our objec-<br />
tive in this note is to find an analogue of H +C for almost<br />
periodic functions and to use it to extend the theory of Coburn,<br />
Douglas, Schaeffer, and Singer in much the same way that Douglas<br />
[6] extended the theory of Toeplitz operators with continuous sym-<br />
bols to cover Toeplitz operators with symbols in H +C. As one<br />
might expect, a completely smooth transference of the theory for<br />
Toeplitz operators with symbols in H +C to the almost periodic<br />
setting is not possible. The obstacles involve some interesting<br />
problems in the value distribution theory of analytic almost peri-<br />
odic functions.<br />
Let ~B denote the Bohr compactification of ' ~ and<br />
let CAP~R) denote the algebra of continuous almost periodic<br />
functions on ~. Then CAP(R) is a commutative C*-algebra9<br />
under the sup-norm and pointwise operations, whose maximal ideal<br />
space is ~B If AAP(~) denotes CAP(E) N H~(~) where H~(~)<br />
is the classical Hardy space of boundary-values of functions that<br />
Supported in part by the National Science Foundation.
Curto, Muhly and Xia 651<br />
are bounded and analytic in the upper half-plane, then AAP(~)<br />
may be viewed as a function algebra on ~B; in fact, it is a<br />
Dirichlet algebra there, i.e., the real parts of the functions in<br />
AAP(~) are uniformly dense in the real-valued functions in<br />
CAP(~). Moreover, Haar measure m on ~B is multiplicative on<br />
AAP(~), and so, for each p, 0< p~ ~, one may build an abstract<br />
Hardy space, HP(m), based on m and AAP(~). The real line<br />
may be viewed as a one parameter subgroup {et~t~<br />
of ~B and<br />
the action of ~ on ~B determined by this subgroup is ergodic<br />
with respect to m. It results from this that for each function<br />
6 H~(m), there is a null set E in ~B, that depends on ~,<br />
such that for x not in E, the function of t, ~Cx+et) ,<br />
belongs to H~(~). For each such x, then, we may extend the<br />
resulting function to the upper half-plane using the Poisson ker-<br />
I Im(z)<br />
nel: ~x(Z) = ~]R~(x§ where Pz(t) = ~ It_z[2. As a<br />
function of z, with x ~ E fixed, ~x(Z) is nice and smooth,<br />
for Im(z) ~ 0, but as a function of x, ~x(Z) need not be<br />
smooth in any sense. 0ur replacement for H ~ is an algebra H<br />
consisting of functions ~ 6 H~(m) such that ~x(Z) is continu-<br />
ous in x and z, or what comes to the same thing,<br />
H ~ {~ E H~(m)I ~* Piy(X) is continuous on ~Bx (0,~)] where<br />
* Piy(X) = ~(x+et)Piy(t)dt" The key feature of H that we<br />
will want to exploit is that we may view functions in H as ele-<br />
ments of both H~(m) and H~(~). Our analogue of H~+C, then,<br />
is H+ CAP(~). When represented on ~, H+ CAP(~) is norm-closed<br />
in L~(~) and when represented on ~B it is norm-closed in<br />
L~(m).<br />
The main result of Section 2 is an invertibility cri-<br />
terion for functions in H+ CAP(~). It is a direct analogue of<br />
the familiar criterion for H +C. In Section 3, we develop a<br />
type II index theory for Wiener-Hopf operators with symbols in<br />
H+ CAP~). Our main theorem here provides a sufficient condition<br />
for a Wiener-Hopf operator to be Fredholm, but unfortunately we<br />
are unable to decide if the condition is necessary. We conclude<br />
by exploring the difficulties that arise in one approach one<br />
might take to find a necessary condition. Here is where contact
652 Curto, Muhly and Xia<br />
with the value distribution theory of analytic almost periodic<br />
functions is made.<br />
In closing this introduction, we note that in addition<br />
to [5], we were motivated by the papers of Carey and Pincus, [3]<br />
and [4], where detailed investigations are made of Wiener-Hopf<br />
and related operators with almost periodic symbols.<br />
w THE FuNcTION ALGEBRA H+ CAP(E)<br />
We begin by recalling how to view ~B as an inverse<br />
limit of tori. Regard the discrete reals ~d' which is the dual<br />
of ~B as a vector space over ~, the rational numbers, and<br />
let A be a basis for ~d over ~, consisting of positive num-<br />
bers. Let I = [(y,n) [ y is a finite subset of A and n is a pos-<br />
itive integer] and direct I by the relation (y,n) < (~,m) if<br />
and only if y ~ ~ and n ~ m. For (y,n) 6 I, we write<br />
rmlYl m2Y2+ mkYk ]<br />
n~!Z= ~-~.~T n! "'" +--~! m i E Z], where y = [yi,Y2,..-,yk ].<br />
Then for (y,n)~< (o,m), is a subgroup of m! " It follows<br />
that we may view ~d as the direct limit li~ n~! ~. If T(y,n)<br />
I<br />
denotes the dual of n-~!~, then T(y,n) is a torus of dimension<br />
equal to the cardinality of y. Moreover, for (y,n) < (o,m),<br />
there is a natural projection from T(o,m) onto T(y,n), and<br />
this yields the representation of ~B as lim T(y,n).<br />
Corresponding to this representation of ~B we obtain<br />
the representation of CAP(E) as lim CAP(y,n) where CAP(y,n)<br />
is the set of all functions in CAP(E) whose Bohr-Fourier coef-<br />
ficients are supported in n~!~. The maximal ideal space of<br />
CAP(y,n) is T(y,n). We may also view AAP(~) as lim AAP(y,n)<br />
where AAP(y,n) = CAP(y,n) a H~(~). However, we want to refine<br />
this limit representation of AAP(~). For each fixed y,<br />
AAP(y,n) is naturally isomorphic to AAP(y,m) for any two posi-<br />
tive integers n and m. So, for the sake of discussion, we<br />
shall consider AAP(y,I) with y = [yl,Y2,...,yk]. A function<br />
f(t)<br />
Z ~(mlYl+---+mkYk)exp~i(mlYl+m2Y2+'''+mkYk)t] in CAP(y,i)<br />
is identified with the function ~ on ~k whose Fourier series
Curto, Muhly and Xia 653<br />
ml m2... mk<br />
is ~ ~(mlYl+'.-+mkYk)Zl z 2 z k Bohr called ~ the spatial<br />
extension of f and proved that the correspondence f ~ ~ ~ is<br />
an algebraic *-isomorphism between CAP(y,I) and c(~k). Under<br />
this representation, then, AAP(y,I) is identified with the al-<br />
gebra A(P) ~ C(~ k) where P = [(ml,''',mk) 6 Z k ] mlY I+ --. +<br />
mkY k 0] and A(P) = If 6 c(~k) [ ~(ml,..j,m k) = 0,(ml,...,m k)<br />
P]. Now P totally orders Z k and (~K,p) may be viewed as<br />
a dimension group in the sense of [7]. Therefore, as is well<br />
known (cf. [7]), we may realize (~k,p) as the inductive limit<br />
of a sequence [(zk,P(s where for each ~, P(~) is lin-<br />
k<br />
early isomorphic to ~+ via a linear transformation in SL(k,Z).<br />
This, in turn, shows that A(P) is the inductive limit of a se-<br />
quence of function algebras on Tk each term of which is iso-<br />
morphic to the polydisc algebra on Tk. The usefulness of this<br />
observation for our purposes is that the functions in each term<br />
of the sequence have smooth extensions into the polydisc. For<br />
technical reasons, it is useful to us to have a slightly differ-<br />
ent representation of A(P) as an inductive limit of algebras<br />
with this property.<br />
For each j, I~ j ~ k, let [~]s (resp.<br />
j ~ )<br />
[~s163<br />
be an increasing sequence (resp. decreasing sequence) of positive<br />
irrationalk numbers such that lim~ ~J yjs = .= lJm~ ~,J and set<br />
P~ = jNl[(ml,-.-,m<br />
k ) = E zk [ mlYl + ...+mjc~+ ..-+mkY k m 0, c js =<br />
~J and ~]. Then it is clear that P~ is isomorphic via a<br />
k<br />
linear transformation in SL(k,Z), to a subsemigroup of Z+. We<br />
now define the subsemigroup P](n) of m+ to be [l(mlYl+-..<br />
+mkYk) [ (ml,''.,m k) s P~] and we set AAP(P~(n)) = [f E CAP(m) [<br />
~(a) = 0, ~ ~ P~(n)]. Then it is clear that AAP(y,n) =<br />
li9 AAP(P~(n)).<br />
Viewing AAP(P~(n)) as a subset of c(~B), we denote<br />
the weak-* closure of AAP(P~(n)) in L~(m) by H~(P~(n)).<br />
From the discussion above, each algebra H (P (n)) is isomorphic<br />
to a subalgebra of H~(~ k) where k = card(y) and for y ~ ~,<br />
n ~ m, and ~ ~ k, H (P (n)) ~ H~(P (m)). The algebra we are
654 Curto, Muhly and Xia<br />
seeking is H, which is defined to be the norm closure of<br />
U H~(P (n)) in L (m), i.e., in an obvious sense,<br />
y,n,~<br />
H = Ig~ H~(P~(n)). As defined, H is a subalgebra of L~(m), in<br />
fact, a subalgebra of H~(m); we want to show that H may be<br />
viewed also as a subalgebra of H~(~).<br />
To this end, recall that we view ~ as acting on ~B<br />
via the dense one-parameter subgroup ~et]tE ~<br />
of ~B that con-<br />
sists of those characters of ~d that are continuous whe~ ~ is<br />
given the usual topology. A function ~ 6 C QR B) restricts to<br />
[et]tE ~ yielding an almost periodic function on ~; i.e., as a<br />
function of t, ~(e t) is almost periodic. And conversely, each<br />
almost periodic function on ~ may be viewed as ~(e t) for a<br />
unique ~ E c(~B). Given a function ~ E LP(m), I~ p~ ~, and a<br />
function f E LI(~), we define ~, f by the formula ~* f(x) =<br />
~(x+et)f(t)dt. Then ~* f E LP(m) and [I~* fIILP(m ) ~<br />
l[~l[ Lp (m)II flJ LI(~ ) 9<br />
For a real number ~, we write [~] for the<br />
character of ~B that X determines; so [X](x) is the value<br />
of the character at x s ~B. The Fourier series of ~* f is<br />
~(k)~(~) [k] where kE~ ~(~)[~] is the Fourier series of<br />
and ~ is the ordinary Fourier transform of f. In particular,<br />
Piy( I y is the Poisson kernel then<br />
if f(t) = t) = ~ t2+y 2<br />
2.1) ~,P. ~ Z $(X)e-YlxlExl.<br />
ly kE~<br />
The thing that prevents this series from converging on ~B is<br />
A<br />
that the support of ~ may have finite cluster points.<br />
LEMMA 2.2. Let ~ E H~(P~(n)). Then the series (2.1)<br />
converges uniformly and absolutely for x E~B and y > O. More-<br />
over, there i8 a C~-function h on ~k (0,~), where<br />
iwlt iwkt<br />
k = card y, such that ~* Piy(et) = h(e ,''',e ;y) for<br />
certain real numbers Wl,-'',w k.<br />
Proof. It suffices to consider P~(1) and, in view of<br />
the preceding discussion, it suffices to prove that if mlY I+ ---<br />
+ mku k E P~(1), then mlYl + ... +mku k ~ 6([ml[+[m2I+.-.+imk[)
Curto, Muhly and Xia 655<br />
0, then + 9 9 9<br />
for some 6 > 0. Fix j, i~ j ~ k. If mj mlY I<br />
+ mkYk = (mlYl +---+mj~j+--.+mkYk)+ mj(yj-~) ~ mj(yj-~), while<br />
J J" j<br />
if m. < 0 then mlY I+ -.. + mkY k ~ (yj-~)mj where ~s and ~<br />
j<br />
are the numbers used to define PY(1). If we take<br />
6 = infj min[yj-a~,~j ~-Yj]/k, then the desired inequality is sat-<br />
isfied.<br />
COROLLARY 2.3. If ~ s H, then for y > O, ~* P.<br />
my<br />
E C(]RB). Moreover, if F is defined on the upper half-plane by<br />
the formula F(t+iy) = ~o* Piy(et)" then F is bounded and analytic<br />
there, and IIFIIH~(:]R ) = IlcgliH~(m). Thus the map ~ ~ F is an<br />
isometric isDmorphism from H onto a norm-closed subalgebra of<br />
H~(~) consisting of functions F such that for every y > O,<br />
F(t+iy), as a function of t, lies in AAP(~).<br />
Proof. The first assertion is true for ~ E H~(P~(n)),<br />
by the lemma, and it is clear that the property is preserved when<br />
taking uniform limits. Therefore ~* P. 6 C(]R B) for all y > 0<br />
my<br />
and ~ E H. The fact that F is bounded and analytic on the<br />
upper half-plane is clear. The only thing that needs checking is<br />
the equality of I[F[IH~(~ ) and II~II<br />
H~(m)<br />
. By [12], for each<br />
E H, lim ~* P. = ~<br />
y~0+ my<br />
a.e. (m). Therefore IIqolI~<br />
lim II~0, Piyll ~ = lira IIF(-+iy)ll~ ~ IIFII~. on the other hand,<br />
y~0+ y~0+<br />
IF(t+iy)I = I~* Piy(t)I ~ II~* PiyiI~ ~ II~oli~ 9<br />
IIH| m)"<br />
Therefore liFil<br />
Henceforth we view functions in H either as functions<br />
-- II<br />
on IN B or as functions on I~ and we will not distinguish be-<br />
tween these two points of view notationally. Likewise, we will<br />
view H+ CAP(I~) as H+ C(I~B).<br />
We note in passing that when viewed as a space of func-<br />
tions on ]R B, H ~ ~ s H=(m) I ~* P. E C(]R B) for all y > 0],<br />
~Y<br />
while on JR, H ~ [~0 s H~(]R) I ~* P-<br />
~y<br />
s AAP(~) for all y > 0]. It<br />
would be very interesting to determine if equality holds in<br />
either case.<br />
The following lemma is due to Rudin [4].
656 Curto, Muhly and Xia<br />
LEMMA 2.4. Let<br />
Banach space X and let<br />
A on X such that<br />
a) AX m Y and AZ ~ Z<br />
b) sup AlllA <<br />
c) for each y E Y and<br />
such that IIAY - Yll ~ r<br />
Then Y+ Z is closed.<br />
Y and Z be closed subspaces of a<br />
be a family of bounded linear maps<br />
for all A s ~;<br />
and<br />
c > O, there is a A 6<br />
We wish to apply this lemma in the context where X is<br />
the closure of H+ CAP(R) in L~(~) or in L~(m), Y = CAP(~),<br />
Z = H, and ~ = lAy]y>0 where Ay~ = ~* Piy' ~ s L~(~) or<br />
L| Since A CAP(~) ~ CAP(~), and A H ~ AAP(~) ~ H, it is<br />
Y Y<br />
easy to see that condition a) of Lemma 2.4 is satisfied. Also,<br />
condition b) is satisfied since each A is a contraction on<br />
Y<br />
both L~(~) and L~(m). Finally, condition c) is satisfied be-<br />
cause translation is continuous on CAP(~) and [Piy]y>0 is an<br />
approximate identity for LI(~). We conclude, therefore, that<br />
H+ CAP(~) is closed in X and so when viewed either as a sub-<br />
space of L~(~) or of L~(m), H+ CAP(R) is closed. This leads<br />
us to<br />
THEOREM 2.5. When viewed either as a subspace of<br />
L~(~) or L~(m), H+ CAP(~) is a norm closed subalgebra.<br />
Proof. We need only prove that H+ CAP(F) is an alge-<br />
bra. For this, it is convenient to work on ~B, and then it<br />
suffices to show that if ~ s H~(P~(n)), for some y, n, and ~,<br />
and if w is a positive real number, then [-w]~ E H+ CAP(~).<br />
(Recall: [-w] is the character of ~B determined by -w.) We<br />
may suppose that w = N(Yl+Y2+...+yk) for some positive integer<br />
N, where y = (yl,...,yk). For if w is not of this form, then<br />
we may write w = N(Yl+..-+y k) - w 0 where w 0 z 0 and<br />
l<br />
[w0]H~(P~(n)) ~ H~(P~e(nl)) for some y1 ~ Y, n I ~ n, and an<br />
appropriate Za. It follows that [-w]~ E H+ CAP(~) if and only<br />
if [-N(Yl+Y2+..-+yk)]~ lies in H+ CAP(~). But by definition<br />
of P~(n), it is easy to see that [-N(Yl+Y2+...+yk)]~ differs<br />
by a trigonometric polynomial from a function in H~(P Y (n))
Curto, Muhly and Xia 657<br />
where ~I > ~" This completes the proof.<br />
THEOREM 2.6. For ~,~ E H+ CAP(~),<br />
~=~+LI(~ *Piy)(* * Piy)-(~)* Piyll~ = o;<br />
i.e., the Poisson kernel is asymptotically multiplicative on<br />
+ CAP(~).<br />
Proof. Since P. is multiplicative on H and since<br />
my<br />
polynomials are dense in CAP(~), it suffices to verify the<br />
assertion of the theorem when ~ = [-w], for any w > 0, and<br />
any<br />
E H~(P~(n)). Using equation (2.1), a simple calculation<br />
shows that (~, Piy)(r Piy)<br />
~(k) [e-Y(W+k)-e -y(w-k)] [k-w]<br />
O~k~w<br />
+ (~)* Piy' where the sum is, in fact, finite. The result fol-<br />
lows immediately.<br />
We come now to the statement of the main theorem of<br />
this section. Its proof is somewhat involved and will be spread<br />
through a series of lemmas. Each will be of interest in its own<br />
right, and the route is scenic.<br />
THEOREM 2.7. View H+ CAP(~) as a subalgebra of<br />
L~(~) and let ~ s H+ CAP(E). Then ~ is invertible in<br />
H+ CAP(E) if and only if there is an r > 0 and 6 > 0 such<br />
that<br />
(2.8) inf[I~, Piy(t)] I 0 < y < r, t s ~ 6.<br />
The necessity of this condition follows easily from<br />
Theorem 2.6. So we may suppose that (2.8) is satisfied and then<br />
show that ~ has an inverse in H+ CAP(~). For 6 as in (2.8),<br />
we can find w > 0 and ~0 E H~(P~(n)), for suitable Y, n,<br />
and s such that ll~(t)-e-Wt~0(t)II~ < -- 88 Then ~(t)<br />
e-iwt~0(t) is invertible in L~(~) and its inverse ~-i has<br />
norm at most 4__ 36" Consequently, I[l-~II~ ~ ~. so, if we can<br />
show that ~-i E H+ CAP(~), then ~ will be invertible in<br />
H But to do this, we need only show that ~I lies<br />
+ CAP(E). in<br />
H+ CAP(~). This is where the lemmas begin.<br />
Recall that the mean motion of a nonvanishing almost<br />
periodic function ~ is defined to be the limit
658 Curto, Muhly and Xia<br />
tim 1 [Arg ~(~)-Arg ~(~)]<br />
(~_~).~ ~-~<br />
whenever the limit exists. If ~ is invertible in CAP(R),<br />
then the limit exists and if the limit is w, then one may<br />
write<br />
~(t) = e i~t exp(~(t))<br />
for an almost periodic ~. (See [9] for a comprehensive treat-<br />
ment of the theory of mean motions of almost periodic functions.)<br />
LEMMA 2.9. If ~ E AAP(~) is invertible in CAP(R),<br />
then the mean motion of ~ is nonnegative and the following<br />
assertions are equivalent:<br />
i) ~ is invertible in AAP(~);<br />
ii) ~ = exp ~ for some ~ E AAP(~); and<br />
iii) the mean motion of ~ is O.<br />
Proof. Let w be the mean motion of ~. Then since<br />
w belongs to the spectrum of ~ [I0, Chapter VI], it follows<br />
that w ~ 0. Clearly ii) implies i) and iii). That i) implies<br />
ii) follows from the fact that the maximal ideal space of AAP(~)<br />
is homeomorphic to the cone over ~B ([2]) and therefore is con-<br />
tractible. On general principles, then, every invertible element<br />
of AAP(~) has a logarithm in AAP(~). Suppose, now, that w =0.<br />
As was noted before the statement of the lemma, we may write<br />
= exp 9 where 9 E CAP(~). We must show 9 6 AAP(~). If<br />
is a polynomial, then 9 is an absolutely convergent Bohr-<br />
Fourier series, and we may write 9 = 91+~ 2 where 91,92 E<br />
AAP(~). But then, on the one hand~ exp @2 ~ AAP(~) (unless 92<br />
is constant), while on the other, exp 92 = ~ exp(-@l s AAP(~).<br />
This shows that ~ E AAP(~). In the general case, we approximate<br />
by a sequence of polynomials that are invertible in AAP(~),<br />
[~n]n:l , and write ~n = exp ~n' ~n s AAP(~). Then<br />
Ill- exp(gn-~)II =<br />
II~-~nIIII~-III~, which implies that there is a<br />
sequence of integers [s I such that I It- ~n + 2nis o.<br />
it follows that 9 s AAP(~), completing the proof.<br />
It is instructive to make explicit the geometric con-<br />
tent of Lemma 2.9. The maximal ideal space of AAP(~) is the
Curto, Muhly and Xia 659<br />
so-called "big disc", ~B [0,~]/~B ~]. If ~ s AAP(~) is<br />
viewed as a function on ~B, and if y > 0 is finite, then the<br />
A<br />
value of the Gelfand transform ~ of ~ at (x,y) is @(x,y) =<br />
~* Piy(X) while the value of @ at [~B ~]] is ~B ~dm.<br />
(See [ii] for an expanded discussion of this.) If ~ is invert-<br />
A<br />
ible in CAP(E) then ~ is bounded away from 0 in some neigh-<br />
borhood of the boundary ~B ~0] of the big disc and the mean<br />
A<br />
motion "counts" the zeros of ~ in the interior of the big disc.<br />
The mean motion of ~ vanishes precisely when there are no zeros<br />
of $ in the interior of the big disc, i.e., precisely when<br />
is invertible in AAP(~).<br />
LEMMA 2.10. Let ~ s AAP(P~(n)) and suppose ~ is<br />
invertible in CAP(JR). Then ~ may be written as ~ = exp ~,<br />
for some ~ E AAP(P~(n)), if and only if the mean motion of<br />
is zero.<br />
Proof. The necessity is trivial. So suppose the mean<br />
motion of ~ is 0, and apply Lemma 2.9 to write ~ = exp<br />
with r E AAP(~). Without loss of generality, we may assume that<br />
~oB ~dm = 0. Then ~II~* PiyiI~ = o. it sufffces to show that<br />
for some y > 0, the spectrum of ~* Piy lies in P~(n). Since<br />
convolving with the Poisson kernel is a homomorphism of AAP(~)<br />
and of AAP(P~(n)), we find that ~* P.y = exp(@ I'PlY) for all<br />
y > 0 Choosing y so that Ill-exp(@l*Piy)II~<br />
9 ~, we find,<br />
upon expanding the logarithm in a power series, that @, P. =<br />
my<br />
log[l-(l-exp(@, Ply)) ] = log[l-(l-~* Ply)] lies in AAP(P~(n)).<br />
This completes the proof.<br />
LEMMA 2.11. Let ~ 6 H~(P~(n)) and suppose that there<br />
are positive numbers, r and 6, such that<br />
inf[l~, Piy(t)l t t E~, 0 < y < r] 6<br />
and that for each y E (0, r), ~, P. has mean motion zero~<br />
IV<br />
Then ~ is invertible in H~(P~(n))[<br />
Proof. For each y > 0, ~* P. E AAP(~) and so we<br />
my<br />
may view ~ as extended to the interior of the big disc,<br />
~B (0,~]/~B ~]. Our hypothesis says that this extension is
660 Curto, Muhly and Xia<br />
is bounded away from 0 by $ in the region ~Bx (0,r) and<br />
that it has no zeros in ~B (0 ~]/~R B x [~]. We would like to<br />
conclude that the reciprocal of the extension represents the in-<br />
verse of ~ in H~(P~(n)).~ To this end, fix Y0 E (0,r). Then<br />
, P. E AAP(P~(n)). Our hypotheses imply that ~, P. is in-<br />
lY 0 ~ lY 0<br />
vertible in CAP(~) and has mean motion zero. Therefore by Lem-<br />
ma 2.10, there is a ~Y0 ( AAP(P~(n))~ such that ~, P.my0 =<br />
exp ~Y0" For Y > Y0' we have ~* Ply = exp(@y 0* Pi(y-y0)) by<br />
the multiplicativity of convolving with the Poisson kernel and by<br />
the fact that the Poisson kernel forms a convolution semigroup.<br />
So, while @Y0 is not quite unique, it is possible to find a<br />
family [~Y]y>0 in AAP(P~(n)) such that ~* Piy = exp(~y) for<br />
all y > 0 and such that @YI* P'iy2 = @yl+y 2 for yl,y 2 > 0.<br />
It follows from this quite easily that the_family [exp(-@y)]y>O<br />
converges in the weak-, topology on H~(P~(n)), as y w 0+, to<br />
an inverse of ~.<br />
COROLLARY 2.12. Let ~ s H satisfy the conditions of<br />
Lemma 2.11. Then ~ is invertible in H.<br />
Proof. Choose u ~, n, and a<br />
Y<br />
~0 E H (P~(n)) such<br />
that II~-~oll ~ m w~' where ~6 is as in the statement of Lemma<br />
2.11. Then I~ 0* Piy(t)I ~ ~6 for y 6 (0,r) and ~* Piy and<br />
* Ply have the same mean motion, namely zero, when y is suf-<br />
~0<br />
-I<br />
ficiently small. By Lemma 2.11, ~0 has an inverse DO in<br />
H~(P~(n)) and of course<br />
-i<br />
so ~0 ' and hence ~,<br />
II~0111 ~ ~6 9 Therefore,<br />
is invertible in H.<br />
Ill-~lll ~ ! 3,<br />
COROLLARY 2.13. Let K be any one of the spaces<br />
y<br />
AAP(P~(n)), H (Ps or H, and let ~ E K satisfy<br />
inf[I~, Piy(t)I I t (~, y > 0] ~ 6,<br />
for some 6 > O. Then ~ is invertible in K.<br />
Proof. We need only check that ~* P. has mean<br />
my<br />
motion zero for each y > 0. However, the map Y * ~* PiY from<br />
(0,~) to CAP(~) is continuous and lim ~* P. is the constant<br />
y~ ly<br />
function whose value is -~ B ~dm, which can't be zero by our<br />
-m
Curto, Muhly and Xia 661<br />
hypotheses. It follows that the mean motion of ~* P. is zero<br />
my<br />
for each y ~ 0.<br />
We know that a function in the disc algebra A(~) or<br />
the algebra H~(T) is invertible if and only if its harmonic ex-<br />
tension to the open unit disc is bounded away from zero. This<br />
fact is, of course, an immediate consequence of the definitions<br />
of these algebras. An analogous fact is true for AAP(~) be-<br />
cause ~B [0,~]/~B {~] is its maximal ideal space. Corollary<br />
2.13 says that the algebras AAP(P~(n)), H~(P~(n)), and H<br />
share this property with AAP(~). However, the proof is rather<br />
roundabout.<br />
The following lemma is the analogue for H of<br />
the fact that if ~ E H~(~) and if (the harmonic extension of)<br />
satisfies [~(z)[ ~ 6 > 0 for r < Iz[ < I, then ~ = ~i~2<br />
where ~I is invertible in H~(~) and ~2 is a finite Blaschke<br />
product. It is the last step needed to prove Theorem 2.7.<br />
LEMMA 2.14. If ~ E H~(P~(n)) and if there are posi-<br />
tive numbers 6 and r such that<br />
inf{]~, Ply(t) [ It Em, o < y < r] 6,<br />
then ~ = ~i~2 where<br />
vergent; and<br />
motion.<br />
a) ~I E H and ~I is invertible in H;<br />
b) ~2 E AAP(~) and ~2 is invertible in CAP(~);<br />
c) the Bohr-Fourier series of ~2 is absolutely con-<br />
d) the spectrum of ~2 is bounded above by its mean<br />
Proof of Theorem 2.7<br />
If ~0+@0 E H+ CAP(~) satisfies the condition of the<br />
theorem, then with a slight decrease in 6 we can replace ~0+@0<br />
with ~+~ where ~ belongs to some H~(P~(n))_ and 40 is a<br />
trigonometric polynomial. But then, for a suitable w E ~, y~,<br />
n l , and ~, eiWt(~ct)+~(t)) lies in<br />
4<br />
H~(P~, (n ~ )) and satis-<br />
fies the estimate of Lemma 2.14. The conclusion of the lemma<br />
implies that eiWt(~(t)+@(t)) is invertible in H+ CAP(~) and<br />
so, therefore, ~+@ is invertib!e in H+ CAP(~). Finally, then,
662 Curto, Muhly and Xia<br />
we conclude that ~0+@0 is invertible in H+ CAP(~).<br />
Proof of Lemma 2.14<br />
Fix Y0 E (0,r). Then ~* P. E AAP(P~(n)) and Lemma<br />
mY0<br />
iwlt iwkt<br />
2.2 implies that ~ , P. (t) = h(e ,..-,e ) for certain<br />
mY0<br />
real numbers Wl,...,w k and a C~-function h on ~k. On the<br />
other hand, the hypothesis that 9" P. is invertible in<br />
IY 0<br />
CAP(~) implies that we may write 9" P. = e iwt exp @ where<br />
mY0<br />
E CAP(~) and w k 0 is the mean motion of ~* Ply 0. Evident-<br />
~k<br />
ly, ~ may also be viewed in terms of a C -function on<br />
and, in particular, we see that @ has an absolutely convergent<br />
Bohr-Fourier series. ~rite @ = @i+r where @i E AAP(~) and<br />
@2 has negative spectrum. Since e iwt exp(@2(t)) =<br />
iwt<br />
)(t) belongs to AAP(~), e exp(r<br />
exp(-@l(t))( 9. Piy 0<br />
may be written as<br />
= ~ akeY0keikt.<br />
Z akeikt where Z ..[akl < ~. Let<br />
0mk~w 0~kmw<br />
Then ~2 E AAP(~) and Z [akle y0~<br />
92(t)<br />
e wyO Z [axl<br />
O~k~w<br />
< ~. On the other hand, it is an easy calculation<br />
to show that for y E [0,Y0) ,<br />
-w(y-YO) iwt<br />
92, Piy(t) = e e ~ exp[@2* Pi(Y0-y )(t)].<br />
Therefore, ~2 is invertible in CA~(~) and each ~2" Piy'<br />
y E [0,Y0) , lies in AAP(~). Now define<br />
Since<br />
[9 * Piy(t)<br />
~l(t+iy ) = 1~ 2. Piy(t ) , t ( m,<br />
~exp[r Pi(y-y O)(t)]'<br />
0 ~ Y ~ Y 0<br />
t E JR, y0 ~ y < ~.<br />
92, FiYo(t ) ; e iwt exp(~2(t)) , it is easy to see that ~o 1<br />
is bounded and analytic on the open upper half-plane and %Ol<br />
-i<br />
boundary values ~2 Since 92 is invertible in CAP(JR),<br />
has<br />
we
Curto, Muhly and Xia 663<br />
have ~i = ~+~, where ~ E H and ~ E CAP(~). But<br />
~, Piy(t ) = (q~@~l) * Piy(t ) _ ~ * Piy(t ) = 91(t+iy ) _ (~ * Piy)(t)<br />
and so ~ has nonnegative spectrum; i.e., ~ E AAP(~). Thus we<br />
-I<br />
see that 91 = ~2 E H. By definition, there is a positive $i<br />
such that I(~ I) * Piy (t)I = 191 (t+iy)I ~ ~i for all t 6m<br />
and y 6 (0,Y0] ; also, by definition, for each y E (0,Y0] , the<br />
mean motion of ~I* Piy is zero. Thus, by Corollary 2.12, ~I<br />
is invertible in H. This completes the proof of the lemma.<br />
We note in passing that we are unable to decide if, in<br />
Lemma 2.14, the function 91 is invertible in H~(P~(n)), al-<br />
though we feel that it ought to be.<br />
w WIENER-HOPF OPERATORS<br />
For 9 E L~(~), we denote by W the Wiener-Hopf (or<br />
Toeplitz) operator on H2(~) defined by the formula<br />
W 9 = PM I H2(~) where M is the operator on L2(~) of multi-<br />
plication by ~, P is the orthogonal projection of L2(~) onto<br />
H2(~), and the vertical bar denotes restriction. If ~(t)= e iXt,<br />
for some X E~, we will write W~ for W 9. If ~ : L2(~) ~<br />
L2(~) denotes the Fourier-Plancherel transform, then we have<br />
W~ ~-I = T~ where TX is the compression of translation (to<br />
the right) by ~ to L2([0,~)); i.e., TX= I[0,~)Lx I L2([0,~))<br />
where I[0 ~ ) denotes the indicator function of [0,~) and<br />
(L~)(x) = ~(x-X). The norm-closed algebra generated by<br />
[L I ~ E CAP(~)] will be denoted by ~. It is a C*-algebra and<br />
is generated by [Wx I ~ E~]. The norm-closed algebra generated<br />
by ~W i 9 E H+ CAP(~)] will be denoted by ~. Of course,<br />
is not a C*-algebra. In [5] and [15] it is discussed how to<br />
represent ~ in a certain Ii factor ~ through a certain<br />
representation p. It is shown that for ~ 6 CAP(~), p(W ) is<br />
Fredholm relative to ~ if and only if ~ is invertible in<br />
CAP(~) and, in this case, the Breuer-Fredholm index of p(W )<br />
is the negative of the mean motion of ~. Our objective is to<br />
extend these results, as far as we can, to the algebra ~.<br />
Let ~ denote the commutator ideal of ~. Then a<br />
moment's reflection reveals that ~ is the norm-closed ideal
664 Curto, Muhly and Xia<br />
generated by the operators of the form W~W_x- W_xW~ =<br />
= W_~W~(WkW_~-l) where k ~ 0 and ~ E H. Moreover, [W +K I<br />
6 H+ CAP(~), K s ~] is dense in ~, and IIw~+~li ~ [I~II~-<br />
LEMMA 3.1. The map ~ W +~ is an isometric isomorph-<br />
ism from H§ CAP(~) onto ~/~. In particular, ~ = [W +K I<br />
E H+ CAP(~), K E ~].<br />
Proof. We need only show that IIW~+KI[ ~ I[~[[~ for any<br />
K E ~; and, in fact, we may assume that<br />
K = ~j kH W jk(WxjW-xj-I)W~jk' where ~jk and ~jk~ have the form<br />
e-ikt@(t), k > 0, @ E H. For such a K, ~N > 0 such that<br />
KW k = 0, if k ~ N. On the other hand, for any Wiener-Hopf<br />
operator W~. and any k > 0 we have W~. = W~W@Wk,~ and conse-<br />
quently k88 ) = (W,g ,n)H2(~ ) . Since l[~[[~ = Ilw~l[,<br />
as is well known, given r > 0, we may choose unit vectors<br />
and ~ in H2(~) such that [I~][~ ~ I(w~g,~)l + ~. But from what<br />
was just noted, [(W g,~)I = liml((W +K)Wkg,Wk~) I ~ HW +KI[. This<br />
completes the proof.<br />
In [5], the von Neumann algebra in which ~ is repre-<br />
sented is essentially the group-measure von Neumann algebra ob-<br />
tained from letting the discrete reals ~d act on ~. The Hil-<br />
bert space is L2([0,~)) | L2(~ d) and the von Neumann algebra is<br />
generated by the operators (1[0,~) | I)(M | I)(1[0,~ ) | I),<br />
s L~(~), and (I[0,~)|174174 , k s where<br />
TX ~ denotes translation by k on L2(~d ). For our purposes, it<br />
is more congenial to take the Fourier transform of this algbra.<br />
So, our Hilbert space will be ~ = H2(~) | L2(m) (recall m is<br />
Haar measure on ~B) and for ~ 6 L~(~ B) we define ~ =~ I~<br />
2 2 ~<br />
where ~ = P| I and M is defined on L (~) | L (m) "="<br />
L2(~x~B,p x m) (~ is Lebesgue measure on ~) by the formula<br />
(~)(t,x) = ~(x+et)~(t,x) , ~ E L2(~) | L2(m). The von Neumann<br />
algebra ~ generated by [W~I ~ ( c(~B)] is unitarily equiva-<br />
lent, via the Fourier transform, to the yon Neumann algebra de-<br />
scribed above that was first used by Coburn, Douglas, Schaeffer,<br />
and Singer; for the proof of this see [15, Section 3]. The alge-<br />
bra ~ is a II~ factor and we denote its canonical trace by 7.
Curto, Muhly and Xia 665<br />
For T = ~ ~ W E ~, write p(T) = S ~ ~ Then as is shown<br />
j k ~jk j k ~jk<br />
in [5] and [15], 9 extends to a faithful *-representation of<br />
onto ~ which is defined to be the C*-algebra generated by<br />
[W% I ~ E c(~B)]. Our objective is to extend p to an isometric<br />
representation of 9 in ~.<br />
To this end, we denote the operator of convolving with<br />
the Poisson kernel on H2(m) and on ~ by Ply" Thus on H2~R),<br />
(Piy[)(x) = f~ [(x-t)Piy(t)dt , while on ~, (Piy~)(t,x) =<br />
f~ ~(x-s,X+es)Piy(S)ds. Then on ~, Ply E ~, and in Lemma 2.1<br />
of [15] it is shown that p may be extended to a faithful repre-<br />
sentation Pl of the C*-algebra ~i generated by ~ and<br />
[Piy ] 9 onto the C*-algebra ~i generated by ~ and<br />
Y<br />
[Piy]y9 in ~.<br />
LEMMA 3.2. If ~ E CAP(m), then [Piy]~9 asymptoti-<br />
cally commutes with W and with %; i.e., lim~P. W-W P II<br />
y~0 ly ~ ~ iy<br />
= 0 and<br />
limIIP. W -U P ~ = 0<br />
y~0 ly ~ ~ iy "<br />
Proof. It suffices to check the case when ~ is a<br />
pure exponential. But then, the result is clear since on H2(m),<br />
PiyWk = e-[k]YwkPiy, while on ~, PiyW[k] = e-lk[Yw[k]Piy.<br />
LEMMA 3.3. The map which sends an operator of the form<br />
Z Hwr<br />
j k jk<br />
to Z ~ ~@ ,<br />
j k jk<br />
~ ~ H+CAP(m)<br />
@~k<br />
extends to an isometric<br />
representation, denoted 92 , of onto a subalgebra<br />
of ~.<br />
Proof. Lemma 3.2 implies that ~01im[IPiyW~-W'k~*zi~ )PiYI~I<br />
N Y<br />
= 0 for all ~ E CAP(m) and likewise for W . On the other<br />
hand, for @ E H, we have PiyW@ = W(@,Piy)Piy and a similar<br />
equation holds for ~@. Now fix a finite family [~jk] g<br />
H+ CAP(m), let S = Z H W@ and let T = Z H ~ . For y 9 0,<br />
j k jk j k @jk<br />
let Sy denote the same combination as S, but with @jk re-<br />
placed by ~jk* Piy' and define Ty similarly. By our initial<br />
observations, we may write<br />
Pi S*SPi = P" S~ S 2 Pi + A<br />
y y my y y y y
666 Curto, Muhly and Xia<br />
and<br />
PiyT*TPiy = P. T~ T 2 P. + B<br />
zy y y my y<br />
where Ayll = ByII = 0. since P s~ S~ P. E ~i and<br />
iy ~y ~y my<br />
Pl (PiyS*2yS2yPiy) = P'myT~yT~Ly P'my (cf. [15]), we see that IISIl 2 =<br />
Hs*sII = s*sP. IL mr- llP. T*TP. LI = L,IT*Tli = llTl 2. This<br />
y~0 my ly ly ly<br />
completes the proof.<br />
Let ~ be the norm-closed ideal in ~ generated by<br />
the trace class elements of ~; i.e., ~ is the closure of<br />
IT E ~I ~((T*T) I/2) < ~]. Then ~ is the ideal of generalized<br />
compact operators in ~. In [5], it was shown that p-l(~N p(~))<br />
is the commutator ideal of ~. Our next objective is to obtain a<br />
similar result for $ and P2"<br />
LEMMA 3.4. r = p~l(~n p2(~)).<br />
Proof. Recall that { is the norm-closed ideal gener-<br />
ated by operators of the form W_xW~(WxW_k-I) where ~ ~ 0 and<br />
E H. Since P2(WkW_k-I) = ~[x]W~x]-I is a finite rank projec-<br />
tion in ~, it is clear that ~ m p~l(~ p2(Z)) " Now by Lemma<br />
3.1, every element of 9 may be written as W$+K where K E {.<br />
So, to prove the reverse inclusion, we need only show that if<br />
P2(W~) E R, then ~ = 0. Define [Ut]ts ~ on ~ by the formula<br />
(Ut~)(s,x) = ~(s-t,x) and set ~t(B) = utlBut , B s m. Then<br />
[~t]tE~ is a o-weakly continuous, one-parameter group of *-<br />
automorphisms of 9 that carries ~ into ~, because [Ut]tE ~<br />
~. Moreover, for A s ~, the map t * ~t(A) from ~ to ~<br />
is norm continuous. [To see this, let r > 0 be given and write<br />
e<br />
A = AI+A 2 where A I is trace class and A 2 E ~ with IIA2II < ~.<br />
Then since [~t]t~ ~ is ~-weakly continuous and ~ is the dual<br />
of LI(~,~) (the completion of the trace class operators in the<br />
trace norm), it follows that it is possible to choose ~ 9 0 so<br />
r Vt It] < ~ and that when this is done,<br />
that II~t(AI)-AI[II < y ,<br />
e e<br />
then II~t(A)-AII ~ IIaI(AI)-AIILI + Ilat(A2)-A211 < y+ 2y = r So, if<br />
A s ~, and f s LI(~), then ~ ~t(A)f(t)dt s ~. If
Curto, Muhly and Xia 667<br />
~@ : P2(W@) E ~, then since ~ ~t(~r = W@*Piy' as an<br />
easy calculation shows, we see that W$,p. s ~ for all y > 0.<br />
ly<br />
Since, however, @* Ply E CAP(~) we conclude from [5] or [15]<br />
that @* Ply = 0 for all y > 0. This implies that @ : 0 and<br />
completes the proof.<br />
We come now to the main theorem of this section, a suf-<br />
ficient condition for P2(W@) to be Fredholm in ~,<br />
s H+ CAP(~), together with a calculation of the index.<br />
THEOREM 3.5. If @ E H+ CAP(~) is invertible in<br />
H+ CAP(~), then P2(W~) is Fredholm in ~. Moreover, there is<br />
an r > 0 such that ~, P. is invertible in CAP(~) for all<br />
ly<br />
y E (O,r) and the index of P2(W~) is minus the mean motion of<br />
~* Ply for any y E (O,r).<br />
Proof. By Lemmas 3.1 and 3.4, P2(W~) is Fredholm in<br />
for every invertible ~ E H+ CAP(~). To calculate the index<br />
of P2(W@), we may assume that r = e-iWt~(t) where w ~ 0<br />
y<br />
and ~ E H (Pz(n)) for suitable y, n, and s By Theorem 2.7<br />
and Lemma 2.14, we may write ~ = ~i~2 where ~I is invertible<br />
in H and ~2 s AAP(~) is invertible in CAP(~). Since ~I is<br />
invertible in H, P2~W i) is invertible on ~ with inverse<br />
P2(W _i ). Also, since P2(W@ ) = P2 (W -iwt )P2(W ]) we conclude<br />
~1 e ~2 "<br />
that the index of P2(W~) is the index of P2 (W -iwt )" Since<br />
e ~2<br />
~2 E CAP(~), the index of P2 (W -iwt ) = p(W -lwt ) is minus<br />
e ~2 e ~2<br />
the mean motion of e-imt~2 by [5]. To see that this mean mo-<br />
tion coincides with that of ~* Piy for each y E (0,r), for a<br />
certain r > 0, simply use Theorem 2.7, the invertibility of ~I'<br />
and Theorem 2.6. This completes the proof.<br />
Conspicuously absent from the statement of Theorem 3.5<br />
is a necessary condition that P2(W@) be Fredholm. One would<br />
expect, of course, that if P2(W~) is Fredholm in ~, then<br />
is invertible in H+ CAP(~). However, while this appears plaus-<br />
ible at first glance, we are unable to find a proof. In fact,<br />
upon deeper reflection, we are not at all certain that it is
668 Curto, Muhly and Xia<br />
true. In the following discussion, we shall indicate briefly how<br />
far we are able to go in proving the assertion that ~ is in-<br />
vertible in H+ CAP(R) whenever Pi(W~) is Fredholm and we<br />
shall identify the obstruction that prevents us from completing<br />
the proof. First, we recall the Fredholm theory of Toeplitz<br />
operators with symbols in H~+C. Let ~ s H ~ and assume that<br />
the Toeplitz operator on Hi(~) determined by ~, T~, is Fred-<br />
holm. To prove that ~ is invertible in H~+C, one first shows<br />
that ~ is invertible in L ~, and then, using the fact that<br />
dim(H 2@~H 2) = dim(H 2GT H 2) is finite, one shows that ~ = ~<br />
where ~ is a finite Blaschke product and @ is invertible in<br />
H . The first step can be carried out in our setting, the second<br />
is where the difficulties lie.<br />
LEMMA 3.6. Let ~ E H and suppose that Pi(W~) is<br />
Fredholm in ~. Then ~ is invert~ble in L~(m); i.e., there<br />
is a 6 > 0 such that l~I ~ 6 a.e.<br />
Proof. If W~ = Pi(W~) is Fredholm in ~, then so is<br />
I~ ~ If 6 is the distance from 0 to the spectrum<br />
12 = (~)*~ .<br />
of ~ + ~ in ~/~, then for each ~ s [0,6), W - ~I is<br />
I 12 I 12<br />
also Fredholm. Consequently, for each ~ E [0,6), there is a<br />
finite projection E E ~ and a positive number ~(~) such that<br />
i i2<br />
- ~I ~ ~(~)(I-E ) Now choose a sequence {~n]::! of pos-<br />
~B<br />
itive continuous functions on such that ~n(x)dm(x) = i<br />
for all n and limI@Ii,~n = I@I 2 a.e.m. (The choice of a<br />
sequence with this property is possible since I$I 2 has only<br />
countably many nonzero Bohr-Fourier coefficients and so may be<br />
viewed as a function on a separable quotient of ~B.) For fixed<br />
n, the ergodic theorem implies that lim i ~T l$12(x_et)~t(0+et)dt<br />
T~ ~ -T<br />
= ~]RBl@li(x-Y)~n(Y)dm = I@l 2. ~n(X) a.e. (m) on the one hand,<br />
while on the other, lim ~-~ l T T at(g I 2)~n(0+et)dt = ~ "<br />
It follows that ~ = ~ - ~! ~ ~(~)(I-E~) where
Curto, Muhly and Xia 669<br />
I T<br />
E no is defined to be t(E~ Tho operators<br />
Ena are not necessarily projections, but since each ~n is posi-<br />
tive, E n~ ~ 0 and Fatou's lemma implies that ~(E~) ~ ~(Ea);<br />
i.e., each E n is trace class. Thus we have shown that<br />
is Fredholm for each n and for each ~ s [0,6).<br />
Since I~12" ~n is continuous on ~B, we conclude from [5] that<br />
I~12. ~n(X) ~ ~ for all x s all n, and all ~ s [0,6).<br />
Since !imI~I2* ~n = I~I2 a.e. m, we ultimately find that<br />
n4~<br />
I~I 2 m 5 a.e. m, and this completes the proof.<br />
COROLLARY 3.7. If ~ s H is such that P2(W@) is<br />
Fredholm in ~, then P2(W~) is bounded below on ~.<br />
Proof. By Lemma 3.6, there is a 5 > 0 such that<br />
I~I a 6 a.e. m. But by definition of ~@ = P2(W~), W@ is a<br />
direct integral of Wiener-Hopf operators with symbols ~(x+et)<br />
in H~(~). Since almost everyone of the integrands is bounded<br />
below by 5, the same is true of W~.<br />
Passing now to the second step, and the obstruction to<br />
finding a converse to Theorem 3.5, we need to investigate how a<br />
function @ in H+ CAP(~) can fail to be invertible there. For<br />
our purposes, it suffices to assume that ~ is in H. Let<br />
Y = {y > 0 I ~* P- vanishes somewhere on ~]. In view of Theo-<br />
ly<br />
rem 2.6, there are three possibilities to consider:<br />
3.8) Y A (0,r) is not dense in (0,r), for any r > 0,<br />
and there is a sequence ~Yn]n= I with Yn %0 such<br />
that each @, Piy n has zeros on ~;<br />
3.9) Y ~ (0,r) is dense in (0,r), for some r > 0; and<br />
3.10) Y ~ (0,r) is empty for some r > 0 and there is a<br />
sequence<br />
{Yn]~=l~ with Yn ~ 0 such that<br />
infn tE~inf It* PiYnI(t) = 0.<br />
In analogy with what happens on the disc, (3.8) and ~.9) corre-<br />
spond to the case where the inner factor of @ contains an infi-<br />
nite Blaschke product while (3.10) corresponds to the case where
670 Curto, Muhly and Xia<br />
the inner factor of $ is the product of a finite Blaschke prod-<br />
uct and a singular inner function. Our objective is to show,<br />
among<br />
other things, that if $ E H~(P~(n)) is such that P2(W~)<br />
is Fredholm, then (3.8) cannot happen. Our analysis employs<br />
classic tools in the value distribution theory of analytic almost<br />
periodic functions.<br />
THEOREM 3.11. Let ~ be a function in H and view<br />
as a function on ~B. Then inI~I ~ Ll(m). If y ~ O, and if<br />
89 is defined to be ~BlnI~* PiyIdm, then ~ is convex.<br />
For each YO 9 0 such that ~(yo ) has a derivative, ~, PiYo,<br />
viewed now as a function on ~, has a mean motion and its value<br />
is -~(y0 )._ Moreover, the functions ~* Ply are free of zeros<br />
on ~ for all y in the interval (a,~) if and only if ~ is<br />
linear on (~,~).<br />
Proof. The only new aspect of this theorem is the<br />
integrability of InI@ I for ~ s H. The rest follows easily<br />
from Theorems 7 and 8 of [9], once one notes that when y 9 0,<br />
so ~, Ply s AAP(~), ~]R BlnI~* PiY Idm = T~limlf~- InI~* PiY I(t)dt<br />
[I]. To show that inI$ I s Ll(m) for ~ E H, it suffices to<br />
assume that @ s H=(P~(n)) for some y, s and n. In this<br />
case, @ may be viewed as a function on a certain torus ~k,<br />
and to be contained in H~(~k). The integral, then, of InI@I<br />
over ~B coincides with the integral of inI@ I over ~k. Since<br />
this is well known to be finite [14], the proof is complete.<br />
Alternatively, the Integrability of inI~ I for ~ E H~(P~(n))<br />
follows from a corollary on page 9 of [8] and the fact that<br />
has a first nonvanishing Bohr-Fourier coefficient.<br />
We note in passing that as a result of Theorem 3.11,<br />
every function @ E H has an inner-outer factorization analogous<br />
to the one for functions on the unit disc. It would be interest-<br />
ing to know whether these factors belong to H. Presumably they<br />
don't.<br />
The function ~@ is commonly called the Jensen func-<br />
tion of ~. At first glance, one might expect the possibilities<br />
for it to be more or less arbitrary. However, this is not
Curto, Muhly and Xia 671<br />
entirely so, as the following corollary illustrates.<br />
y<br />
COROLLARY 3.12. If @ s H (P~(n)), with Jensen func-<br />
tion ~, then in each interval (~,~), a ~ O, there are only<br />
finitely many subintervals on which 89 is linear. Moreover, on<br />
any one of these, -~' lies in P~(n).<br />
Y<br />
Proof. The first assertion is essentially contained in<br />
Theorem 22 of [9]. The second follows from the fact that the<br />
spectrum of $ has only finitely many points in any interval of<br />
the form [0,r), r < ~, therefore the greatest lower bound of<br />
the spectrum of $ belongs to the spectrum of @. The same is<br />
true for $* Piy' for any y > 0, and so the result follows<br />
from Theorem 3.11 and Lemma 4 on page 271 of [i0].<br />
LEMMA 3.13. Let $ E H and let E be the projection<br />
onto ~ @ P2(W@)~. Also, for y > O, let Ey be the projection<br />
onto ~ @ P2(W~.Piy)~. Then ~(Ey) ~ T(E).<br />
Proof. Since the operator P. on ~ is Hermitian<br />
ly<br />
and has zero kernel, it suffices to prove P. E ~ ~ E~. This<br />
my y<br />
implies that E is equivalent in ~ to a subprojection of E,<br />
Y<br />
and so T(E ) ~ T(E). But the containment relation follows from<br />
N<br />
the fact that for ~ s Ey~ and ~ s F~g, (Piy~,W@~) =<br />
(g,W( Piyn) = 0<br />
@,ply)<br />
With the preliminaries attended to, we come now to our<br />
final result relating the value distribution theory of $ to the<br />
Fredholmness of P2(Wo).<br />
THEOREM 3.14. Let ~ (H~(P~(n)) and suppose P2(W0)<br />
is Fredholm in ~. Then ~ has only finitely many intervals of<br />
linearity in (0, ~) and, in particular, ~ does not satisfy<br />
(3.8). If ~ takes on only countably many values on (O,r),<br />
for some r 9 O, then ~ takes on only finitely many values in<br />
(0,r) and so, if ~ is not invertible in H+ CAP(~), ~ satis-<br />
fies (3.10).<br />
Proof. Since P2(W$) is Fredholm, the projection E<br />
of ~ onto ~ O P2(W~)~ is finite. Therefore, by Lemma 3.13,<br />
for each y 9 0, the projection E onto ~ ~ P2(W],p~ )~ is<br />
y<br />
ly<br />
also finite with ~(Ey) ~ ~(E). Suppose y belongs to an
672 Curto, Muhly and Xia<br />
interval of linearity of 89 Then by Theorem 8 of [9], @* Piy<br />
is invertible in CAP(m). Consequently, P2(W@,Piy) is Fredholm<br />
with index equal to minus the mean motion of 4' Piy" By Theorem<br />
3.11, then, the index of P2(W~,p ) is ~(y). But since<br />
iy<br />
@*Piy is analytic, the index of P2(W@,Piy) is ~(Ey) and we<br />
conclude that ~(y) ~ T(E) < ~ by Lemma 3.13. Since ~ is<br />
convex by Theorem 3.11, ~(y) exists for all but countably many<br />
values of y and is increasing. Also, by Corollary 3.12, @~(y)<br />
E P~(n) for y in any interval of linearity. Since P~(n) has<br />
only finitely many points in any finite interval, we conclude<br />
that 89 can have only finitely many intervals of linearity. Of<br />
course the fact that (3.8) is ruled out for @ now follows from<br />
Theorem 3.11, as do the other assertions of the theorem.<br />
i.<br />
2o<br />
3.<br />
.<br />
.<br />
.<br />
7.<br />
.<br />
9.<br />
I0.<br />
REFERENCES<br />
R. Arens, The boundary integral of logI~ I for gener-<br />
alized analytic functions, Trans. Amer. Math. Soc. 86<br />
(1957), 57-69.<br />
and I. Singer, Generalized analytic func-<br />
tions, Trans. Amer. Math. Soc. 81(1957), 379-393.<br />
R. Carey and J. Pincus, Mosaics, principal functions,<br />
and mean motion on yon Neumann algebras, Acta Math.<br />
138(1977), 153-218.<br />
and , Mean motion, principal func-<br />
tions, and zeros of Dirichlet series, Integral Equa-<br />
tions and Operator Theory 2(1979), 484-502.<br />
L. Coburn, R.G. Douglas, D. Schaeffer, and I. Singer,<br />
C*-algebras of operators on a half-space, II. Index<br />
theory, Inst. Hautes ~tudes, Sci. Publ. Math. No. 40<br />
(1971), 64-81.<br />
R.G. Douglas, Toeplitz and Wiener-Hopf operators in<br />
H~+C, Bull. Amer. Math. Soc. 74(1968), 895-899.<br />
E. Effros, Dimensions and C*-alsebras, CBMS Regional<br />
Conference Series in Mathematics, Amer. Math. Soc.,<br />
Providence, R.I., 1981.<br />
H. Helson, Ana!yticity on compact abelian groups, Alge-<br />
bras in Analysis, Academic Press, N.Y., 1975.<br />
B. Jessen and H. Tornehave, Mean motions and zeros of<br />
almost periodic functions, Acta Math. 77(<strong>194</strong>5), 137-279.<br />
B. Ja. Levin, Distribution of Zeros of Entire Functions~
Curto, Muhly and Xia 673<br />
ii.<br />
12.<br />
13.<br />
14.<br />
15.<br />
Translations of Mathematical Monographs, Vol. 5, Ameri-<br />
can Mathematical Society, Providence, R.!., 1964.<br />
P. Muhly, Function algebras and flows II, Arkiv f~r<br />
Mat. 11(1973), 203-213.<br />
, Function algebras and flows III, Math. Z.<br />
136(1974), 253-260.<br />
W. Rudin, Function Theory i_~n Polydiscs, Benjamin, New<br />
York, 1969.<br />
, Spaces of type H~+C, Ann. Inst. Fourier<br />
(Grenoble) 25(1975), 99-125.<br />
J. Xia, Wiener-Hopf operators with piecewise continuous<br />
almost periodic symbols, to appear.<br />
Department 0f Mathsmatics<br />
I01 MacLean Hall<br />
The University of Iowa<br />
lowa City, Iowa 52242<br />
Submitted: January 2, 1985
Integral Equations<br />
and Operator Theory<br />
Vol. 8 (1985)<br />
BASE CHANGE AND THE FREDHOLM INDEX<br />
Mihai Ptrtinar<br />
INTRODUCTION<br />
0378-620X/85/050674-1951.50+0.20/0<br />
9 1985 Birkh~user Verlag, Basel<br />
The present paper deals with a relationship between 2.L.Taylor's functional cal-<br />
culus and the multidimensional Fredholm index, both referring to commutative n-tuples<br />
of operators. We develop an enlarged frame, with the purpose of making the Fredholm<br />
objects more flexibile under analytic transformations. In particular two base change<br />
results, Theorem 3.1 and Theorem 4.2, are obtained in this paper.<br />
Let us recall for the beginning some terminology and facts from multidimen-<br />
sional spectral theory. Let H be a complex Hilbert space and let T = (T1,... ,T n) be a<br />
commutative n-tuple of linear bounded operators on H. 3.L.Taylor defined in [19] the<br />
joint spectrum Sp(T,H) of the n-tuple T on H as the subset of those points ), r C n, with<br />
the property that the Koszul's complex K(T- X, H) is not exact. In other words,<br />
X I~ Sp(T, H) iff the condition<br />
(1) Hq(T- X, H) = O, qZO,<br />
is fulfilled, where we have denoted by H.(T- , H) the homology spaces of the complex<br />
K(T - ~, H). This joint spectrum has many of the expected properties, when compared<br />
with the spectrum of a single linear operator, among which we recall only Taylor's theo-<br />
rem [20, Theorem 4.3] about the existence of a continuous functional calculus f ~+ I(T),<br />
with analytic functions f defined in neighbourhoods of Sp(T,H).<br />
Motivated by Taylor's ideas, several authors developed (in convergent direc-<br />
tions) the Fredholm theory for commutative n-tuples of operators. Thus it is unanimously<br />
accepted that the point X e C n doesn't belong to the essential joint spectrum SPe(T, H)<br />
if the condition<br />
(2) hq=dim Hq(T-)t,H)< ~, q_> O,<br />
holds true. Then the Fredholm index of T is the integer
Putinar 675<br />
n<br />
ind(T)= ~. (-l)qh .<br />
q=0 q<br />
These definitions extend the corresponding notions for a single linear operator<br />
and the expected stability results hold (see [22, Chap.HI, w 7]). However, in the multi-<br />
dimensiona! case new phenomena appear (see for instance [6]) and the aim of this paper<br />
is to investigate one of them.<br />
More precisely, the computation of the index, or of some finer invariants as the<br />
numbers h of the m-tuple f(T), where f:U + C m is an analytic map defined on an<br />
q<br />
open neighbourhood U of Sp(T, H), is the motivation of this paper. It turns out that,<br />
instead of working on C m with the above usual definitions it is more convenient to lift<br />
the objects to U, to investigate there a Tor-intersection condition and to compute the<br />
corresponding intersection number between the analytic fibre f-l(0) and the n-tuple T.<br />
Let us explain this procedure in more detail.<br />
The Fredholm condition (2) can be interpreted as follows. Let L(H) denote the<br />
algebra of bounded linear operators on H, with its two sided closed ideal K(H) of com-<br />
pact operators. Then condition (2) is equivalent by [5, w 3] with<br />
(2)' Hq(T - X, L(H)IK(H)) = O, q >_ O,<br />
where the operators T i -X i act by left multiplication on the CaIkin algebra L(H)/K(H).<br />
For another interpretation consider O(C n) the algebra of holomorphic functions<br />
on C n and let m X be its maximal ideal associated to k :<br />
m X : {f c o(cn)[ fG1) = 0}.<br />
The quotient o(cn)-module o(cn)/m k admits a canonical free resolution given by the<br />
Koszul complex associated to the system z -X, where z = (Zl,... ,z n) stands for the co-<br />
ordinate n-tuple in C n :<br />
K.(z - ~, o(cn)) --* o(cn)Imx ~ 0.<br />
Note that the quotient algebra o(cn)/mx is canonically isomorphic with C. Then<br />
regarding the n-tuple T as the topological representation<br />
0 : o(cn) -* L(H)/K(H)<br />
which associates T i to z i, the condition (2)' can be reformulated independently of the<br />
coordinates and of the Koszul resolution, as follows:<br />
(2)" Tot O(cn) (o(cn)/mk, L(H)/K(H)) = O, q > O.<br />
q
676 Putinar<br />
This equivalent definition for the Fredholm property of the n-tuple T - X can be<br />
easily extended to more general objects, a fact which will turn out to be extremely use-<br />
ful.<br />
Thus~ let F be an analytic coherent sheaf on C n, which admits a finite globally<br />
free resolution with finite type O-modules, like the sheaf O/m k above. For explana-<br />
tions concerning such sheaves, see w I. Throughout this paper O denotes the sheaf of<br />
holomorphic functions on C n, Then the vanishing condition<br />
(3) Tor O(cn) (F(Cn), L(H)IK(H)) = 0, q > 0,<br />
q<br />
will be interpreted as a generalized Fredholm "incidence" relation between the sheaf F<br />
and the n-tuple T. A natural K-theoretic difference construction will provide under the<br />
assumption (3) an integer<br />
which generalizes the Fredholm index.<br />
ind O(cn) (F, L(H)/K(H)) ~ Z,<br />
Let us illustrate these notions with two examples.<br />
EXAMPLE 1. Consider the n-tuple T and an integer k, I < k < n. Let us denote<br />
by / the ideal of the sheaf O generated by Zl,... ,z k. Then the sheaf F = 0/I, which is<br />
supported on the complex (n-k)-dimensional linear subspace {z[ = z 2 ..... z k = 0} of<br />
C n, has a finite free resolution given by the partial Koszul complex K.(Zl,... ,Zk; O).<br />
In this case condition (3) is equivalent with the Fredholmness of the k-tuple (TI,... ,T k)<br />
and<br />
ind O(cn) (O/(Zl, .... Zk) , L(H)/K(H)) = ind (T 1 .... ,Tk).<br />
Remark that the n-tuple T is Fredholm if (3) holds, but the n-dimensional index ind(T)<br />
vanishes when k < n.<br />
EXAMPLE 2. Let f be a non-constant analytic function of C n and let F be the<br />
f<br />
quotient sheaf O/(f). It has a finite free resolution, 0 -+ O --* O --+ F --~ 0, and it<br />
is supported on the hypersurface { f = 0}. Then (3) means exactly that the operator f(T)<br />
is Fredholm, and in this case<br />
ind O(cn) (O/(f), L(H)/K(H)) = ind I(T).<br />
Other examples will be presented in w 3 and w 5,<br />
The present paper is an amplified version of an earlier paper, with the same<br />
title, circulated as INCREST preprint (April, 1981). Meanwhile two recent complemen-
Putinar 677<br />
tary papers dealing with Fredholm theory for n-tuples have appeared. R.Carey and a.<br />
Pincus have announced in [4] an interesting analytical method of checking9 in our termi-<br />
nology, condition (3) and of computing the corresponding index for some special classes<br />
of n-tuples of operators and certain coherent sheaves with support of complex codimen-<br />
sion 1. Secondly, R.Levi announced in [14] a K-theoretic framework for multioperato-<br />
rial Fredholm theory. The reader is referred to the final part of Section 4 for more<br />
details.<br />
Stein spaces.<br />
The present paper has five sections.<br />
The first section contains some preliminaries on analytic coherent sheaves on<br />
In the second section we define the notion of a Fredholm sheaf with respect to<br />
a Stein algebra representation into a Banach algebra with a distinguished two-sided<br />
closed ideal. Some stability results for Fredholm sheaves and for the associated index<br />
under geometrical and algebraic operations are then presented.<br />
In the third section the notions and the results of w 2 are applied to the case of<br />
commutative n-tuples of operators on Banach spaces. The main result is Theorem 3.1.<br />
The fourth section is expository and contains another base change result, Theo-<br />
rem 4.2, which refers finally also to commutative n-tuples of operators.<br />
In the fifth section a non-operatorial example is presented, which illustrates in-<br />
tuitively the general construction of the index.<br />
w 1. PRELIMINARIES<br />
A class of analytic sheaves, which is stable under some algebraic and geometric<br />
transformations, is described. The reader is referred to the recent book [13] for an<br />
excellent introduction to analytic geometry.<br />
Let X be a Stein space with structure sheaf O X. We denote by FR(X) the class<br />
of analytic coherent sheaves F on X, which admit a finite globally free resolution with<br />
finite type Ox-modules:<br />
(4) 0--+ L n--+ Ln_ 1--+ ... --~ L 0 -+ F-+ 0~<br />
m.<br />
where L i = OXI for some non-negative integers mp 0 < i < n.<br />
In view of the equivalence between Stein modules and coherent sheaves on X<br />
established in [Ill, an element F r FR(X) is completely determined by the Fr~chet<br />
O(X)-module F(X). Similarly, the exact sequence (4) can be derived from the free O(X)-<br />
-resolution<br />
(4)' 0--+ L n -+ Ln_ l--~ ...-+ L 0-+ F(X)-+ 0,
678 Putinar<br />
O(X)mi.<br />
where L i = Thus we shall use, depending on the context, one of these two<br />
equivalent descriptions of the same object.<br />
For the sake of completeness we sketch the proof of the following.<br />
LEMMA I.I. Let 0-+ FI-+ F2-+ F3-+ 0 be a short exact sequence of<br />
analytic coherent sheaves on a Stein space X. If two of the sheaves F l, F 2, F 3 belong to<br />
FR(X), then the third also belongs to FR(X).<br />
PROOF. Let M i denote the finite type O(X)-module Fi(X). If two of the modules<br />
M I' M2' M3' which are related by a short exact sequence<br />
0 -+ MI--+ M2-+ M3-+ 0,<br />
belong to FR(X), then the third admits by [3, Proposition III.6.3] a finite resolution with<br />
finite type projective modules.<br />
Because each projective module is a direct summand of a free module, the proj-<br />
ective resolution can be modified up to a special form:<br />
-+ -+ -+ L -+ 0,<br />
(5) 0 -+ Pn Ln-I "'" o<br />
where Pn is projective and Lj, 0 < j < n-l, are finite type free O(X)-modules. Then by<br />
comparing the three resolutions (of M 1, M 2 and M 3) through the short exact sequence,<br />
one gets that Pn is stably free. Therefore the complex (5) is equivalent to a free resolu-<br />
tion.<br />
Ox/mx r FR(X).<br />
We describe now some examples of elements in FR(X).<br />
LEMMA 1.2.[11, Proposition 4.6]. Let X be a Stein manifold and Zet xr X. Then<br />
We have already remarked in the introduction that this lemma is valid when X =<br />
= C n, because of the existence of a canonical resolution of Ox/m x. We mention also<br />
the following example. Let Z be a locally complete intersection in a contractible Stein<br />
manifold. Then the structure sheaf 0 Z = OXfl(Z) belongs to FR(X).<br />
LEMMA 1.3. Let f " X --~ Y be a flat morphism of Stein spaces and let F be a<br />
coherent sheaf on Y. Then f~F c FR(X) whenever F ~ FR(Y).<br />
Recall that the morphism f : X --~ Y is said to be flat if the Oy,f(x)-module<br />
OX, x is flat for any point x r X.<br />
The proof of Lemma 1.3 is immediate because a flat morphism preserves by<br />
pull-back the exactness of a resolution.
Putinar 679<br />
LEMMA 1.4. If F and G belong to FR(X), then F(~G has the same property,<br />
provided that F is fiat over G.<br />
PROOF. Let L' and L." be finite free resolution of F, respectively G. Then the<br />
simple complex associated to the bicomplex Lr(~L~ ' is a finite free resolution for the<br />
sheaf F(~)G.<br />
w 2. THE MAIN NOTIONS<br />
In this section a generalized Fredholm theory for coherent sheaves of FR(X) and<br />
relative to a Banach algebra representation of O(X) is described.<br />
Let A be an unita[, not necessarily commutative, Banach algebra and let I be a<br />
two-sided closed ideal of A. The basic facts concerning the K-theory of Banach algebras<br />
can be found in [20], where the commutativity is inessential. Thus the boundary opera-<br />
tor<br />
8 : KI(A/I) -+ K0(I)<br />
in the long exact sequence of K-theory for the pair (A,I) can be interpreted as a genera-<br />
lized index. We recall the classical situation A = L(H), I = K(H), where H is a Hilbert<br />
space. Then the group K0(K(H)) coincides with Z and 8 is, after this identification, the<br />
Fredholm index map.<br />
DEFINITION 2.1. Let X be a Stein space, let F c FR(X) and let p : O(X) -+ A be<br />
a unital representation of the Stein algebra O(X) into a Banach algebra A with a distin-<br />
guished two-sided closed ideal I. The sheaf F is said to be Fredholm relative to p and to<br />
I, if<br />
(6) TorO(X)(F(X) A/I) = 0, q_> 0.<br />
q<br />
Formula (6) should be compared with relation (3) in the introduction.<br />
Let us assume (6) holds and let L be a finite free resolution of F(X) with finite<br />
type O(X)-modules. Then the complex of free (A/I)-modules L (~O(x)A/I is exact, hence<br />
homotopicalty trivial, that is, if the boundary operator d of L has degree -1, then there<br />
exists an operator c of degree +l on L.(~)O(x)A/I , such that (d~l) c + e (d~)l) = id.. The<br />
A/I - linear map<br />
is invertible, where<br />
d(~lA/I + e : Le(~O(x)A/I --~ Lo(~O(x)A/!<br />
L e =~L2p and L ~<br />
= nL2p+l 9
680 Putinar<br />
The free O(X)-modules L e and L ~ have the same rank, and, after choosing bases<br />
in these two O(X)-modules, the above map gives an element [d~IA/I + el in the group<br />
KI(A/I). We set<br />
(7) indO(X)~, A/I) = a [d~)lA/I + el,<br />
where ~ : K I(A/I) --~ KD(1) is the generalized index.<br />
PROPOSITION 2.2. The definition of the index is independent of the choice of<br />
the resolution L., the homotopy ~ and the basis in L. .<br />
PROOF. If e' is another homotopy of the complex L.~)O(x)A/I between the null<br />
morphism and the identity, then for each t ~ [0,I], te + (I - t)e' remains a good homoto-<br />
py, thus the classes [d~)IA/I + e] and [d(~IA/I + e'] coincide.<br />
Changing the bases in the components of the complex L. the element<br />
[d~)IA/I + e] becomes [u(~)IA/I] [d~IA/I + e] [V~IA/I] , where ucAuto(x)(L e) and v~<br />
c AUto(x)(Lo). If we denote by ~ : A -~ A/I the natural projection, then [ux IA/I] =<br />
= Ir~[u~)IA], therefore ~ [u~)IA/I] = 0 and analogously 8 [v~)IA/I] = 0. Then the additi-<br />
vity of ~ implies the invariance of (7) with respect to the choice of bases.<br />
Let L'. and L'.' be two finite resolutions of F(X) with finite type free O(X)-modu-<br />
les. There is a morphism of complexes of O(X)-modules h : L'. -+ L'.' which is a quasi-<br />
isomorphism (i.e. it induces isomorphisms in homology). Let L. be the cone of h, that is<br />
the complex with components Lp = L'-~b~L p'~ '' p-l and boundaries<br />
dp : (-1)PhP<br />
The complex of free O(X)-modules L is exact by [3, Corollary 1.6.6], hence it is hom~-<br />
topically trivial via a homotopy e and consequently<br />
indO(X)(L, A/I):= a [df~lA/I + e~lA/I] = ~ ~,[d~l A + e~l A] = 0.<br />
Thus we have reduced the proof of Proposition 2.2 to the following lemma.<br />
LEMMA 2.3. Let 0 -+ L': -+ L. -+ L'. -+ 0 be a short exact sequence of finite<br />
complexes of finite type free O(X)-modules. If the three complexes are exact over A/I,<br />
then<br />
indO(X)(L: ' , A/I) + indO(X)(L '. , A/I) = indO(X)(L., A/I).<br />
PROOF. The boundary operator d of L. is of the form
Putinar 681<br />
d<br />
where (-l)Ph : L' -+ L" is a morphism of complexes.<br />
P P P<br />
Consider the complex of continuous functions C([O,l], L.(~O(x)A) with the<br />
boundary operator<br />
d t= ~1 A, re[0, 1].<br />
d'<br />
By [18, Theorem 2.1] the complex C([0,1], L I~O(x)A/I) is exact, and being a<br />
complex of free C([0,1], A/D-modules, there is a homotopy e t which depends continu-<br />
ously on t and which trivializes it. Then the elements d0Q1A/I + e 0 and dl~)lA/I + e l<br />
define the same class in KI(A/I), and<br />
indO(X)(L., A/I) ; ~ [dll~)lA/I + e l] - ~[d0~)IA/I § e 0] =<br />
; indO(X)(L'.~L'o', A/I) ; indO(X)(L'o , A/I) + indO(X)(u ,. ' A/I), q.e.d.<br />
COROLLARY 2.4. Let 0 -+ F' -+ F -+ F" -+ 0 be a short exact sequence of<br />
Ox-modules of FR(X). If two of the sheaves F', F and F" are Fredholm relative to P and<br />
I, then the third has the same property and<br />
indO(X)(F, A/I) = indO(X)(F', A/I) + indO(X)(F", A/I).<br />
PROOF. The first assertion follows from a long exact sequence of tor's, while<br />
the second assertion was proved in Lemma 2.3.<br />
THEOREM 2.5. Let f : X + Y be a flat morphism of Stein spaces and let P :<br />
: O(X) + A be an unital representation, where A is a Banach algebra with a distin-<br />
guished two-sided closed ideal I.<br />
If F ~ FR(Y) is Fredholm relative to po f* and I, then f~ F is Fredholm relative<br />
to P and I, and then<br />
(8) indO(X)(f*F, A/I) = indO(Y)(F, A/I).<br />
PROOF. Let us assume F r FR(Y) is Fredholm relative to p e f* and I, and let L.<br />
denote a finite free resolution of F(Y) by finite type free O(Y)-modules. Then I*F<br />
FR(X) by Lemma 1.3 and L.~)O(y)O(X) is a resolution of (f*F)(X) by free O(X)-modu-<br />
les. The sheaf F is Fredholm relative to p and I iff the complex L.~)O(y)A/I :<br />
: (L.(~O(y)O(X))~)O(x)A/I is exact, hence iff f*F is Fredholm.
682 Putinar<br />
The complexes which appear in the computation of the two sides of (8) are iso-<br />
morphic) so that the proof is complete.<br />
w 3. AN APPLICATION TO COMMUTATIVE N-TUPLES OF OPERATORS<br />
Let E be a complex Banach space and let T = (TI,... ,Tn! be a commutative n-<br />
-tuple of linear bounded operators on E. First of all we explain the definitions of the<br />
previous section in the particular case of the representation<br />
p:O(C n) -+ L(E), p( ~ a z (~)= ~ a(T c(,<br />
~>0 ~>0<br />
and relative to the ideal K(E) of compact operators.<br />
Let X ~ C n be fixed. Then the sheaf O/m~, which belongs to FR(Cn), is Fred-<br />
holm relative to p and K(E), iff the Koszul's complex K(T-X, L(E)/K(E)) is exact,<br />
because K.(z- X) O) provides a finite free resolution of O/m I. This definition of a<br />
Fredholm n-tuple on a Banach space is similar with (2) t. However, it is not known<br />
whether the condition (2) is equivalent with (2)' on an arbitrary Banach space.<br />
As for the abstract index defined in the preceding section) we shall replace the<br />
morphism 8 with the usual Fredholm index map ind : KI(L(E)/K(E)) -+ Z) in order to<br />
obtain numerical results. It is important to notice that this morphism of groups still<br />
factorizes through Coker(K 1 ~ (E)) -+ K 1 (L(E)/K(E))).<br />
In order to compute the index of T-l, let e be a homotopy which splits the<br />
exact complex K(T - X ) L(E)/K(E))) that is de + r = id. It turns out that) denoting by e<br />
a lifting for r with coefficients in L(E)) the operator<br />
is Fredholm, and consequently<br />
d+e:p~)->0K2p(T-X,E) -+ ~ K2p+I(T- X, E)<br />
ind O( C n )(O/m), L(E)/K(E)) = ind (d + e).<br />
An induction argument, e.g. ris, Proposition 2.6]) shows then that<br />
ind (d + e) = p~0 (-I)p dim Hp(T - )., E).<br />
This integer will be by definition the index of the n-tuple T - X) and it will be denoted<br />
ind(T - ).).<br />
The invariance theorems under small norm or compact perturbations of this<br />
multidimensional index are proved for instance in [15) w 2].
Putinar 683<br />
Adopting the weaker definition (2) for n-tuples on Banach spaces and defining<br />
the index by the same Euler characteristic all the stability results remain true, except<br />
for the invariance of the index under compact perturbations which is still an open<br />
question (see [10] and [21] ior details).<br />
The joint essential spectrum is by definition the set:<br />
5Pe(T,E) = {X r cnl T - ~ is not Fredholm}.<br />
In view of the above interpretation of Fredholmness, the joint essential spectrum<br />
coincides with Taylor's joint spectrum with coefficients in the quotient algebra<br />
L(E)/K(E):<br />
SPe(T,E) = Sp(T,L(E)/K(E)).<br />
This equality implies, as in the Hilbert space case, some properties of the joint essential<br />
spectrum, like for instance the spectral mapping property.<br />
The Fredholm index vanishes on the unbounded connected component of the<br />
open set on\ 5Pe(T,E) , so that Fredholm domains of T, corresponding to a non-vanish-<br />
ing index, contain only 0-dimensional and n-dimensional analytic subsets'<br />
Let U be a connected Stein open neighbourhood of Sp(T,E) and let f : U -* C m<br />
be an m-tuple of non-constant analytic functions defined on U. Then Taylor's functional<br />
calculus produces an m-tuple of commuting operators f(T)= (fl(T),...,fm(T)), with<br />
joint spectra related to those of T by the spectral mapping relations:<br />
We distinguish three cases:<br />
5p(f(T),E) = f(Sp(T,E)), 5Pe(f(T),E) = f(SPe(T,E)).<br />
l) m < n. Then f(SPe(T,E)C)= Sp(f(T),E), where K c denotes the union of the<br />
compact K and of the bounded connected components of cn\K. In particular f(X)r<br />
c SPe(f(T),E) whenever )~ I~ SPe(f(T),E) but ind(T - X) ~ 0.<br />
One might say that in this case the Fredholm behaviour of the m-tuple f(T) can<br />
be read on the analytic fibres of the map f. Indeed, let Us assume in order to apply the<br />
general scheme of w 2, that f is a flat map. If 1J r cm\ SPe(f(T),E), then by Theorem<br />
2.5, i*(O/mlj) e FR(U) is Fredholm relative to T and<br />
ind (f(T) - IJ) = indO(U)(f*(O/m ),L(E)/K(E)).<br />
The analytic sheaf f*(O/m ) is concentrated on the analytic fibre f-l(lJ), which, as we<br />
tJ<br />
already remarked, is disjoint of SPe(T,E)C.<br />
2) m > n. Then the set f(U) has no interior points, hence Sp(f(T),E) does not<br />
contain open sets. Therefore ind (f(T) - I J) = 0 for any 1J e (:m\ SPe(f(T),E).
684 Putinar<br />
3) m = n. This is the case when numerical relations between the indices of T and<br />
I(T) are expected to hold. We prove further on such a formula in a particular but<br />
generic case.<br />
THEOREM 3.1. Let T be a commutative n-tuple of linear bounded operators on<br />
a Banaeh space E and let f : U --~ C be an analytic flat map with finite fibre at 0 c C n,<br />
where U is a Stein open neighbourhood of Sp(T,E).<br />
and<br />
If the n-tuple f(T) is Fredholm, then the n-tuples T - ~, f(~) = O, are Fredholm<br />
(9) ind f(T) = fO,~=0 "~k(f) ind(T - k),<br />
where v)(f) denotes the multiplicity of f at<br />
PROOF. We are under the hypotheses of Theorem 2.5, with F = Ocm/m 0 and<br />
the functional calculus representation p : O(U) --+ L(E) associated to T. Hence the O U-<br />
-module f*F is Fredholm relative to p, and<br />
ind f(T) = indO(U)(f*F, L(E)/K(E)).<br />
Thus we have to analyse the right term of this equality.<br />
The sheaf f*F is concentrated on the zeroes set of f. More precisely, at the<br />
level of global sections, there are isomorphisms<br />
(f*F)(U) = O(U)/(fl,...,fm ) = X) ~ O~/fem O.<br />
f( : 0<br />
The multiplicity v~(f) is by definition the integer dim O),/m (O)/f*m 0) and it is<br />
finite by the assumption on finite fiber at 0.<br />
By induction on N = ~vk(f) =/(f'F), the length of the Ou-modules f'F, we shall<br />
prove the equality<br />
(I0) indO(U)(f*f, L(E)/K(E)) : I(XI: 0 V)(f) indO(U)(o/m~,L(E)/K(E)),<br />
and the Fredholmness of the n-tuples T - )., f(k) = 0.<br />
If N = l, then f*F = O/ink and (10) is true. Moreover we claim that the sheaf<br />
O/m is Fredholm relative to p iff the n-tuple T- lJ is Fredholm, 1J e U. Indeed<br />
IJ<br />
because Koszul's complex K.(z - p, O U) provides a finite free resolution of Ou/m l~ and<br />
because p is a functional calculus representation for T, both conditions are equivalent<br />
with the exactness of the complex K.(T - H, L(E)/K(E)).<br />
Let us assume relation (10) holds for N = I(f*F)-I. There exists an exact se-<br />
quence of Ou-modules 0-+ G -+ f*F-+ Ou/mk0--~ 0, where f(),0 ) = 0. By the<br />
additivity of the length, l(G)=/(f'F)-1 and by Lemma l.l, G r FR(U). Then (10)<br />
follows from Corollary 2.# and the n-tuple T - k 0 is Fredholm by the above remark.
Putinar 685<br />
The proof is complete after noticing that<br />
indO(U)(o/mz, L(E)/K(E)) = ind(T - k).<br />
Let us illustrate the theorem with some examples of analytic flat and finite<br />
maps. With the above notations the n-tuple T is Fredholm whenever f(T) is Fredholm, if<br />
the analytic map f : U -+ C n, f(0) = 0, belongs to one of the following classes."<br />
- f is one to one,<br />
- f(z) = (Pl(Zl),... ,Pn(Zn)) with PI,... 'Pn non-constant polynomials,<br />
- f(z)= (Sl,.~ denoting by Sj the elementary symmetric polynomials in<br />
Zl,...,z n. In each of the three cases the multiplicity of f at its zeroes can be easily<br />
computed.<br />
R E M A R K S 3.2. a) Contrary to the case 1) above, in formula (9) only points<br />
contained in non-vanishing index Fredholm domains of T occur.<br />
b) The theorem still holds true when U is allowed to be an arbitrary open neigh-<br />
bourhood of Sp(T,E), but then the flatness assumption muCt be replaced with the follow-<br />
ing condition<br />
TorO(Cn)(o(cn)/m0 ,<br />
(1 1) q O(U)) = 0, c~l.<br />
This means that the comple x K.(f, O(U)) is exact in positive dimensions.<br />
In this case the proof consists in passing to the envelope of holomorphy 0 of U,<br />
and to make use there of coordinateless arguments, in the spirit of [16]. Namely, let<br />
: 0 -+ C n be the analytic extension of f. Then the flatness of f is replaced by (11),<br />
while the finiteness assumption which is not necessarily valid for f, is corrected in<br />
formula (9) by the vanishing relations<br />
indO(U)(oo/m l, L(E)IK(E)) = O, ~ c U\ U.<br />
These are in turn consequences of the inclusions of the coordinateless spectra in the<br />
generating part U of U:<br />
a(O ,L(E)/ K(E)) u a(U ,E)CU,<br />
see [16] for details. Then the proof of Theorem 3~ can be adapted to this more general<br />
situation~<br />
w 4. BASE CHANGE FOR FREDHOLM COMPLEXES<br />
A central role in the last section was played by the family of complexes<br />
K.(T - X, E)) parametrized on X r C n. In order to analyse separately the behaviour of<br />
the homology spaces Hq(T- X)E) under analytic transformations we shall reduce ;this<br />
problem to a finite dimensional one and there we shall use the methods of analytic
686 Putinar<br />
geometry,<br />
Let us recall for the beginning some terminology. A complex (L, d) of Banach<br />
spaces and linear operators ~ dq<br />
9 .. "-* Lq+ 1 Lq Lq_l--'* ...<br />
is said after G.Segal [17] to be Fredholm if the homology spaces are finite dimensional,<br />
dim Hq(L) < % and if every boundary operator dq is direct, that is Ker(dq) and Im(dq)<br />
are closed complemented subspaces in Lq, respectively in Lq_ 1"<br />
With the notations of the preceding section, the n-tuple T is Fredholm iff the<br />
complex K (T,E) is Fredholm in the above sense.<br />
Let X be an analytic space with structure sheaf O X. The (topological free) O X-<br />
-module associated to the presheaf U -~ Ox(U)QF , where F is a Banach space will be<br />
A<br />
denoted by OxQF. An analytically parametrized complex of Banach spaces is by<br />
definition a sequence (Lq)qr z of Banach spaces and of elements dq r r(x, Ox(~L(L q,<br />
Lq_l)) , such that dq o dq+ 1 = 0, q oZ. The image of dq through the natural projection<br />
F(X, Ox~)L(Lq, Lq_ 1) -'* OX,x(~L(Lq, Lq_ 1)/mx~)L(L q, Lq_ 1) =L(Lq, Lq_ 1)<br />
is a linear bounded operator dq(X)cL(Lq, Lq_l). Thus one gets for every x r X a<br />
complex (L. , d.(x)) of Banach spaces whose boundaries depend on x. This complex will<br />
be denoted in short by L(x) and its homology spaces by Hq(L.(x)). On the other hand for<br />
an analytically parametrized complex there is a corresponding complex of sheaves<br />
(OxQL" , d) which will be denoted by L , and its homology sheaves by Hq(L ). When X<br />
is a complex manifold, the above definition of an analytically parametrized complex<br />
means nothing more than that the operators dq(X) depend analytically in the norm<br />
topology.<br />
Finally, by a right bounded complex L. we mean a complex with vanishing nega-<br />
tive terms, Lq -- 0 for q < O.<br />
There are many stability results for analytically parametrize d complexes of<br />
Banach spaces ([18, Lemma 2.2], [21], [22, Chap.III]). As a by-product of them one<br />
obtains rather directly by a descending procedure the following result (see also [l/#,<br />
Proposition l]).<br />
PROPOSITION 4.1. Let L be a right bounded complex of Banach spaces, analy-<br />
tically parametrized on a reduced analytic space Y, and let N be a non-negative inte-<br />
ger. Suppose that the complexes L(y) are Fredholm for y c Y. Then there exists locally<br />
on Y a right bounded complex P of finite type free Oy-modules and a morphism of<br />
complexes
Putinar 687<br />
which induces the isomorphisms<br />
for 0 < q < N and locally on y c Y.<br />
qb : P. --+ L<br />
H CP) ~- Hq(L), Hq(P(y)) -~Hq(L(y))<br />
This proposition reduces many statements of the infinite dimensional analytic<br />
Fredholm theory to the finite dimensional case. In order to illustrate this principle we<br />
present without proof the following infinite dimensional version of Grauert's continuity<br />
theorem [2, Theorem 3.3.#].<br />
THEOREM 4.2. Let L. be a right bounded complex of Banach spaces, analyti-<br />
cally parametrized on a reduced analytic space Y and let q be a non-negative integer.<br />
Suppose that the complexes L.(y) are Fredholm for every y ~ Y. Then the fol-<br />
lowing assertions are equivalent:<br />
a) The function y -~ dim Hq(L.(y)) /s locally constant on Y.<br />
b) The families of subspaces {Ira dq+l(y)} and {Ker dq(y)} form analytic Banach<br />
subbundles of the trivial bundle Y x Lq.<br />
c) If f : X -+ Y is a morphism of analytic spaces, then the natural map<br />
is an isomorphism.<br />
f*Hq(L) ...+ Hq(f* L)<br />
The proof of the theorem reduces the statement, via Proposition 4.1, to the<br />
finite dimensional case and then uses Theorem 3.3.4 from [2]. We point out only the<br />
importance of the splitting assumption in the definition of a Fredholm complex, in order<br />
to reduce the property c) to the finite dimensional case.<br />
As a continuation of the preceding section we derive some applications of<br />
Theorem 4.2 to multioperatorial Fredholm theory.<br />
Let T be a commutative n-tuple of linear bounded operators on a Banach space<br />
E and let Dc::C n be a Fredholm domain of T. Then K.(T-I, E) is an analytically<br />
parametrized family of Fredholm complexes on 1 e D. Let q be a fixed non-negative<br />
integer.<br />
Then Proposition 4.1 implies that the iumping points of the function<br />
form a thin analytic subset S of D.<br />
-+ dimHq(T- X,E) , teD,<br />
Applying Theorem #.2 to the complex K.(T- ~,E), one gets that the homology<br />
spaces Hq(T - t,E) form in a natural way an analytic vector bundle on D\S. Indeed, let
688 Putinar<br />
X be the simple point X and let f: X--* D\S be the inclusion map. Then<br />
Hq(L.)/mxHq(L.)~. Hq(L.(X)) by c), hence Hq(L.) is a locally free OD\ S - module, where<br />
L = K.(z - T, ODt~E).<br />
A<br />
Let Tq denote the homology sheaf Hq(K.(z- T,O x E)), defined on C n. Then,<br />
again by Proposition #.1, Tq is a coherent sheaf on the open set U = cn\SPe(T,E). This<br />
fact was used by R.Levi [14] in order to define the following abstract index<br />
n<br />
r(T) := y. (-l) i IT i lu],<br />
i=0<br />
as an element of the Grothendieck group K0(U) of analytic coherent sheaves on U.<br />
Levi's index K(T) contains much more information than the usual Fredholm index.<br />
Notice that the set which carries K(T),<br />
n<br />
X := Sp(T,E)\SPe(T,E) = U Supp(Ti l U)<br />
i=0<br />
is an analytic subset of U. The element K(T) has an integer multiplicity in every point<br />
x r X, which is up to a sign the local index computed in some particular cases by<br />
R.Carey and J.Pincus in terms of traces of commutators. This local index can be<br />
expressed as follows:<br />
ix T' X<br />
xcV i=0<br />
where V runs over the irreducible components of the analytic set X, and the multiplicity<br />
of the sheaf T i along V is defined by the formula<br />
where y is a generic point of V.<br />
mv(T i) =length- (T. ~)~ O.. ))<br />
UV,y I,y Uy v,y<br />
Let us interpret from this point of view the indices relative to coherent sheaves<br />
defined in Section 2. Let F c FR(C n) be a Fredholm sheaf relative to the representa-<br />
tion in the algebra L(E) associated to the n-tuple T and relative to the ideal of compact<br />
operators. We assume in addition that the set Supp(F) is disjoint of SPe(T,E); this<br />
assumption follows from the Fredholm condition of F, when for example this sheaf is a<br />
locally complete intersection ideal of Ocn.<br />
Because the joint spectrum of T is bounded, the analytic set Sp(T,E)~<br />
]7 Supp(F)= XNSupp(F) doesn't contain components of dimension greater than zero,<br />
hence it is a finite subset of U. For any pair (i,j) of integers, the sheaf Tort~ (F, Ti[U) is<br />
supported on XNSupp(F). Then Definition 2.1 of the index of T relative to F gives rise<br />
to the equality
Putinar 689<br />
indO(Cn)(F, L(E)IK(E)) =..~ (-l)i+Jdim TorU(F, Ti)(U),<br />
i,]<br />
which in turn can be interpreted as follows:<br />
C n<br />
ind O{ )(F, L(E)/K(E)) : [F]" K(T).<br />
Concluding, the index of the n-tuple T relative to the sheaf F and to the ideal<br />
of compact operators represents the Tor-intersection multiplicity of F against the class<br />
K(T).<br />
w A FINAL EXAMPLE<br />
We exemplify the general notions of w by representations of Stein algebras<br />
into commutative C*-algebras.<br />
Let X be a Stein manifold of complex dimension n and let x be a fixed point of<br />
X. Consider a pair (A, B) of compact CW-complexes, BCA, and let ~:A -+ X be a<br />
continuous map. We investigate the Fredholmness of the sheaf O/m x relative to the<br />
representation<br />
p : O(X)-~ C(A), p(f) = ~*(i) = fo~, f ~ O(X),<br />
and relative to the ideal I = V(B) = {g r C(A) [ glB = 0}. We have denoted as usually by<br />
C(K) the commutative C* - algebra of continuous functions on the compact K.<br />
By rll, Proposition 4.6] the simple point x is a complete intersection in X,<br />
hence there exists a n-tuple f = (fl'" " " 'fn ) of holomorphic functions on X, such that the<br />
augmented Koszul complex<br />
is exact.<br />
K.(f, O X) -+ Ox/m x --~ 0<br />
Thus the sheaf Ox/m x is Fredholm relative to p and I iff the complex:<br />
K.(f o r C(A)/I)<br />
is exact. But the C*-algebra C(A)II is naturally isomorphic with C(B), therefore we<br />
have proved the following fact:<br />
A) The sheaf Oxlm x is Fredholm relative to 0 and I = V(B) iff x ~ ~(B).<br />
In order to compute the index we identify the groups K I(C(A)/I) with KI(B) and<br />
K0(1) with K0(A, B). Then the boundary operator 8 : KI(C(A)/I) --+ K0(1) becomes the<br />
usual coboundary 8 : K I(B) --* K0(A,B)in topological K-theory, see [20].<br />
By its definition in w 2, the index indO(X)(O/mx , C(B)) is the class in K0(A, B) of<br />
the complex of trivial vector bundles K.(f o~, A which is exact on B, in short
690 Putinar<br />
indO(X)(O/mx , C(B)) = X(K(f o ~, A x C)),<br />
where X : Cn(A, B) § K0(A, B) stands for the generalized Euler-characteristic map [l,<br />
w 2.6].<br />
Let U be an open neighbourhood of x in X, such that U is diffeomorphic equi-<br />
valent by f with a ball B(0, 6) of radius 6 > 0. We may assume that the n-tuple f - X =<br />
= (fl-~l''"'in-Xn) is still a regular sequence in O X for every XcB(0,6). Let e be<br />
chosen such that 0 < 6 < 6. Then ~(5)~ V = r where V = f-lg(0, r<br />
Let i : ({(A), {(B)) -~ (X, X\V) denote the inclusion map and let<br />
: KO(u, U\V) --* KO(A, B) be the composition of the morphisms<br />
i* ~*<br />
: K0(U, U\V) = K0(X, X\V) --+ K0(#(A), {(B)) --* K0(A,B),<br />
where the first one is the excision isomorphism [1, w 2.#]. Then for any X r B(0, ~) the<br />
sequence f - ;k is still regular, hence<br />
indO(X)(O/my, C(B)) = ~ f(y), OxC)), y r V.<br />
But K0(O, 0 \V) = K0(B 2n, 8 B 2n) = t~0(S 2n) --7., and x(K.(f - f(y), 0 x C)) is the<br />
positive generator of this group, for every y r V, [1, p.llS]. Concluding we can state<br />
the following assertion.<br />
B) The function x-+ indO(X)(O/mx , C(B))c K0(A, B) is locally constant on<br />
X\~(B) and equals r * (Tx) , where T x stands for the positive generator of the group<br />
K0(X,X\ {x}) = 7, inherited from the complex structure of X.<br />
Notice that indO(X)(O/mx , C(B)) = 0 if x ~ #(A). Indeed, in this case the complex<br />
K.(foq~, Axe) is exact. Therefore the index vanishes identically when X\~(B) is a<br />
connected subset of X.<br />
Let us conclude with an example of non-vanishing index. Let X = C, A =<br />
: {Izl < 1} and B= {Izl = 1}, If(~: A-~ C is a continuous map, then the sheaf Olin 0<br />
is by A) Fredholm relative to the representation p = @* and the ideal I = V(B) iff (~<br />
doesn't vahish on B', The group K0(A, B) coincide with Z, and in this case assertion B)<br />
has the following numerical interpretation:<br />
indO(C)(O/m0 , C(B)) = deg(~ I B).<br />
Notice the formal analogy with the classical theory of Toeplitz operators.<br />
Acknowledgment. I would like to thank Dan Yoiculescu for very useful convers-<br />
ations about this material.
Putinar 691<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
10.<br />
11.<br />
12.<br />
13.<br />
14.<br />
15.<br />
16.<br />
17.<br />
18.<br />
19.<br />
20.<br />
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Acta Math. 125 (1970), 1-38.<br />
TAYLOR, 3.L. : Banach algebras and topology, in "Algebras in Analysis",<br />
Academic Press, 1975, pp.I18-186.
6 9 2 Putinar<br />
21.<br />
22.<br />
VASILESCU, F.-H. : Stability of the index of a comlex of Banach spaces, J.<br />
Operator Theory I (1979), Ig7-205.<br />
VASILESCU, F.-H. : Analytic functional calculus and spectral decomposition,<br />
Ed. Academiei and D.Reidel Co., Bucharest and Dordrecht, 1982.<br />
Department of Mathematics, INCREST<br />
Bd. P~cii 220, 79622 Bucharest<br />
Romania<br />
Submitted: November 28, 1984
Integral Equations<br />
and Operator Theory<br />
Vol. 8 (1985)<br />
TRANSVERSALITY AND THE INDEX THEOREM I)<br />
N. Teleman<br />
INTRODUCTION<br />
0378-620X/85/050693-2751.50+0.20/0<br />
9 1985 Birkh~user Verlag, Basel<br />
In this paper, we give a new proof of the index theorem<br />
based on transversality.<br />
The topological index of elliptic operators involved in<br />
the Atiyah-Singer [3'] ,[14] and Sullivan-Teleman [18],[19],[20]<br />
index theorems, have two main ingredients : the Chern character of<br />
the symbol of the operator and the L-character (the Hirzebruch<br />
[10] polynomials in the Pontryagin classes) of the base manifold.<br />
The expression for the L-character, in terms of Pontrygin classes,<br />
is complicated.<br />
The Thom construction [21] provides a very elegant,<br />
simple, geometric realization of the L-character of homology<br />
manifolds; the rational Pontryagin classes appear to be an alge-<br />
braic by-product of the L-character. The Thom construction -<br />
based on transversality - shows that the L-character is built-in<br />
naturally, geometrically.<br />
It is spontaneous then to use the Thom construction<br />
for a proof of the index theorem.<br />
We are going to show here that by realizing this idea,<br />
I)<br />
Research supported by the N.S.F. Grant N ~ MCS 8102758. Paper<br />
revised while the author was visiting I.H.E.S., France.
694 Teleman<br />
the signature operator with coefficients in an arbitrary aspheri-<br />
cal complex vector bundle ~ ~plits-up in a product of two<br />
operators : a signature operator with constant coefficients over<br />
the open submanifold (of the base manifold) consisting of the<br />
sapport of the bundle ~ , and a signature operator over an even<br />
dimensional sphere; the bundle ~ is called aspherical if it is<br />
the pull-back of some bundle over a sphere. The Serre theorem [15]<br />
states that any complex vector bundle is, essentially, over %he<br />
rationals, an aspherical bundle.<br />
The operator splitting above allows us to prove the in-<br />
dex formula for such a signature operator by direct eomputation~<br />
on the base manifold of the operator. This computation involves<br />
the following pieces of information :<br />
i) Hirzebruch-Thom formula<br />
2) Bott-isomorphism<br />
3) Hodge theory<br />
4) Excision invariance of the analytical index<br />
5) Multiplicativity property of the index<br />
6) The index formula for spheres<br />
The proof of the index theorem we give here provides a<br />
proof of the index theorem for Lipschitz and topological manifolds<br />
[18] ,[19],[20] independent of the Atiyah-Singer index theorem for<br />
smooth manifolds. Our method allows also to recapture the index<br />
theorem for signature operators on admissible pseudomanifolds,re-<br />
sult due to J. Cheeger [7].<br />
The presentation of the background material for this<br />
paper is presented in such a manner as to allow a possible axioma-<br />
tization of the index theory. For the reading of this paper, fami-<br />
liarity with [6],[18],[19] and [20] is necessary.<br />
The author thanks Jeff Cheeger for useful discussions<br />
about pseudomanifolds.<br />
2. Basic notations; recall of basic results. The aim of<br />
this section is to introduce the basic notations and recall basic<br />
results which are necessary for the rest of the paper. Secondarily,<br />
by doing so, we might be able to produce an axiomatization of the
Teleman 695<br />
index theory.<br />
Let HT (i.e. Hodge and Transversality) denote one of<br />
the categories : Diff, PL or Lip (of Lipschitz manifolds). In<br />
w we explain,, brielfly, how our considerations extend to the<br />
category of admissible pseudomanifolds introduced by J. Cheeger<br />
[6].<br />
Any object in HT satisfies the following properties<br />
(A.I)-(A.12). The notations and definitions here are as in [19].<br />
(A.I) Any object in HT is a finite CW-complex. The<br />
geometrical dimension is the dimension of the object.<br />
(A.2) Any object in HT has a Lebesque measure<br />
induced by a Riemannian metric.<br />
(A.3) Let C(X) denote the Banach algebra of all<br />
real continuous functions on X . For any pair (X,~) as in (A.I),<br />
(A.2), the sequence of separable real Hilbert spaces,<br />
{Ri}o
696 Teleman<br />
is a bounded operator which decreases supports, i.e.<br />
Supp Af(~) c Supp ~ (The support of ~ , Supp ~, can be defined<br />
because ~r-- is a C(X)-module)<br />
tries :<br />
A.6. The Hilbert spaces ~r are related by isome-_<br />
,r : r m-r<br />
§ ~ , m = dim X<br />
(derived from the Riemannian metric);<br />
These are C (K) -homomorphisms, and<br />
,m-r ,r r (m-r)<br />
o = (-i)<br />
(A. 7) The operator r , with domain :<br />
r<br />
~ = , ~m-r<br />
d<br />
Br = (_l)m(r+l)+l dm-r ,<br />
is the formal adjoint to d (Stokes' formula).<br />
(A.8) The pairing :<br />
~r x ~m-r § m<br />
~,a + (~'*S)r ' (d~ = 0 ,d~ = O)<br />
where ( ' )r denotes the scalar product on er , is non-degene-<br />
rate (Poinear~ duality pairing).<br />
r ~r<br />
(A.9) Let W 1 c be the subspace<br />
r r<br />
~i = nd n nB<br />
endowed with the norm II II<br />
The inclusion<br />
2=<br />
Ill II 2+lld ll 2 II II 2
Teleman 697<br />
(w~,II Ill ) + (~r, II If)<br />
is compact. (Relliah lemma).<br />
(A. IO) The spheres S r , O < r < ~ , belong to the<br />
category HT , and H~(S r) = H*(Sr,~) .<br />
Any morphism :<br />
f : xm+ S m-r<br />
in HT is homotopic to a transverse regular mapping<br />
: xm+s m-r<br />
to a point Po . This means that there exists an object<br />
(vr,u v) 6 HT of dimension r , and an embedding<br />
F : V r Bm-r § m<br />
(~m-r is the unit disc in ~m-r) such that<br />
V r ]Bm-r ~? F ~Bm-r<br />
v r "f<br />
{Po }<br />
= _ ~-r)+<br />
is a bundle map (S m-r is thought as S m-r Bm-r U m.r<br />
Bm-r<br />
being the upper (lower) hemispheres, and<br />
~(X
698 Teleman<br />
in HT<br />
(Xl,U I) , (X2,~ 2)<br />
, their product (X 1 X2,Ul x u2 )<br />
(i) ~r(x I X2,~ 1 ~2 ) = @ ~P(Xl,P I) ~ gq(x2,~ 2) ,<br />
p+q=r<br />
(ii)<br />
(iii)<br />
(@ denotes the Hilbert tensor product)<br />
d(~ @ ~) = d~@ ~ + (-i) deg ~ @ dG ,<br />
,(~ @ ~) = (_l)deg ~. deg~(,~) @ (,~)<br />
belongs to HT , and :<br />
(in (iii) , X 1 and X 2 are even dimensional).<br />
(A.12) The KUnneth formula holds for H*<br />
w Review of Signature Operators.<br />
3.1. Hodge Theory. (A.I)- (A.8) provide Hodge theory. As cus-<br />
tomary, the space of harmonic forms of degree r on an object X<br />
is defined :<br />
is an isomorphism<br />
r d~ = O Be = O}<br />
Hr(X) = {~I~ 6 W 1 , , ..<br />
Theorem 3.1. For an~ ob/ect X g HT<br />
(i) the~a~ural Hodge homomor~hism.<br />
~r r<br />
: H + ~r(x)<br />
(ii) the operator * induces an isomorphism :<br />
*: Hr(x) ~ HdimX-r(x )<br />
(iii) the spaces d~-i , d~+l , Hr<br />
and mutually orthogonal in ~r ; in addition<br />
Proof. See w [19].<br />
~r = H r @ d~-i @ ~+i<br />
are closed
Teleman 699<br />
3.2. Signature operator.<br />
The operator<br />
+ ~ D = d+~ = D + -<br />
D-( WI§ O , 8 D ,<br />
+<br />
are defined in the usual way, see [3'] and [19]; the operator D<br />
is called signature operator.<br />
Theorem 3.2. The signature o~erator<br />
+ +<br />
D : W ! § W O<br />
is a Fredholm operator, and<br />
Index D + = Sig X ,<br />
where Sig X denotes the signature of the Poincar~ pairing on the<br />
middle eohomology of X (we suppose here dim X ~ 0 mod. 4).<br />
Proof. For the smooth case, see Atiyah-Singer [3']; for the<br />
Lipschitz case, the Telemam [19].<br />
Let ~ § X be any continuous complex vector bundle.<br />
We suppose that the bundle ~ has an A(X)-structure, i.e. a<br />
vector subspace A(X,~) of the space of continuous sections in<br />
~,C(X,~) , is prescribed, and A(X,~) is a finitely generated<br />
projective A(X)-module, such that :<br />
A(X,~I 8 ~2 ) = A(X,~I) 8 A(X,~ 2)<br />
A(X,~) = A(X) ~...8 A(X) (N times)<br />
where N denotes the product bundle of rank N<br />
The operators :<br />
D : W (X,~) +W (X ~)<br />
D~<br />
= D + D[<br />
are defined as in w [3'] and ~6 [19]; D + (X,~) is called signature<br />
operator with values in<br />
+<br />
Theorem 3.3. D~ is a Fredholm operator.<br />
The Proof of Theorem 3.3. invokes in addition to<br />
(A.I) - (A.8), the Rellich lemma (A.9).
700 Teleman<br />
3.3. The excision theorem.<br />
(A.4) -(A.7) and (A.9) provide the necessary structure<br />
to carry over all considerations from w167 [19] into the<br />
category HT. As a consequence we have the :<br />
a common o~en part U<br />
Theorem 3.4. (Excision)<br />
Let (M,~) ,~ = 1,2 , be two HT-Spaczs which have<br />
i I i 2<br />
M 1 ~ U , M 2<br />
Let W c U be a closed set, and let<br />
§<br />
@ -~ M ~ = 1,2 ,<br />
be complex bundles of the same rank. Suppose that :<br />
(i) PllU = ~21u<br />
(ii) the following bundle isomorphisms<br />
A21<br />
qlu 2Ju<br />
A89<br />
81 I g ' e21u<br />
~1 (M I "-W) ~21 (M2"-W)<br />
811 (M I ~W) 021 (M2"~W)<br />
are given such that the diagram
Teleman 701<br />
be commutative. Then<br />
(i) (3.3)<br />
(ii)<br />
(3.4)<br />
~11(u ~ w)<br />
e11lu ~ w)<br />
A21<br />
!<br />
A21<br />
~21 (U~W)<br />
821 (U~W)<br />
Index D + - Index D~2 + - Index D +<br />
~I = Index D81 82 ;<br />
if 81 and e 2 are trivial bundles, then<br />
Index D~I - Index D~2 = (Rank ~i ) Sig M 1 -<br />
- (Rank ~2 ) Sig M 2<br />
As a corollary of this theorem we get :<br />
Theorem 3.5. For any complex vector 5undle ~ § X 2m<br />
in the category HT , Index D~ + does not depend on the chosen<br />
measure on X .<br />
It is clear that<br />
+<br />
Index D~<br />
and that<br />
+<br />
+ Index D~<br />
induces a homomorphism.<br />
K~ +<br />
Any element u s K~ is a formal difference<br />
u = [~i ] - [~2 ] of vector bundles, where ~2 may be chosen to<br />
be trivial (this will be tacitly used, later, in some occasions).<br />
By definition, the (virtual) rank of u is :<br />
Rank u = Rank ~I - Rank {2<br />
If ~ is a bundle, let I{I denote the product bundle<br />
of the same rank as { on the same base space.<br />
depends only on [~] 6 K~<br />
We can rewrite the formula (3.4.) for ~1,~2,eI,82<br />
replaced by I~ll, [[21, 1811, 1821 , where the bundle isomor-<br />
phisms A21,A 89 are each the identity; (3.4.) becomes
702 Teleman<br />
(3.4 ') Index D + +<br />
. 1611- Index DI~21 =(Rank I~I) Sig M I -<br />
the<br />
-(Rank I~21) Si s M 2<br />
If we substract (3.4) and (3.4') side by side, we get<br />
Corollary 3.6. For ~1,~2 bundles as in theorem 3.4 (ii), we<br />
have :<br />
+ +<br />
(3.4") Index D~ = Index<br />
1-1~11 D 2-1 21<br />
of virtual rank zero).<br />
(Notice that ~i-[(iI , i = 1,2 , are vector bundles<br />
w The Thom's construction; L-classes.<br />
4.1. For this exposition, see R. Thom [21]<br />
Theorem 4.1. (Serre [15]). Let X be any finite CW-complex.<br />
Let g 6 Hr(sr,~) be a generator. Then the mapping<br />
~r(x) | ~ ~ Hr(X,~)<br />
If] | p/q , p/q f*(g) ,<br />
(for f : X § S r a continuous map) is an isomorphism for<br />
2r > dim X+ 1 . Here ~r(x) stands for the cohomotopy group of<br />
x in dimension r , i.e.<br />
~r(x) = {homotopy classes of maps x + S r} ;<br />
this is an abelian group.<br />
4.2. Suppose now that X m E HT , dim X = m is even. Take any<br />
continuous map :<br />
f : X m § X r , m+1 < 2r , m-r ~ 0 (mod 4)<br />
By (A. IO) , f is homotopic to a map ~ which is transverse regu-<br />
lar to a point p 6 S r . Let F and V be as in the statement<br />
of (A.IO) :<br />
vm-r ~r~F X m
Teleman 703<br />
We associate with f the signature of V m-r , or<br />
Sigvm-r,when m-r ~ 0 mod 4 ; by definition, Sig(V m-r) is the<br />
signature of the Poincar4 pairing (A.8.) on the middle L 2-<br />
cohomology of V m-r . We require :<br />
(A. IO') Sig V m-r is independent of the transverse<br />
regular mapping belonging to the homotopy class of f , and<br />
r (X) + ~ ,<br />
[f] § Sig V<br />
is a homomorphisms. []<br />
By definition, (see Thom [21], and Goreski-MacPherson<br />
[8]) the homology L-class Lr(X m) is that rational homology class<br />
Lr(Xm) 6 Hr(Xm,~)<br />
with the property that :<br />
(4.1) f*(g) (Lr(Xm)) = Sig V m-r ,<br />
where g 6 Hr(sr,~) is a generator. We set :<br />
L~(X m) = ~ Lm_4i(xm) 6 H4,+e(xm,~ ) ,<br />
i=O<br />
E ~ dim xm(mod. 4)<br />
As X m has a Poincar4 duality, relative to the ratio-<br />
nal singular cohomology ~) , the cohomology class<br />
L4i(x m) 6 H4i(xm,~)<br />
may be defined by the formula<br />
(4.2.)<br />
X m<br />
L* (X m)<br />
(L4i(x m) [J f*(g)) [X m] = Sig V m-4i ,<br />
denoting the fundamental class of X m and<br />
L*(X m) = Z L4ilX m)<br />
i=o<br />
is the Hirzebruch L-character of x m<br />
*)This is not the case for pseudomanifolds.
704 Teleman<br />
The formulas (4.1), (4.2) show that L4i(x m) and<br />
Lm_4i(xm) are Poincar4 dual :<br />
(4.3) Lm_4i(xm) = L4i(x m) n [x m]<br />
This construction allows us to define Lm_4i(xm) f<br />
resp. L4i(xm) in dimensions above, resp. below, the middle dimen-<br />
sion of X m . The other classes are defined by taking the direct<br />
product of X m by S m , see [8] and [12].<br />
Proposition 4.2.<br />
(i) L~ m) = 1<br />
L4n(x 4n) X 4n = Sig X 4n (Hirzebruch signature<br />
formula)<br />
L4n(x Y) = Z L4P(x) x L4q(y)<br />
p+q=n<br />
L*(S r) = 1<br />
(ii) L*(X4n)([x4n]) = Sig X 4n (Hirzebruch signature<br />
o<br />
formula)<br />
L4n(X 4n) = [X 4n]<br />
Li(X x y) = Z L (X) L (Y)<br />
p+q=i P q<br />
L,(S 2r) = [S ,2r]<br />
Proof. The two groups of relations (i), (ii) are related by<br />
Poincar4 duality. For the proof of the multiplicative property of<br />
L, , see w Appendix, given here for the benefit of the reader.<br />
4.3. The even dimensional rational cohomology of any finite com-<br />
plex is generated, over ~ , by the Chern character of all complex<br />
vector bundles over it. We would like to use this fact in order<br />
to reformulate the defining formula for the L-classes.<br />
To do this, we have to suppose first that X m is even<br />
dimensional, m = 2s .<br />
Let<br />
f: X2s §<br />
S2s
Teleman 705<br />
be a continuous map, and let { § S 2s be a complex vector<br />
bundle over S 2Z-4i . As the Chern character is natural, and<br />
K(S "25-4i) ~ Z (Bott isomorphism), the defining formulas (49<br />
and (4.2) for L,(X) and L*(X) become :<br />
Theorem 4.29 (Hirzebruch-Thom formula)<br />
(i) Ch(f*{) (L2s163 = Sig v4i.Ch~([S2Z-4i])<br />
or, for Poincar~ duality spaces :<br />
(ii) (L4i(x 2s [] Ch(f~)) [X 2s = Sig V 4i<br />
9 Ch6([$2s<br />
A virtual vector bundle over x of the form f*~ , where<br />
f : X § S 2r<br />
is a continuous map, and ~ 6 K~ 2r) , will be called aspherical.<br />
Notice that the Chern character of an aspherical ele-<br />
ment f*~ is of the form<br />
where<br />
f*~ , and<br />
of dimension<br />
Ch(f*~) = n + U2r ,<br />
n 6 H~ is the virtual dimension of the bundle<br />
U2r 6 H2r(x,~)<br />
Proposition 4.3. Let X m be a finite CW complex<br />
m . Then for any cohomology classes<br />
a o 6 H~ , a r 6 Hr(xm,~) ,<br />
r = even, 2r > m+l , there exists one and only one<br />
~ 6 K~ m) | ~ having the properties :<br />
(i) ~ is aspherical<br />
(ii) Ch n = a + a<br />
o r<br />
Proof. 1) Existence. The cohomology class a r is aspherical.<br />
Serre's theorem 4.1. implies that there exists a continuous map<br />
f : X m § S r<br />
such that f*(gr ) = a r , where gr 6 Hr(sr,~) is a generator.
706 Teleman<br />
As the Chern character (on S r) is an isomorphism, there exists<br />
an element ~ 6 K~ r) | ~ such that<br />
Theorem 5.1. (Index Theorem for signature operators in<br />
HT). Let X 2s 6 HT and let ~ + x 2s be a complex vector bundle<br />
over x 2~ . Then :<br />
(5.1)<br />
Ch ~ = gr<br />
Let e = I | a 6 K~ r) | ~ , where I is the trivial<br />
o<br />
bundle of rank I on S r . Then q = f*(e+~) has the desired pro-<br />
perties.<br />
isomorphism. []<br />
2) Uniqueness. The Chern character (on X m) is an<br />
Let ~ denote the external tensor product of bundles;<br />
the same symbol will be used to denote external products of<br />
K-theory classes.<br />
property that<br />
Let 82r 6 K~ 2r) | ~ denote that element with the<br />
Ch e2r = g2r 6 H2r(s2r,~)<br />
Proposition 4.4. For any element ~ 6 K~ m) 8<br />
having the property that the Ch ~ is a homogeneous cohomology<br />
class, the element<br />
s em+ E 6 K~ mx S m+~) ~<br />
(where ~ = O,1 such that m+~ = even) is aspherical.<br />
Proof. Ch(~+ ) = Ch ~ x Chem+ E = (Ch~)xgm+ C ,<br />
which is a homogeneous cohomology class whose degree exceeds the<br />
middle dimension of X m x S m+e . The proposition 4.3. completes<br />
the argument.<br />
w The Index Theorem for Signature Operators.<br />
5.1.<br />
Index D + = (Chq) ( Z 2s<br />
i=O 2~-4i(X2~))<br />
We are going to show that the formula (5.1) follows
Teleman 707<br />
from<br />
i) Theorem 4.2. (Hirzebruch-Thom formula)<br />
2) Bott isomorphism : K~ = ~ ,<br />
+<br />
3) Theorem 3.2. (Index D = Sig X, Hodge theory)<br />
4) Theorem 3.4. (ii) (Excision)<br />
5) Theorem 5.2. (Multiplicativity of the Index).<br />
6) Theorem 5.3. (Index formula for spheres).<br />
2s<br />
Theorem 5.2. Let X i 6 HT , i = 1,2 ; let<br />
be complex vector bundles and<br />
~1 [] ~2 § X I x X 2<br />
their external product.<br />
Then<br />
5.2. Index D~I [] ~2 = Index D~I " Index D~2<br />
~i § Xi<br />
Theorem 5.3. (Index formula for spheres). The index<br />
formula (5.1) holds for even dimensional spheres, i.e. for ~ a<br />
bundle over a sphere s 2r :<br />
(5.3) Index D + =~Ch ~ [S 2r]<br />
The proof of theorem 5.2. will be postponed to w In<br />
order to prove theorem 5.3., it is enough to check it for the<br />
generator of the reduced K-theory group, K~ = ~ , for<br />
r = 1,2, .... Because the index of signature operators is inde-<br />
pendent on the measure on the base space, we can take the stan-<br />
dard Lebesque measure on S 2r and therefore the problem can be<br />
reduced to the smooth case. For this computation see [14] ch. XV,<br />
w (The author thanks A. Connes for the 2 r factor in(5.3.)).<br />
Proof of Theorem 5.1.<br />
5.2. The aspherical case.<br />
We will prove (5.1) first for n an asphericalbundle:<br />
n = f*~ ,
708 Teleman<br />
where<br />
X2f.~ S 2-~ -4i $2~-4i<br />
(5.4) f : , $ +<br />
By virtue of (A.10), we may suppose<br />
the point P 6 IB 2s c S 2~-4i<br />
o<br />
Let<br />
(5.5) V 4i f-l{po}<br />
and if we thought of the embedding F<br />
tion, we may consider that<br />
5.6) V 4i x IB 2g-4i c X 2~<br />
5.7) fl ({v} 163 : {v} 2s § I]32s<br />
is the identity mapping for any v 6 V 4i , and<br />
(5.8) f(X2s v4ix IB 2s _c S 2s ~{po}<br />
f transverse regular to<br />
(see 4.2.) as identifica-<br />
The mapping f can be deformed continuously on the subset of<br />
X2Z~ v4i x IB 2Z-4i consisting of those points x which are sent<br />
by f into the lower hemisphere IB 2s (and, therefore, in<br />
IB 2 s {po } ; this will be done by moving f(x) along the me-<br />
ridian passing through f(x) northward, until the point reaches<br />
the equator (think of P as the South pole); the function f<br />
o<br />
will be kept unchanged everywhere else). As a consequence, we may<br />
strengthen (5.8.) and suppose that :<br />
2~-4i S2s<br />
(5.8.') f(X 2~ ~V 4i IB 2~-4i)_ _c IB+ c<br />
This last homotopy does not modify the isomorphism class of f ~.<br />
At this point, it is very easy to describe the bundle f~<br />
For, we think of ~ as obtained by clutching together two product<br />
bundles<br />
2s ~n pr~ 21-4i<br />
IB IB<br />
2s<br />
along the equator ~IB_ by a clutching function :<br />
2~-4i<br />
: ~IB § GL (n,~)<br />
As f carries V 4i IB 2s onto IB 2Z-4i , and the complemen-<br />
tary onto B+ Z-4i- , and as ~ has a product bundle structure on
Teleman 709<br />
IB 2s f~ is a product bundle both over V 4i IB 2s<br />
x and<br />
9<br />
i~complementary. If we take into account (5.7.), we can see that<br />
f*~ is obtained by clutching together two product bundles with<br />
fiber ~n<br />
(V 4i x IB 2~-4i) x ~N pr~ V4ix IB 2~-4i<br />
(X2s V 4i x IB2s x ~N<br />
with a clutching function f~ :<br />
v4i -4i<br />
f*~ : x~IB 2 § GL(n,~)<br />
(f~q0) (v,x) = ~(x)<br />
pr~ X2s ~V 4i IB 2s<br />
+<br />
We are preparing to use excision for showing that Index Df~<br />
depends only on V 4i and ~ For, let X ~2s be<br />
~2s = v4i $2s ;<br />
~2z may be thought as glueing together V 4i x IB 2s and<br />
V 4i IB~ s along V 4i a IB 2Z-4i_ by using the identity map-<br />
ping. Let ~ § ~2~ be the bundle obtained by clutching together<br />
the product bundles with fibre ~r over the two pieces<br />
2~-4i v4i 2~-4i<br />
V 4i x IB , along x a IB , and with the clutching<br />
isomorphism<br />
therefore<br />
V 4i a IB 2s<br />
: x ~ GL(n,~)<br />
~(v,x) = ~(x) ;<br />
(5.9) ~ = I , ~ ,<br />
V 4i<br />
where ~4i + V4i is the product bundle of rank 1 o<br />
We are almost ready to invoke the Corollary 3.6. for :<br />
M1 = X2s , M2 = ~2s<br />
~i = f~ ' 62 = ~<br />
el= If* l , o2 =
710 Teleman<br />
U = V 4i IB 2s , and<br />
W = Int U<br />
of course, U and W so chosen are not exactly what we need;<br />
notice, however, that these two sets are necessary to show the<br />
existence of bundle isomorphisms A21,A21,EI,E 2 which make a com-<br />
mutative diagram (3.2); this is a homotopic problem in nature and,<br />
if we can solve it for this choice of U and W we can solve it<br />
for correct U and W obtained by thickening. This saves us<br />
some more notation.<br />
It will be convenient to say that the product bundles<br />
used to describe f*$ and ~ are coo~dinas charts (in the sense<br />
of Steenrod [16]); those two product bundles with bases<br />
V 4i IB 2s will be called first coordinate charts, and they<br />
will be denoted (f ~)i ' rasp. ~i ; the other two charts will<br />
be called second coordinate charts, and will be denoted (f ~)2'<br />
rasp.<br />
~2 " The same conventions and notations will apply to the<br />
product bundles If*s1, I~I- thought as obtained by coordinate<br />
charts, respectively with the same bases, but with clutching func-<br />
tions equal to the identity.<br />
Now we are in a position to define the isomorphisms<br />
A21,A21, EI,E2 involved in the Corollary 3.6.<br />
We set :<br />
A21 = identity : (f*~)l + ~i<br />
A~I = identity : If*~.ll §<br />
l I = identity : (f*~)N§<br />
12 = identity : ~2 +I~12<br />
Notice that on U ~W = V 4i ~IB 2~-4i these isomorphisms are<br />
compatible with the clutching functions, and hence the diagram<br />
(3.2) commutes for these bundles and isomorphisms.<br />
From Corollary 3.6, (5.9) and Theorem 5.2, we get<br />
+ +<br />
Index = Index D =<br />
Df*E-If*~ I 7- I~' I<br />
+ +<br />
(5.10) Index DIV[]~_IIv~ ] = Index Dlv [](~-]~I) =
Teleman 711<br />
+ .Index D +<br />
Index D~v ~-[~I ;<br />
the bundle ~-I~I is over a sphere ; therefore, Theorem 3.2<br />
and Theorem 5.3. will give :<br />
ther :<br />
Index D + = Sig V 4i 2 i-2i Ch(~- I~I)[S 2~-4i] =<br />
= Sig V 4i 2 ~-2i Ch $ ([$2~-4i])<br />
From the Hirzebruch-Thom formula (Theorem 4.2) we get fur-<br />
Index D + = 2 s<br />
f*~-I f*E I<br />
Ch(f*~)(L2s i (X 2s ))<br />
or, using again the Theorem 3.2 and Theorem 4.2. :<br />
+<br />
Index D<br />
f<br />
= Rank ~'Sig X 2s + Ch(f*~) (2s =<br />
= Ch(f*~) (Lo(X2s + Ch(f*~) --s (2 L2s (X2~)) =<br />
Ch(f*~) ( Z 2 ~-2i (X 2~<br />
= L2s i ) ) ;<br />
i=O<br />
here we have used the Hirzebruch signature formula (Proposition<br />
4.2 (ii)), and the fact that f*~ is an aspherical bundle. This<br />
proves the index theorem for aspherical bundles.<br />
5...3 Reduction to the aspherica ! case. Let ~ 6 K~<br />
be such that Ch ~ is homogeneous. From Proposition 4.4. we know<br />
that ~ ~ g2~ s K~163 xS2~) ~ is aspherical; Theorem 5.1.<br />
for aspherical elements applies, and we have :<br />
Index D+m82s (5_1) Ch(~82s ( E<br />
i=O<br />
22s163 (X2s S 2s )<br />
IITheorem 5.2) ll(ch and L, are multiplicative<br />
Index D~-Index D +<br />
e2s<br />
II (Theorem 5.3)<br />
Index D~- + 24 .<br />
00<br />
(Ch~) E (2%-2iL2s -<br />
i=O<br />
Ch~( ~ 2s163 j(S 2s<br />
j~<br />
II (Ch82s ( ~ $2 s ~] )=l'The~ 5.3,Prop.4.2)<br />
2Z-2i L 2 s<br />
Ch ~ (i=EO 2s (X2s "
712 Teleman<br />
which proves the Index Theorem for signature operators in the whole<br />
generality.<br />
6. Proof of Theorem 5.2.<br />
6.1. Review of Baaj-Julg [4] and Hilsum [9]results.<br />
S. Baaj and P. Julg [4] defined the elements of Kasparov's [II]<br />
KK(A,B)-groups in terms of unbounded operators. Kasparov's cons-<br />
truction applies to oth-order operators, while the Baaj-Julg cons-<br />
truction is very natural for ISt-order operators ; the important<br />
examples of operators arising from geometry are operators of order<br />
> O For this reason at least, the Baaj-Julg construction is<br />
very important. In a consecutive important development along these<br />
lines, M. Hilsum [9] shows that the signature operators D +<br />
constructed by the author in [19] fit naturally into the Baaj-<br />
Julg construction, and gives functional-analysis new proofs to two<br />
basic theorems from [19] and [20] .<br />
For what we are concerned, the group KK(C~(X),~) suffi-<br />
ces our needs (C~ = C -algebra of continuous complex-valued func-<br />
tions on X) ; it is isomorphic to Ko(X ) , the K-homology group<br />
of X in dimension 0 .<br />
Let A be a C~-algebra. A cycle into the Baaj-Julg<br />
KK(A,~)-group consists of a triple (H,~,D) , where :<br />
i) H is a ~2-graded, i.e. H = H + ~ H- , infinite dimensional<br />
separable Hilbert space,<br />
2) 9 : A § L(H) is an unital C~-algebra homomorphism, compatible<br />
with the graduation of H , i.e. ~(a) (H c H , Va 6 A ,<br />
m<br />
3) D is an unbounded linear operator on H such that<br />
a) D(D), D(D ~) are dense in H (D = domain)<br />
b) D is closed<br />
c) 1 + D~D has dense range in H<br />
d) D is an operator of degree I, i.e. D(D(D) n H c H ~<br />
4) D = D ~<br />
5) ~(a) (I+D2[ 1 is a compact operator, for any a s A ,<br />
6) Do~(a) - ~(a) oD is an operator which can be extended to a<br />
bounded operator on H for any a s A D , where A D is a dense
Teleman 713<br />
subalgebra in A<br />
perty.<br />
The requirement 5) is equivalent to a Rellich-type pro-<br />
To any Baaj-Julg cycle (H,~,D) there corresponds a<br />
Kasparov cycle [ii] by the canonical mapping :<br />
B : (H,~,D) + (H,~,D(I+D2) -I)<br />
notice that D(I+D2) -I is a bounded, self-adjoint, Fredholm ope-<br />
rator of degree i. The index of this cycle is by definition :<br />
Index{ (D(I+D2)-IIH+) : H + § H-}<br />
Kasparov defines a pairing (external product) :<br />
KK(AI,~ ) x KK(A2,~ ) § KK(A 1 ~ A2,~ )<br />
The Kasparov external product is difficult to perform, but the<br />
corresponding Baaj-Julg product is very simple, because it is al-<br />
gebraic :<br />
THEOREM 6.1. (Baaj-Julg [5]) Let (Hi,~i,Di) , i = 1,2<br />
be Baaj-Julg cycles over the C*-algebras A 1 , resp. A 2 .<br />
(i) Then the external product given by the formula :<br />
(HI,~I,DI)) (H2,~2,D 2) = (H 1 ~ H2,~I ~2,DI @i + 1 QD2)<br />
is~ s~till a ~aaj-Julg cycle, and via B , it corresponds to<br />
Kasparov's external product; in addition,<br />
(ii) Index{(Hl,~l,Dl) x (H2,~2,D2) } =<br />
= Index(Hl,~l,Dl).Index(H2,D2,~2 )<br />
The statement (ii) follows from the fact that it is true<br />
for the Kasparov multiplication, and from (i).<br />
THEOREM 6.2 (M. Hilsum [9]) For any compact oriented<br />
Lipschitz manifold of even dimension,<br />
{W+o ~ wo , multiplication by Lipschitz functions, D + ~ D-}<br />
is a Baaj-Julg cycle over the algebra of continuous functions.<br />
The same result holds for signature operators with values<br />
in a vector bundle.<br />
6.2. Let Xi,~i , i = 1,2, be as in the Theorem 5.2. We
714 Teleman<br />
consider the corresponding signature operators D + i = 1,2. We<br />
~i '<br />
can keep track of these operators by considering tlie graduated<br />
Hilbert spaces<br />
W O(xi,~i ) = W~(Xi,~i ) @ Wo(Xi,~i ) ,<br />
in which W+(Xi'gi ) o , resp. Wo(X i,~i ) , has degree O, resp. i, and<br />
D~i = D+~i ~ D~i- is thought as an operator of degree 1.<br />
A direct calculation still valid for differential forms<br />
with values in vector bundles, see [14], Ch. XV, w 6, Lemma 2,<br />
involving the information given by the axiom A.II, shows that:<br />
(6 .i)<br />
where<br />
X 1 with values in ~i :<br />
@ ~r(x I) @A(XI ) A($ I) ;<br />
r<br />
on the differential forms of degree r , ~i<br />
by (-i) r , see [14], p. 230.<br />
we get<br />
= 81 + ~i @ ,<br />
D~im~ 2 D~ 1 D~ 2<br />
~l is an involution on the space of differential forms on<br />
From (6.1) and from the fact that ~i<br />
(6.2) D~Im~2 = ~I~176 | 1 @ I | D~2]<br />
is the multiplication<br />
is an involution,<br />
~i is a hermitian isomorphism which preserves the graduation, so<br />
~lOD~l is still a self-adjoint, degree 1 operator.<br />
We know from Theorem 6.2. and Theorem 6.1 (i) that the<br />
bracket from (6.2) is the operator-component of a Baaj-Julg cycle;<br />
by invoking hhat ~i is an isomorphism, and the Theorem 6.1 (ii),<br />
we get<br />
Index D~I~2 = Index[(~l~ | i + I | D~2J=<br />
which proves Theorem 5.2.<br />
= Index(~lOD l).Index D~2 =<br />
Index "Index<br />
= D~I D~ 2
Teleman 715<br />
w Admissible Pseudomanifolds vs.HT. In this sec-<br />
tion, we intend to check, briefly, (A.I.) - (A.12.) for admissible<br />
pseudomanifolds. The fulfillment of these properties for these<br />
spaces constitutes an important series of results due to<br />
J. Cheeger [5] , [5'], [6] , [7] and M. Goreski-MacPherson [8].<br />
(A.I.) Recall [6], that a closed pseudomanifold of<br />
dimension n is a finite simplicial complex of dimension n in<br />
which any (n-1)-simplex is a face of precisely two n-simplice~<br />
(A.2.) If M n is a pseudomanifold of dimension n ,<br />
let M (k) denote its K-skeleton . M n may always be thought<br />
as a smooth Riemannian manifold away from the codimension two<br />
skeleton of it. It is possible to introduce natural metrics on<br />
M n , e.g. piecewise flat metrics, which are intimately related<br />
to the geometry of M n . Such metrics define the required Borel<br />
measure.<br />
(A.3.) It makes sense, then, to speak about smooth<br />
differential forms on M with support in M n\ M (n-2) . Here,<br />
the classical operators d , , , 6 , A are defined.<br />
~i(M) is, by definition, the space of L2-forms of<br />
degree i on M , obtained by the completion of the space of<br />
smooth differential forms with support away from the codimension<br />
two skeletons.<br />
(A.4) 9 The operator d i is densely defined in h i<br />
let d i denote also its closure, and ~d i its domain.<br />
i and homomorphisms d i<br />
The spaces ~d ' , i = 0,1,2,...,<br />
form the L2-cohomology complex ~(M)<br />
If the pseudomanifold M satisfies an extra local<br />
property regarding the metric geometry of the link of any simplex<br />
in M , the pseudomanifold M is called admissible, see [6] ,<br />
p. 127.<br />
9 For example, the pseudomanifolds having only<br />
even codimension strata, see [8], and cone-like metrics times<br />
flat metrics, are admissible.
716 Teleman<br />
The homology Hi(~(M)) , for M an admissible pseudo-<br />
manifold, is the dual of the middle homology group IH~(M n)<br />
defined in [8], see theorem 6.1. [6].<br />
tiable functions on M<br />
(A.5.) A(M) is, e.g. the algebra of PL-differen-<br />
(A.6.) The operator * , densely defined on smooth<br />
forms with compact support in M n \ M (n-2) , extends isometrical-<br />
ly to the L2-spaces.<br />
(A.7.) This is the property B) w [6] , p. 95 ; it<br />
is verified in the proof of Theorem 5.1. [6]<br />
(A.8.) The L2-cohomology is represented by harmonic<br />
forms. The *-operator induces isomorphisms between the spaces<br />
of harmonic forms in complementary dimensions. The pairing (A.8.)<br />
is given by integration<br />
M<br />
this pairing is clearly nondegenerate because the ,-operator is<br />
an isomorphism.<br />
(A.9.) This follows from Theorem 3.1. [5] or Theorem<br />
3.3. [5'3, which states that there exists an orthonormal system<br />
of eigenforms for the Laplacian, in any dimension, and the corres-<br />
ponding eigenvalues converge to +~ . The Rellich lemma is one of<br />
the basic ingredients of the Baaj-Julg cycle.<br />
(A. IO.), (A.IO'.). The middle intersection (co)homo-<br />
logy of manifolds, is isomorphic to the singular (co)homology,<br />
ef. 4.3. [8].<br />
For the transversality arguments, at least for pseudo-<br />
manifolds with even codiemnsion strata, see w [8]<br />
(A.11.) , (A.12.). This is Theorem 7.1. [7]<br />
bundle<br />
X 2s<br />
To summarize, we have the :<br />
Theorem 7.1. (J. Cheeger [7]). For any complex vector<br />
over the admissible closed, oriented pseudomanifold
Teleman 717<br />
+<br />
Index D = (Ch ~) (<br />
n i= O<br />
0o<br />
2s163 ))<br />
The author is in debt to Jeff Cheeger for useful<br />
discussion regarding pseudomanifolds.<br />
w Appendix. The multiplicativity property of the<br />
L-classes follows from two basic facts<br />
r.<br />
(i) , (ii) explained here.<br />
(i) If f. : M 9 § S 1 , i = 1,2, are two mappings,<br />
1 1<br />
consider the mapping<br />
r1+r 2<br />
fl ^ f2 : MI M2 § S<br />
where<br />
with<br />
fl ^ f2 = p o (flxf2) ,<br />
r I<br />
p : S x S r2 § S rl A S r2 = S r1+r2<br />
is the projection mapping; here A denotes the 6mash product<br />
rl = r2 r 2<br />
S ^ S r2 S rl x S /{a I} x S U S rl x {a2} ,<br />
r.<br />
1<br />
a. 6. S<br />
1<br />
If f.<br />
1<br />
is transverse regular to<br />
r.<br />
1<br />
x 9 6 S , and<br />
1<br />
x i r a i , i = 1,2, then fl ^ f2 is transverse regular to the<br />
point (Sl,X 2) E S rl ^ S r2 , and<br />
I .<br />
3.<br />
(fl A f2)-1(X1'X2 ) = f11(Xl ) x f21(X2 )<br />
(ii) The KHnneth formula (A.12.)<br />
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718 Teleman<br />
3'.<br />
4.<br />
5.<br />
5f o<br />
6.<br />
7.<br />
8.<br />
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Teleman 719<br />
18.<br />
19.<br />
20.<br />
21.<br />
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folds, Acta Mathem., Vol. 153 (1984), 117-152.<br />
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des Vari4t4s Triangul4es, Symp. Int. Top. Alg. Mexico,<br />
(1956), 54-67.<br />
Department of Mathematics<br />
State University of New York<br />
Stony Brook, New York 11794<br />
U.S.A.<br />
Submitted: January i, 1985