FRITZ NOETHER (1884 - 194?)

FRITZ NOETHER (1884 - 194?)

The index of a linear operator may be defined as the
difference between the dimension of its kernel and the co-
dimension of its image. The index of an integral operator was
first introduced by Fritz Noether in his important pioneering
paper "Uber eine Klasse singul~rer Integralgleichungen,"
Math. Ann. 82 (1920), 42-63. In the same paper he proposed the
first index formula. During the 65 years which have passed
since then, the topic of the index has become one of the central
topics in mathematics.
This issue of Integral Equations and Operator Theory
is dedicated to the memory of Fritz Noether. It contains a
selection of up-to-date papers on the index theory. Included
is the biography of Fritz Noether with its very tragic end.
This biography is written by his son for this issue. There is
also a list of his publications compiled by A. Dick.
J.A. Dieudonn~ has written for this issue a paper about the
early history of the index theory.
I wish to thank all authors for their contributions
and cooperation.
I. Gohberg

Integral Equations
and Operator Theory
Vol. 8 (1985)
FRITZ NOETHER (1884 - 194?)
Gottfried E. Noether
0378-620X/85/050573-0451.50+0.20/0
9 1985 Birkh~user Verlag, Basel
Fritz Noether was born on October 7, 1884, in the
university town of Erlangen in southern Germany, where his father,
Max Noether, was University Professor of Mathematics. Though
Erlangen was the smallest of the three universities in the State
of Bavaria, it had a high reputation in the field of mathematics.
In 1872, in his inaugural lecture, Felix Klein had developed
what became known as the "Erlanger Programme." Two years later,
Paul Gordan, who was to become known as the "King of Invariants,"
came to Erlangen. He was joined one year later by Max Noether,
at first as ausserordentlicher (associate) professor, and from
1888 until his retirement in 1918 as Ordinarius (full professor).
The NSther ancestors had been well-to-do Jewish
tradesmen in the Black Forest area of Germany. (The spelling of
the name was changed to the present spelling by Max.) In 1838,
two brothers, Joseph and Hermann, moved to Mannheim to start a
wholesale iron business that remained in NSther hands for i00
years until its "aryanization" in 1937 by the Nazi regime. Of
the three sons of Hermann NSther, two continued in the family
business, but the third son, Max, handicapped since the age of
fourteen from the effects of infantile paralysis~ turned to the
study of mathematics, an interest that seems to have come from
his mother's family, Jewish merchants in Mannheim. In 1868,
I am greatly indebted to Dr. August Dick in Vienna, author
of Emmy Noether (1885 - 1935), Boston: Birkhauser, 1981,
for compiling detailed information about the life of
Fritz Noether, partfedlarly his early life.

574 Noether
after study in Mannheim, Heidelberg, Giessen, and G~ttingen,
Max Noether was awarded the Ph.D. in Mathematics from the
University of Heidelberg. After five years as Privatdozent in
Heidelberg during which he began to establish his reputation as
an algebraic geometer, he was called to Erlangen, where he would
stay until his death in 1921.
In 1880, Max Noether married Ida Kaufmann, who came
from a well-to-do Jewish family in the Cologne area. They had
four children. Emmy, the oldest, born in 1882, was to become
Paul Gordan's only doctoral student. Of her, Edmund Landau
would say that "she was the coordinate origin of the Noether
family."
The third child of Max and Ida Noether was Fritz.
Fritz received the customary education for boys of the bourgeois
middle class, elementary school from age six and humanistic
Gymnasium from age ten. In 1903, he passed the Abitur entitling
him, after fulfilling his military duty, to admission to the
university. Starting with the winter semester 1904/5, he
studied mathematics and related subjects for nine semesters,
first at the University of Erlangen and then in Munich. In
March 1909, he defended his doctoral dissertation Uber rollende
Bewegung einer Kugel auf Rotationsfl~chen. His Ph.D. adviser
at the University in Munich was Professor A. Voss. But in the
dissertation, the candidate gave equal thanks to Professor
A. Sommerfeld for valuable inspiration. A close working relation-
ship between Fritz Noether and A. Sommerfeld is indicated by the
fact that Part 4 (Die technischen Anwendungen der Kreisel-
theorie) in F. Klein's and A. Sommerfeld's Uber die Theorie des
Kreisels, published in 1910 by Teubner in Leipzig, bears the
notation: Prepared for publication and supplemented by Fritz
Noether. Indeed, Sommerfeld's introductory remarks refer to
Fritz Noether as collaborator for this fourth and final part of
the Klein and Sommerfeld work on gyroscopes.
While Max and Emmy Noether were primarily interested
in pure mathematics, Fritz Noether embarked on a career in

Noether 575
applied mathematics and mechanics. After postdoctoral studies in
G~ttingen, he became assistant to Karl Heun at the Technische
Hochschule in Kar!sruhe. In the summer of 1911, on presentation
of Uber den G~Itigkeitsbereich der Stokesschen Widerstandsformel,
he obtained the venia legendi (permission to lecture) and was
appointed Privatdozent for Theoretical Mechanics and Mathematics.
In December 1911, he married Regina W~rth of Randegg on the
Bodensee, daughter of a customs official. There were two sons,
Hermann, born September 21, 1912, and Gottfried, born January 7,
1915.
During World War I, Fritz Noether saw active service on
the German-French front and, after being wounded, was assigned to
doing research in ballistics. At the end of the war, he returned
to teaching at the Teehnische Hochsehule in Karlsruhe, where in
the meantime he had been promoted to associate professor. In
1921 he took a leave of absence to do industrial research for
the Siemens-Konzern in Berlin. In the fall of 1922 he accepted
a full professorship at the Technische Hochschule in Breslau,
which he held until 1934, when, though not yet 50, he was forced
into retirement for "racial reasons." He thus shared Emmy
Noether's fate, whose permission to teach - along with that of
several other Gottingen mathematicians - had been withdrawn one
year earlier. While Emmy Noether accepted a guest professorship
at Bryn Mawr College in the United States, Fritz Noether decided
to accept the offer of a professorship at the Institute of
Mathematics and Mechanics at the University of Tomsk in the USSR.
In the summer of 1934, brother and sister met for a last time
before departing in opposite directions. Less than a year later
Emmy Noether died very suddenly in Bryn Mawr following a tumor
operation. Fritz Noether was the honored guest at the Emmy
Noether Memorial Session held September 5, 1935, in Moscow with
P.S. Alexandrov, then President of the Moscow Mathematical
Society, delivering the eulogy. Barely two years later, in
November 1937, at the height of the Stalinist purges, Fritz
Noether was arrested by the NKVD in his home in Tomsk.

576 Noether
Informa about Fritz Noether after his arrest is
fragmentary. At some time, most likely before the end of 1939,
he was in Orel. In December 1939 and January 1940, he was seen
in Butyrka Prison in Moscow. There is a report that he was seen
in the center of Moscow toward the end of 1941 or the beginning
of 1942. But numerous attempts, official and unofficial, to
find out more about his fate remain unsuccessful.
At the time of his arrest, Fritz Noether had completed
the manuscript of a book on Bessel functions. The manuscript
had been translated into Russian and was scheduled for publica-
tion in the Soviet Union. It is not known whether the book has
ever been published.*
The University of Connecticut
Storrs
CT 06268
U.S.A.
The author would appreciate receiving any relevant
information

Integral Equations
and Operator Theory
Vol. 8 (1985)
i.
2.
3.
4.
5.
6.
7.
8.
9.
i0.
Ii.
12.
13.
14.
15.
LIST OF PUBLICATIONS OF FRITZ NOETHER
0378-620X/85/050577-0351.50+0.20/0
9 1985 Birkh~user Verlag, Basel
(Assembled by Dr. Auguste Dick, Vienna)
Uber rollende Bewegung einer Kugel auf Rotationsfl[chen,
Leipzig, Teubner, 1909. Inaugural Dissertation
Zur Kinematik des starren K~rpers in der Relativtheorie,
Annalen der Physik (4) 31 (1910~ 919-944
Uber den G[itigkeitsbereich der Stokesschen Widerstands-
formel, Leipzig, Teubner, 1912. Habilitationsschrift
Z.f. Mathematik und Physik, 62 (1914) 1-39
Uber die Entstehung einer turbulenten Fl[ssigkeits-
bewegung, Sber. Bayerische Akad. Wiss 1913, 309-329
Zur Theorie der Turbulenz, Jber. DMV 23 (1914),
138-144
Zur Theorie der Turbulenz, Nachr. Ges. Wiss. G@ttingen,
1917, 199-212
Uber analytische Berechnung der Geschosspendelungen,
Nachr. Ges. Wiss. G6ttingen, 1919,373-391
(Uber Geschosspendelungen) ~
Artilleristisehe Monatshefte, Mai/Juni, 1919, 170-204
Bemerkung ~ber die L@sungszahl zueinander adjungierter
Randwertaufgaben bei linearen Differentialgleichungen,
Sber. Heidelb. Akad. Wiss., 1920, 1-12
Uber eine Klasse singul~rer Integralgleichungen,
Math. Ann. 82 (1920), 42-63
Das Turbulenzproblem, ZAMM i (1921), 125-138; 218-219
Uber Stromaufnahme in Metallrohrleitungen und
verwandte Erdungsfragen, Wiss. Ver@ff. aus dem
Siemenskonzern, i (1921), 35-60
Uber die Abstimmung der L6sehdrosseln I, Elektroteehn.
Zeitsehr. 1921, 1478-1482
Uber die Abstimmung der L6sehdrosseln II, Elektrotechn.
Zeitschr. 1922, 385-388
Uber eine Aufgabe der Kapazit[tsberechnung, Wiss.
Ver@ff. aus dem Siemenskonzern, 2 (1922), 198-202
ohne Titel erw~hnt in (7), S. 17
Titel wie (7)[12.3.82 A.D.]

578 Dick
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
Uber ein Problem der Strom!eitung , ZAMM 2 (1922),
274-278
Zur asymptotischen Behandlung der station[ren LSsungen
im Turbulenzproblem, ZAMM 6 (1926), 232-243~ 497-498
Uber Stromverdr[ngung in zylindrischen Leitern von
allgemeiner Querschnittsform, Annalen der Physik (4)
84 (1927), 775-778
Bestimmung der Stromverteilung in zylindrischen
Leitern von allgemeiner Querschnittsform, ZAMM 7
(1927), 452-454
Uber den "Strahl" begriff in der Fresnelschen und der
Maxwellschen Krystalloptik, Probleme der modernen
Physik, Hsg. P. Debye, Leipzig, Hirzel, 1928, 58-64
Berechnung von elektrisehen StrSmungsfeldern,
Handbuch der Physik, 13 (1928) 76-102
Kerl Heun T (Naehruf), ZAMM 9 (1929) 167-171
Bemerkungen zur 0seenschen Hydrodynamik, ZAMM 9 (1929)
5O9
Zur statistischen Deutung der K~rm~nschen Ahnlich-
keitshypothese in der Turbulenztheorie, ZAMM, ii
(1931) 224-231
Uber die Verteilung des Energiestroms bei der Total-
reflexion, Annalen der Physik (5) ii (1931), 141-146
Ausbreitung elektrischer Wellen 6ber der Erde
Funktionentheorie und ihre Anwendung in der Technik,
Hsg. R. Rothe u.a. Berlin, Springer, 1931, 154-170
Integrationsprobleme der Navier-Stokesschen Differen-
tialgleichungen, Handbuch der Physikalischen und
techn. Mechanik, Bd. 5, 3. Leiferung, Leipzig, Barth,
(1931), 719-796
Anwendung der Hillschen Differentialgleichung auf
die Wellenfortpflanzung in elektrischen oder
akustischen Kettenleitern, Verh. des 3. Internat.
Kongr. f. Techn. Mechanik, Stockholm, 3 (1931), 143-149
mit E. Waetzmann: Uber akustische Filter, Annalen
der Physik (5) 13 (1932), 212-228
Dynamische Gesichtspunkte zu einem statistischen
Turbulenztheorie, ZAMM 13 (1933), 115-120

Dick 579
31.
32.
33.
34.
35.
36.
Bemerkungen 6ber das Ausbreitungsgesetz f6r lange
elektrische Wellen und die Wirkung der Heavisides-
chieht, Elektr. Nachrichtentechnik, i0 (1933), 160-172
Das station[re (und quasistation[re) elektromagnetische
Feld Differential- und Integralgleichungen der
Mechanik und Physik, hsgg. von Phillip Frank und
Richard von Mises. 2., vermehrte Auflage, zugleich 8.
Aufl. von Riemann-Webers partiellen Differential-
gleichungen der mathematischen Physik. Vieweg,
Braunschweig, 1935; im 2. (physikal.) Teil 627-755,
Unver[nderter Nachdruck, New York, Dover Pub.,
Braunschweig, Vieweg, 1961"
Elektromagnetische Wellen an einem Draht bei konzen-
trierter Energiequelle, Physikal. Zeitschr. der
Sowjetunion, 8 (1935) (1-24)**
Uber die Rekursionsformeln der Besselschen und
Hermiteschen Funktionen. (In russischer Sprache mit
deutscher Kurzfassung), Mitt. des Forschungsinstituts
f~r Mathematik und Mechanik an der Kujbyschew-
Universit~t Tomsk, i (1935), 120-125
Asymptotische Darstellungund geometrische Optik
(In deutscher Sprache mit russischer Kruzfassung)
Mitt. des Forschungsinstituts fir Mathematik und
Mechanik an der Kujbyschew-Universit~t Tomsk, i
(1935), 175-189"**
Uber elektrische Drahtwellen, C.R. Congr. Internat.
Math., 0slo, 2 (1937), 234-235
Zweiter/physikalischer/Teil, unter Mitarbeit von
G. Beck-Kansas (USA), R. F6rth-Prag, R. v. Mises-
Istanbul, F. Noether-Tomsk, G. Schulz-Berlin,
A. Sommerfeld-M~nchen, E. Trefftz-Dresden
Paginierung nach dem Separatum
gezeichnet ist diese Arbeit mit: Tomsk, 25. 2. 1936
und enth[it einen Hinweis des Autors auf ein demn[chst
in russischer Sprache erscheinendes Buch 6ber
Besse!sche Funktionen.

Integral Equations
and Operator Theory
Vol. 8 (1985)
0378-620X/85/050580-I051.50+0.20/0
9 1985 Birkh~user Verlag, Basel
THE INDEX OF OPERATORS IN BANACH SPACES
J.A. Dieudonn@
The notion of index of an operator is a central one in
the theory which leads to the famous Atiyah-Singer theorem, one
of the highlights of modern Global Analysis. In this paper I
want to give an historical sketch of the development of that
notion during the first half of the 20th century, providing the
tools of "Functional Analysis" used in the Atiyah-Singer proof.
I. The Fredholm Alternative
The fundamental result proved by I. Fredholm in 1900 is
called the "Fredholm alternative." Let I = [0,i] and suppose
for simplicity that we start with a continuous "kernel" K(s,t)
in I x I. Consider the linear integral operator K in the
space C(~) of continuous functions, defined by
(i) (K.~)(t) = flK(s,t)~(s)ds ;
then the kernel N = Ker(l + K) of the operator I + K in
C(I) has finite dimension, and the image F = Im(l + K) is the
space of functions ~[C(I) satisfying the "orthogonality
relations"
(2) f1~(t)ki(t)dt = 0
for the functions k. forming a basis of the kernel N' of the
i
"transposed" integral operator I + K' , where
(3) (K'.~)(t) = fiK(t,s)~(s)ds 9
Furthermore the dimensions of N and N' are equaZ, which
implies that F is a closed subspace of C(1), of finite
codimension given by
(4) eodim F = dim N
This result did not seem at all surprising, since relation (4)
was well known for endomorphisms of finite dimensional spaces,
and I + K appeared as a kind of "limit" of such operators.

Dieudonn4 581
II. The Riesz-Fredholm theory of compact operators.
Already Fredholm had seen that his results would still
be valid if it was only assumed that Ix-yleK(x,y) is continu-
ous for some e < i After the introduction of the Hilbert
space L 2, it was easily seen that the "Fredholm alternative"
also held when Ks X I) and the operator K in L2(1) is
defined by (i) for ~6L2(I).
The big step forward in the theory of these operators
was taken by F. Riesz. He was inspired by Hilbert's theory of
"bounded operators" in Z2, and in particular by his concept of
"completely continuous" operators, which included the integral
operators K defined by (i) for a symmetric kernel, but were
more general. Hilbert's definition was given in the language of
bilinear forms~ F. Riesz translated it in terms of operators:
at first, for him, a completely continuous operator on L 2 (or
in L p for i < p < + ~) was defined by the condition that it
transforms aweakly convergent sequence into a strongly conver-
gent one. Then in 1915, he realized that this definition is
equivalent to the following one: the operator transforms any
bounded set into a relatively compact one. But then that defini-
tion obviously can be extended to a linear map u: E -> F when
E and F are arbitrary normed spaces, and this justifies the
name compact operator, later adopted for such maps. In a beauti-
ful paper published in 1918, F. Riesz showed that the Fredholm
theorems were valid for any endomorphism v = i + u of the space
C(1), where u is a compact endomorphism; at that time
general normed spaces had not yet been defined, but Riesz's
arguments apply without change to an arbitary normed space.
For future reference, it is convenient to rapidly
describe Riesz's procedure (which, even for the integral opera-
tor (i), gives more information that Fredholm's). Initiating
in Analysis, a method first used in algebra by E. Weyr in 1890
to obtain the Jordan normal form of a complex matrix, he
considers, for the iterated operators v m, the subspaces
F m = Im(v m) and N m = Ker (vm). By an elementary (but clever)

582 Dieudonn4
use of the definition of compactness, he shows that the increas-
ing sequence
(5) 0cNICN2 c'''cNmc''"
consists of finite dimensional subspaces, and the decreasing
sequence
(6) E~FI~F2 ~'''~Fm~''"
consists of closed subepaces of finite codimension.
Furthermore, both sequences are stationary: there is
a smallest integer k for which Nk+ j = N k and Fk+ j = F k for
all j > 0 ; E is the topological direct sum F k ~ N k ; the
restriction of v to F k is an automorphism of F k , and the
restriction of v to N k is a nilpotent endomorphism of that
finite dimensional space, such that v m is not identically 0
k
in N k for m < k, but v = 0.
Several authors later found more general endomorphisms
of Banach spaces for which the preceding decomposition holds;
for these operators v, one therefore always has the relation
(7) dim Ker (v) = codim Im(v) (= dim Coker(v))
The most interesting ones were described by Ruston in 1954; in
the Banach space End(E), let K be the closed subspace con-
sisting of compact operators; Ruston's operators u are
"close to K ', in the fol]owing sense: if d is the distance
in End(E),
(8) lim (d(un,K)) I/n = 0.
Then the Riesz theory applies without change to these operators:
the spectrum of u is at most countable, each point h # 0 in
the spectrum is an isolated eigenvalue of finite multiplicity,
and there is a decomposition in a topological direct sum
E = F(h) ~ N(h), such that the restriction of u -X.I E to
F(1) is an automorphism of that subspace, N(~) has finite
dimension and the restriction of u-X.l E to N(X) is a nil-
potent endomorphism of N(~).

Dieudonn4 583
III. The F. Noether paper of 1921
The first paper in which appeared in a natural way an
operator v : E + F for which Ker(v) and Coker(v) are both
finite dimensional, but have different dimensions, was published
in 1921 (of course, it was later easy to concoct ad hoe examples,
such as the shift in Z2). Its author was Fritz Noether, Emmy
Noether's brother, who later was a professQr in a Russian uni-
versity and disappeared in Siberia before World War II.
In his paper he considers a "singular integral equa-
tion" which had first been studied by Hilbert, in his address to
the International Congress of Mathematicians at Heidelberg in
1904. He wanted to solve the "Riemann problem" for holomorphic
functions: given a bounded plane domain A, limited by a
smooth curve F, find a function ~ + i~ continuous in A O F,
holomorphie in A and such that on F
(9) a(s)~(s) + b(s)~(s) : f(s)
where s is a parameter varying in [-~, + z] and proportional
to arc length in F , and a, b, f are continuous periodic
functions of period 2z. Using classical properties of the
Green function in A, Hilbert showed that, if
Cl0)
k(s,t) : ~b(s).cot~-e + A(s,t)
where A
equation
is continuous, then ~ must be a solution of the
r (K.~)(s) ~ a(s)~(s) + v.p.J~K(s,t)~(t)dt- = f(s)
where v.p. means the Cauchy "principal value" of the integral
(since the integrand has a pole of order I at the point t = s).
Hilbert's solution of (i0) consists in applying again the
operator K to both sides, and proving by direct computation
that the equation
r K.(K.~) = K.f
is a Fredholm integral equation (i) with continuous kernel.
This may look like a trick, but in the light of modern Function-
al Analysis it is quite natural: K is a pseudo-differential
operator of order -i on the circle, hence I K.X is a pseudo-
differential operator of order -2, and it is well known that

584 Dieudonn4
such operators are of type (i).
Hilbert probably thought that the "Fredholm alter-
native" still held for the operator K, for when he wanted to
prove uniqueness of a solution of (Ii), he only checked that
~.~ = 0 had no nontrivial solution, whereas he should have con-
sidered instead the "transposed" equation tK.~ = 0, K(s,t)
being replaced by K(t,s).
F. Noether's discovery was that Hilbert's assumption
was not always true. He observed that if ~ is a nontrivial
solution of K.~ = 0, the corresponding holomorphic function
+ i~ should satisfy
(13) fFd(log(~+i~)) = fFd(log(a-ib)) = 2~n
where n is a positive or negative integer. If n < 0, this
would mean that ~ + i ~ has poles, in 4, so there is no
solution to K.~ = 0; on the contrary, if n Z 0, ~.~ = 0 has
2n linearly independent solutions, so that the difference
between both sides of (7) is 2n, given by the integral along F
~fFd(log(a-ib))
(first example of expression of the index by topological means).
IV. Small Perturbations
It is probably an old idea to study an equation which
"differs slightly" from another one by assuming that the solu-
tions of both equations also "differ slightly" from one another:
it is for instance the principle which has been used in celestial
mechanics since the beginning of the 18th century. In spectral
theory, very soon the same idea was found useful, for instance
in Hilbert's and E. Sehmidt's work on linear operators in
Hilbert space, which they "approximate" by operators of finite
rank; or in the investigations of G.D. Birkhoff-Kellogg and
Schauder on fixed point theorems for operators in function
spaces, with the help of Brouwer's fixed point theorems in
finite dimensional spaces.
A more general viewpoint may be found in the series
of papers which Rellich, in 193Y-1942, devoted to operators in

Dieudonn4 585
Hilbert space which depend "continuously" or "analytically" on a
complex parameter; they were essentially concerned with the way
in which the spectrum of the operator varies with the parameter.
In 1943, in connection with Bourbaki's program of study
of topological vector spaces, I was led to investigate what
happens in general when a continuous linear map u : E + F
between normed spaces is "perturbed" by the addition of a "small"
map w : E § F . I naturally gave to the word "small" the mean-
ing that the norm llwll would be small; the investigation was
chiefly directed to finding, for v = u + w, relations between
Ker(u) and Ker(v) , and between Im(u) and !m(v)
An elementary case had been known and used for a long
time: if u is an isomorphism of E onto a subspace of F ,
then the same is true for v if I I w I I < ]I u-ll I 9 In
addition, if u(E) is closed in F , and F is the topological
direct sum of u(E) and a closed subspace G , then F is also
the topological direct sum of G and v(E) (which is therefore
closed in F).
The theory of normed spaces had introduced a generali-
zation of the notion of isomorphism onto a subspace: any
continuous linear map u : E + F factorizes in
E P > E/(Ker(u)) v_~_> F
and it was natural to consider the case in which v is an iso-
morphism of E/Ker(u) onto a subspace of F ; such maps u are
now called strict homomorphisms. Even when E and F are
complete (i.e. Banach spaces), it is trivial to see that there
exist strict homomorphisms u such that v = u + w is not a
strict homomorphism for arbitrary small llwll (for instance
u = 0). In this ease Ker(u) has infinite dimension and Im(u)
infinite codimension; I realized that only by strong assump-
tions on Ker(u) or Im(u), results similar to the elementary
theorem on isomorphisms could be ohtained for strict homomorphisms.
For Banach spaces E, F, Banach had proved two deep
results on strict homomorphisms:
i) A continuous linear map u : E + F is a strict
homomorphism if and only if u(E) = Im(u) is closed in F.

586 Dieudonn4
2) If E ~ and F* are the Banach spaces dual to E
and F (with the strong topology), then in order that
Im(u) = F, it is necessary and sufficient that the transposed
t
map u : F* § E* be an isomorphism of F* onto a (necessarily
closed) Subspace of E*
For Banach spaces E,F, these theorems and the
elementary isomorphism theorem imply that if u : E § F is such
that Im(u) = F, then, for llwll small enough, one has also
Im(v) = F for v = u + w ; if in addition E is the topologi-
cal direct sum of Ker(u) and a closed subspace G, then it is
also the topological direct sum of Ker(v) and G.
None of these properties holds for non complete normed
spaces E,F, even if u is a strict homomorphism.
The main special cases of applications of these resultS
to a continuous linear map u : E § F and its perturbed map
vmu+w are:
(i) If E and F are Banach spaces, u a strict
homomorphism and dim Coker (u) < + ~ then, for I lwl I small
enough, v is a strict homomorphism and dim Coker (v)
dim Coker (u). This is not true if E and F are not complete.
(ii) If E and F are Banach spaces, u a strict
homomorphism and dim Ker (u) < + ~ then, for Ilwll small
enough, v is a strict homomorphism and dim Ker (v) !
dim Ker (u) This is still true if E and F are not
complete.
(iii) If E and F are Banach spaces and both
dim Ker (u) and dim Coker (u) are finite, this already implies
that u is a strict homomorphism; if
C14) ind(u) : dim Ker(u) - dim Coker(u) (the index
of u ) then ind(u + w) = ind(u) for I lwl I small enough (I
did not explicitly introduce the number ind(u) ).
V. Compact perturbations
Published in occupied France during the war, my paper
remained unnoticed until 1951; in that year three papers on
the index of strict homomorphisms of Banach spaces were published,
one in the United States and two in the USSR.

Dieudonn4 587
(a) The Yood paper. B. Yood first added some
interesting complements to my results (which he knew) on "small"
perturbations. He separately studied the strict homomorphisms
u : E + F between Banach spaces for which dim Ker(u) < +
(which he called "property A") and those for which dim Coker(u)
< + ~ (which he called "property B"). He gave an important
equivalent characterization of property A: for each compact
-i
set K c F , all bounded sets in u (K) are relatively
compact. To the fact that
property B for u property A for
t
u
Which I had proved in my paper), he added the similar result
t
property A for u property B for u
(the two results are not equivalent when E and F are not
reflexive).
Furthermore, he showed that if two strict homomorphisms
between Banaeh spaces u : E + F , v : F + G have property A
(resp. B), then v Ou has property A (resp. B). Conversely, if
v ~ u has property A (resp. B), then u has property A (resp. v
has property B).
The main novelty in Yood's paper is his idea of con-
sidering as a "perturbation",instead of a "small" map, a compact
map. Using his criterion for property A, he showed that if u
has property A (resp. B) and if w : E + F is compact, then
u + w has property A (resp. B)~ if u has both properties A
and B, then ind (u + w) = ind (u)
Strict homomorphisms of Banach spaces which possess
both properties A and B are now usually called Fredholm opera-
tors. Yood showed that Fredholm operators are characterized by
the following condition: there exist continuous maps v : F + E
and w : F § E such that i X - wo u and iy - uo v have
finite rank.
Finally, he observed that the technique of Weyr-F.Riesz,
which succeeds for maps v = 1 X + u , where u is a compact
endomorphism of E , cannot always be applied when 1 X is
replaced by an arbitrary a~tomorphism f of E ; he gave an
example in which the kernels of (f - u) n have dimensions

588 Dieudonn4
which increase indefinitely with n.
(b) The Atkinson and Gohberg papers. In 1948, the
British mathematician F. Atkinson submitted a paper to Mat.
Sbornik, but it was published only in 1951. He could not of
course mention Yood's paper, but he was not aware of mine either.
He independently reproved the results of Yood and mine, and added
a property which had not been proved by Yood: if both u : E § F
and v : F + G are Fredholm operators, then v c u : E + G is
also a Fredholm operator, and
(15) ind (v o u) = ind (v) + ind (u)
In the same year, 1951, I. Gohberg independently obtained the
same results, and announced them in a Doklady Note.
VI. Further results
The theorems published in the three preceding papers
are the only ones needed for the proof of the Atiyah-Singer index
formula, where they are applied to Sobolev spaces. But many
papers Were published after 1951, extending these theorems in
various ways, in particular to unbounded operators and to more
general topological vector spaces, such as Fr6chet spaces. We
mention some of them in the Bibliography.
BIBLIOGRAPHY
i. Atkinson, F.V.: The normal solubility of linear
operators in normed spaces, Mat. Sbornik, N.S.28 (70),
pp. 3-14 (Russian)
2. Dieudonn6, J.: Sur les homomorphismes d'espaces
norm6s, Bull. Sci. Math. (2), 67 (1943), pp. 72-84.
3. Gohberg, I.C.: On linear equations in normed spaces,
Dokl. Akad. Nauk, SSR (N.S.) 76 (1951), pp. 477-480
(Russian)
4. Gohberg, l.C. and Krein, M.: The basic propositions on
defect numbers, root numbers and indices of linear
operators, A.M.S. Transl. (2), 13 (1960), pp. 185-264
(Uspehi Mat. Nauk (N.S.), 12 (1957), pp. 43-118).
5. Kato, T. Perturbation theory for nullity, deficiency
and other quantities of linear operators, Journ. Anal.
Math., 6 (1958), pp. 281-322.

Dieudonn4 589
6.
7.
8.
9.
i0.
Noether, F.: Uber eine Klasse singul~rer Integral-
gleichungen, Math. Ann., 82 (1921), pp. 42-63.
Rellich, F.: St~rungstheorie der Spektralzerlegung, I,
Math. Ann., 113 (1937) pp. 600-619.
Ruston, A.F.: Operators with a Fredholm theory,
Journ. London Math. Soc., 29 (1954), pp. 318-326.
Schwartz, L.: Homomorphismes et applications eompl~tement
continues,
pp. 2472-2473.
C.R. Aead. Sci. Paris, 236 (1953),
Yood, B.: Properties of linear transformations
preserved under addition of a completely continuous
transformation, Duke Math. Journ., 18 (1951),
pp. 599-612.
10 Rue du G~n~ral Camou
75007 Paris
France
Submitted: January 7, 1985

Integral Equations
and Operator Theory
Vol. 8 (1985)
0378-620X/85/050590-2451.50+0.20/0
9 1985 Birkh~user Verlag, Basel
FREDHOLM THEORY OF WIENER-HOPF EQUATIONS
IN TERMS OF REALIZATION OF THEIR SYMBOLS
H. Bart, I. Gohberg and M.A. Kaashoek
The Fredholm properties (index, kernel, image, etc.) of
Wiener-Hopf integral operators are described in terms of realiza-
tion of the symbol for a class of matrix symbols that are analy-
tic on the real line but not at infinity. The realizations are
given in terms of exponentially dichotomous operators. The re-
sults obtained give a complete analogue of the earlier results
for rational symbols.
0. INTRODUCTION
This paper concerns the Fredholm theory of Wiener-Hopf
integral equations. Consider
(0.i) r - f k(t - s)r s : f(t), 0 _< t < ~,
0
where k(.) is an m m matrix function of which the entries are
in LI(-~,~). For 1 ~ p ~ ~, p fixed, let I - K be the operator on
Lm[0,~) defined by the left hand side of (0.I). It is well-known
p
(see [7]) that the operator ! - K is a Fredholm operator if and
only if the symbol of the equation (0.1) has no zeros on the real
line, i.e.,
(0.2) det(l m - k(l)) ~ O, -~ < I < ~.
Here k(-) is the Fourier transform of k(-). Furthermore, in that
case the Noether index of I - K is equal to the winding number
relative to the origin of the curve parametrized by
det(l m - k(~)), -~ < ~ < ~.
In the framework of the general theory additional in-
formation about I - K, its kernel, its image, etc. is available

Bart, Gohberg and Kaashoek 591
only in terms of a Wiener-Hopf factorization and the partial in-
dices of the symbol I m - k('). However, as is well-known, for
non-scalar symbols, the factorization and the indices cannot be
constructed explicitly except for some rare cases, and even then
the final answers are depending on complicated operations like
the Fourier transform and its inverse.
For the case of rational matrix symbols the difficul-
ties referred to above were overcome in [1,2], where the analysis
of the Wiener-Hopf equation (0.&) is based on a representation of
the symbol in the following form:
(0.3) ! m - k(1) : ! m + C(I - A)-IB, -~ < I < ~.
Here A is a square matrix with no eigenvalues on the real line of
which the order n may be different from m. Further, B and C are
matrices of sizes n x m and m x n, respectively. In systems theo-
ry ([i0,11], also [9,1]) the right hand side of (0.3) is called a
state space realization. By analyzing systematically the operator
I - K in terms of a realization of its symbol the following re-
sults have been obtained (see [1,2,3,6]). First of all let us
mention that condition (0.2) is equivalent to the requirement
that the matrix A x :: A - BC has no eigenvalues on the real line;
in other words, I - K is Fredholm if and only if A x has no real
eigenvalues. Assuming this to be the case, then
Ker(l - K) = {~ ~ Lm[0,~) I ~(t) Ce -itAx
= X
p
x ~ Im P n Ker pX},
9 X
L m 0,~) I f pXelSA Bf(s] d s
Im(l - K) : {f ~ P 0
c Im P + Ker pX},
Ind(l - K) = dim(Im P n Ker P - codim(!m P + Ker pX),
where P and pX are the Riesz projections corresponding to the
eigenvalues in the upper half plane of A and A x, respectively.
Furthermore (see [3,6]), a generalized inverse of the Fredholm

592 Bart, Gohberg and Kaashoek
operator I - K is given by the integral operator
with
((I - K)+~)(t) = ~(t) + f y+(t,s)~(s)ds,
0
9 x + 9 x
f iCe -ltA (I - ~ )e lsA B,
y+(t,s) = . . . .
-iCe -itA H+e IsA B,
s < t,
s > t,
0 _< t < ~,
where H+ = I - pX _ (I - Px)(p P)+PX and (pxllm P)+ is a
generalized inverse of the finite dimensional operator
pXllm P: im P + !m P.
When the symbol of (0.1) is analytic on the real line
and at infinity the above results remain true ([2,3,6]). The main
change is that for A one has to take a bounded linear operator
acting on a possibly infinite dimensional Banach space which has
no spectrum on the real line.
The main aim of the present paper is to extend the con-
structive approach described above to a more general class of
symbols, namely to matrix symbols that are analytic on a strip
containing the real line but not at infinity. This requires that
in the realization (0.3) one allows the operator A to be an un-
bounded linear operator. But then the following questions arise.
How to build the projections P and pX and how to give a meaning
to the exponential expressions appearing in the description of
the kernel, the image and the generalized inverse? To answer
these questions we have developed in [5] (see also [4]) a little
theory of exponentially dichotomous operators. In the present
paper we show that with the machinery of exponentially dichoto-
mous operators also a constructive Fredholm theory can be devel-
oped for Wiener-Hopf equations with non-rational symbols. In fact
in this way we get a complete analogue of our results in the
rational case for matrix symbols of Wiener-Hopf equations of
which the kernel k(.) is continuous one\ {0}, exponentially de-
caying and has a jump discontinuity at the origin.

Bart, Gohberg and Kaashoek 593
The paper is organized as follows. In the first section
we review the results of [4,5] about exponentially dichotomous
operators and realization as far as they are needed in the pres-
ent paper. Section 2 contains the main theorems. In Section 3 the
Wiener-Hopf integral operator is reduced to an operator whose
Fredholm properties can be analyzed easily and which appears as a
natural generalization of the indicator (see [3,6]) of the ratio-
nal case. In Section 4 the proofs of the main theorems are given.
The final section contains an application to the Riemann-Hilbert
boundary value problem.
Canonical factorization of symbols of the type con-
sidered here and the corresponding inverse formulas for Wiener-
Hopf equations on the half line and the full line appear in [5].
The study of the partial indices and non-canonical factorization
are planned for another publication.
We shall use the symbol ! to denote an identity opera-
tor. The symbol I with a subscript X stands for the identity
operator on the space X. Instead of I m we write I m.
operator
(1.1)
i. REALIZATION AND SPECTRAL EXPONENTIAL REPRESENTATION
In this paper we deal with the Wiener-Hopf integral
(I - K)~(t) : ~(t) - S k(t - s)~(s)d s, 0 s t < ~,
0
assuming its symbol I - k(-) admits a realization of the type
m
(1.2) I m - k(1) = I m + C(l! X - A)-IB, -~ < I < ~,
where X is a (possibly infinite dimensional) complex Banach space,
B: ~m § X and C: X § ~m are bounded linear operators and -iA
(X § X) is exponentially dichotomous in the sense of [4,5]. So,
by definition, A is a densely defined closed linear operator and
the space X in which A acts admits a topological direct sum de-
composition

594 Bart, Gohberg and Kaashoek
(1.3) X : x_ ~ x+
with the following properties: the decomposition reduces A, the
restriction of iA to X is the infinitesimal generator of an ex-
ponentially decaying strongly continuous semigroup and the same
is true for the restriction -iA to X+. These requirements deter-
mine the decomposition (1.3) uniquely and the projection of X
onto X_ along X+ is called the separating projection for -iA. In
case A is bounded, -iA is exponentially dichotomous if and only
if the spectrum o(A) of A does not meet the real line and then
the separating projection is just the Riesz projection correspon-
ding to the part of o(A) lying in the upper half plane. In gen-
eral, the condition that -iA is exponentially dichotomous in-
volves a more complicated spectral splitting of the (possibly
connected) extended spectrum of A. The details (including a char-
acterization of exponentially dichotomous operators in terms of
two-sided Laplace transforms) may be found in [5] or [4].
For what follows it is essential that the assumption
concerning the symbol made above can be rephrased in terms of the
kernel k appearing in (1.1). What it amounts to is that k has the
spectral exponential representation
k(t) : iCE(t;-iA)B, t # 0,
where E(.;-iA) is the bisemigroup generated by -iA, i.e.,
(1.4)
E(t ;-iA)x :
[
mieWit ~AI Xm~px~
{ ie -it(AI X+) (I - P)x,
t < O,
t > O.
Note that the function E(.;-iA) takes its values in L(X), the
space of all bounded linear operators on X.
As was proved in [5] (also [4], Section 7) the (matrix)
function k admits a spectral exponential representation if and
only if k is continuous one\ {0}, exponentially decaying (in
both directions) and has a jump discontinuity at the origin in

Bart, Gohberg and Kaashoek 595
the sense tha~ the left and right limits exist but need not be
equal. Given a function k subject to these conditions, a spectral
exponential representation for k can be constructed in the
following expliclt way. Let X denote the complex Banach of all
bounded [m-valued functions on ~\ {0} having uniformly continuous
restrictions to the half lines (-~,0) and (0,~). The algebraic
operations in X are defined pointwise and the norm is the usual
supremum norm. Take -q < ~ < 0, where q is a positive constant
suchthat
sup e qltl llk(t)kl < ~,
t~0
and define A (X § X), B: ~m § X and C: X § ~m by
D(A) : {f c X I f is differentiable and f' E X}.
[Af](t) = I -i~f(t) + if'(t), t < O,
[ i~f(t) + if'(t), t > 0,
[By](t) = e-Wltlk(t)y, t # 0,
Cf = i(lim f(t) - lim f(t)).
~t§ t+0+
Here, of course, D(A) stands for the domain of A. A straighfor-
ward argument shows that -iA is exponentially dichotomous, B and
C bounded and k(t) : iCE(t;-iA)B, t # 0. For details, see [5]
(also [4], Section 7).
The following lemma, taken from [5] (also [4], Section
8), plays an important role in this paper.
BASIC LEMMA. Suppose W(1) has the realization
W(1) : I m + C(hI X - A)-IB,
where (as above) B and C are bounded and -iA is exponentially

596 Bart, Gohberg and Kaashoek
dichotomous. Then the following three statements are equivalent:
(i) det W(h) # O, h { ~,
(ii) A X :: A - BC has no spectrum on the real line,
(iii) -iA := -i(A - BC) is exponentially dichotomous.
The non-trivial part of the lemma is the fact that (i)
implies (iii). The proof given in [4,5] is based on Wiener's
theorem and involves the spectral splitting of a possibly connec-
ted (extended) spectrum.
For later use we recall a few simple facts about bi-
semigroups. Suppose -iA (X § X) is exponentially dichotomous, and
let E(.;-iA) be the corresponding bisemigroup. Take x e X. The
function E(.;-iA)x is continuous on ~\ {0} and exponentially de-
caying (in both directions). It also has a jump (discontinuity)
at the origin and in fact
(1.6) lim E(t;-iA)x = -Px, lim E(t;-iA)x = (I - P)x,
t§ ~ t+0+
where P is the separating projection for -iA. If x belongs to the
domain D(A) of A, then E(.;-iA)x is differentiable one\ {0} and
(1.7) d E(t;-iA)x = -iE(t;-iA)Ax = -iAE(t;-iA)x,
dt
t # 0.
Obviously, the derivative of E(';-iA)x is continuous one\ {0},
exponentially decaying (in both directions) and has a jump at the
origin. From (1.4) it is clear that
(1.8) E(t;-iA)P = PE(t;-iA) = E(t;-iA), t < 0,
(1.9) E(t;-iA)(l - P) = (I - P)E(t;-iA) = E(t;-iA), t > 0.
Moreover, the following semigroup properties hold:
(1.10) E(t+s;-iA) = -E(t;-iA)E(s;-iA), t,s < 0,

Bart, Gohberg and Kaashoek 597
(1.11) E(t+s;-iA) = E(t;-iA)E(s;-iA), t,s > O.
2. MAIN THEOREMS
First we fix some notation. As before K denotes the
integral operator on L~[0,~) given by (1.1), i.e.,
[K~](t) = S k(t - s)~(s) d s, t ~ 0.
0
Here p is given, 1 s p s ~, and the kernel k is an integrable
m x m matrix function. It will be assumed that k has the spectral
exponential representation
(2.1) k(t) = iCE(t;-iA)B, t # 0.
Recall that A x stands for A - BC. The space in which A and A act
is denoted by X.
THEOREM 2.1. Assume the kernel k of the integral
operator K has the spectral exponential representation (2.1).
Then I - K is a Fredholm operator if and only if -iAx
(:= -i(A - BC)) is exponentially dichotomous. In that case,
letting P and px denote the separating projections for -iA and
-iA x, respectively, the following statements hold true:
(i) Im P n Ker pX is finite dimensional, Im P+ Ker pX
is closed with finite codimension (in X) and
ind(l- K) = dim(Im P n Ker pX) _ codim(Im P+ Ker pX),
(ii) a function ~ belongs to Ker(l - K) if and only if
there exists a (unique) x ~ Im P n Ker pX such
that
~(t) = CE(t ~-mA . x )x, t a 0,
(iii) dim Ker(l - K) = dim(Im P n Ker pX),

598 Bart, Gohberg and Kaashoek
(iv)
(v)
a function ~ in L~[0,~) belongs to Im(l - K) if
and only if
oo
r j ~ v "
0
;-zA )B~(t)d t ~ Im P + Ker P ,
codim Im(l - K) = codim(Im P + Ker pX).
An operator T + is called a generalized resolvent of an
operator T if T = TT + T. As soon as such an inverse is known,
+
then T y is a solution of Tx = y whenever this equation is solv-
able. Each Fredholm operator has a generalized inverse.
THEOREM 2.2. Assume the kernel k of the integral
operator K has the spectral exponential representation (2.1), and
let -iA x (:= -i(A - BC)) be exponentially dichotomous with
separating projection pX. Then a generalized inverse of the
Fredholm operator ! - K is given by
[(I - K)+~](t) = ~(t) + f y+(t,s)~(s)ds,
0
oo
t a O,
+
u (t,s) = iCE(t-s;-iAX)B - iCE(t;-iAX)H+E(-s;-iAX)B
= iCE(t-s;-iAX)B + iCE(t;-iAX)(l - H+)E(-s;-iAX)B,
where H + = I - pX _ (I - Px)(pXllm p)+pX and (pXllm P) is a
generalized inverse of the Fredholm operator pXllm P: Im P § ImpX.
+ .
The operator H zs easily seen to be a projection of X
onto Ker pX. Theorem 2.2 remains true if "generalized inverse" is
replaced by "left inverse", "right inverse" or "(two-sided) in-
verse". In fact, left invertibility occurs if and only if
Ker H + c Im P, right invertibility if and only if Im P c Ker H+
and (two-sided) invertibility if and only if Ker H+ = Im P. So
I - K is invertible if and only if X = Im P r Ker pX and in that
case H + is the projection of X onto Ker pX along Im P. Via
factorization the "if part" of the latter result was also proved

Bart, Gohberg and Kaashoek 599
in [5]. In general these inversion results look very similar to
those for the case of rational symbols (see [i], Section 4.5,
[2], Theorem 1.3.4, [6], Section 2).
3. COUPLING AND RELATED RESULTS
In this section we establish two lemmas that will play
an important role in the proofs of the main results. The first is
a coupling lemma (cf. [3]). Notation is as in Section 2. In
particular A x stands for A - BC.
LEMMA 3.1. Assume the kernel k of the integral
m ~ ~
operator K on L_[O,~) has the spectral exponenttal representation
P x
(2.1), and let -iA be exponentially dichotomous. Denote the
separating projection for -iA and -iA x by P and pX, respectively,
and introduce
U: Im pX § L [0,~),
uX: im P § L~[0,~),
R: L~[0,~) + Im P,
R x ~ pX
: L [0,~) § !m ,
J: Im pX § Im P,
[Ux](t) : iCE(t;-iA)x,
[U : iCE(t ;-IA " )x,
R~ = # E(-t;-iA)B~(t) d t,
0
X . X
R ~ = - I E(-t;-iA )B~(t) d t,
0
Jx = Px,
jx x x
: Tm P + Im P , jXx : P x,
K X : L [0,~) § L [Q,~), [KX~o](t) = ~ kX(t - s)~o(s)d s,
where k x is given by the spectral exponential representation
kX(t) = -iCE(t;-iAX)B, t # 0.
Then all these operators are well-defined, linear and bounded.
Moreover
: L [0,~) ~ Im pX + Lm[O ~) $ Im P
P
0

600 Bart, Gohberg and Kaashoek
is invertible with inverse
K x ]
I U Lm[0,~) ~ Im P § Lm[0,~) ~ !m
R x jx p p
In terms of [3], the lemma says that the operators
I K and jx
- are matricially coupled with coupling relation
II- I
The operator J is also called an indicator for I - K. Note that
jx = pXll m P: im P § Im P already appeared in Theorem 2.2.
PROOF. All operators appearing in Lemma 3.1 are well-
defined, linear, bounded and acting between the indicated spaces.
In this context two observations should be made. The first is
that since ~m is finite dimensional, the operator functions
E(.;-A)B and E(';-iA are continuous one\ {0} with a jump at
the origin, all with respect to the norm topology. The second is
that in view of (1.8), we have R = PR and R x = pxRX. In particu-
lar R maps into Im P and R x maps into Im pX.
Checking the coupling relation (3.1) amounts to verify-
ing eight identities. The computations are long and tedious,
which is the reason why we restrict ourselves to proving only
that RX(l - K) + jXR = 0. The proof of the other seven identities
follow the same pattern.
Take ~ ~ L~[0,~). Then
RXK~ = -i 55 E(-t;-iA -iA)B~(s) d s d t.
00
Observe that it is permitted to change the order of integration.
Doing this, we get
: f [-i f E(-t iA S.
0 L 0 J
For the moment, let x c D(A). In view of (1.7) and the definition
pX

Bart, Gohberg and Kaashoek 601
of A x, we have
d (E(-t ;-iA " )E(t-s;-iA)x)
dt
: -iE(-t;-iA " )BCE(t-s;-iA)x
where t # 0,s. Taking advantage of (1.6), it follows that
co
-i J E(-t . x : _
;-iA )BCE(t-s;-iA)x dt P E(-s;-iA
0
Since D(A) is dense in X, this identity actually holds for all
x c X. But then
co co
RXK~ = f pxE(-s;-iA)B~(s) d s - f E(-s;-iAX)B~0(s) d s
0 0
= P + R
This can also be written as RX(l - K)~ + jXR~ = 0. D
The second lemma is concerned with the difference of
the separating projections for -iA and -iA x where it is assumed
that both these operators are exponentially dichotomous.
LEMMA 3.2. Let P and pX be the separating projections
for -iA and -iA x, respectively, and define F: Lm[O,~) § X and
P
AX: X + Lm[0,~) by
P
co
r~ = I E(-s;-ia)B~(s) d s,
[AXx](t) iCE(t . x
= ;-iA )x.
Then F and A x are well-defined bounded linear operators and
FA x = p _ pX. Moreover F and P - pX are compact.
PROOF. It is easy to see that s and A x are well-
defined bounded linear operators (cf. the first paragraph of the
proof of Lemma 3.1). Take x ~ D(A). In view of (1.7) and the
definition of A x, we have

602 Bart, Gohberg and Kaashoek
and so
Hence, by (1.6),
d (E(-t;-iA)E(t;-iAX)x) = iE(-t;-iA)BCE(t;-IA" X)x, t ~ 0,
dt
X
tAx : i ~ E(-t;-iA)BCE(t;-iAX)xdt
--oo
9 ;1.
FAXx : -(I - P)pXx + P(I - P : Px - P
Since D(A) is dense in X, we may conclude that FA x = P - P
It remains to prove that F is compact. Write
We shall establish the compactness of F+. For F_ the argument is
of course analogous.
Put h(0) = -PB and h(s) = E(-s;-iA)B, s > 0. Then h is
an exponentially decaying continuous function from [0,~) into
L(~m,x), the Banach space of all bounded operators from {m into
X. Note that
oo
F+~ : # h(s)~(s)d s, ~ 9 Lp[O,~).
0
For 0 < T < ~, let H T be the bounded linear operator from L~[0,~)
into X given by
T
HT~ = # h(t)o(t) d t.
0
Then ILE+ - HTll does not exceed the Lq-norm of h on the interval
[T,~), where, as usual, p-i + q-1 : 1. Since h is exponentially
decaying, it follows that H T § F+ when T § ~. Hence it suffices
to show that the operators H are compact.
T
Fix T > 0. For notational convenience, we consider H T

Bart, Gohberg and Kaashoek 603
as an operator on Lm[0,T). Let n be a positive integer and define
P
the function h n on [0,T) by giving it the constant value h(kn-IT)
on the interval [kn-&T,(k+l)n-IT). As h is uniformly continuous
on bounded intervals, we have that h n
+ h uniformly on [0,T) But
then the bounded linear operator H : Lm[0,x) § X given by
~,n p
T
HT,n~ : f h (t)~(t) d t
0 n
tends to H when n § ~. Observe that
T
n-1 (k+l)n-lT
= ~ h(kn-l~) f-1 ~(t) dt.
HT'n~ k=0 kn
Now all the values of h are of finite rank. Thus HT, being the
limit of a sequence of operators of finite rank is compact.
4. PROOFS OF MAIN THEOREMS
In view of the Basic Le~ma of Section 1, the first part
of Theorem 2.1 can be rephrased as follows. The operator I - K is
Fredholm if and only if the determinant of the symbol ! - K does
not vanish on the real line. This fact is known from [7] for
arbitrary integrable matrix valued kernels. Here, we present a
more direct argument based on the special form (2.1) of k, there-
by aZso proving the rather explicit second part of Theorem 2.1.
PROOF OF THEOREM 2.1 (FIRST PART). Suppose -iAx is
exponentially dichotomous, and let P and P be the separating
projections for -iA and -iA x, respectively. Then P - pX is com-
pact by Le~na 5.2. It follows that I - P + pX and I - pX + p are
Fredholm operators. In particular ~er(l - P + pX) is finite
dimensional and Im(I - pX + p) is closed with finite codimension
(in X). Since Im P n Ker pX c Ker(l - P + pX) and
Im(l - pX + p) c Im P + Ker pX, the same conclusions hold for
Im P n Ker F x and Im P + Ker pX, respectively.
By Lemma 3.1, the operator jx = pXllm P: Im P § Im pX
is an indicator for I - K. Taking into account the special form
of the operators appearing in the coupling relation (3.1) and the

604 Bart, Gohberg and Kaashoek
results obtained in Section 1.2 of [3], we see that (ii) and
(iii) are satisfied. For (iv) and (v) the situation is slightly
more involved. In first instance we obtain that codim Im(! - K) =
= dim Im pXllm J and
Im(I - K) = {~ c Lm[o,oo) I RX9 E !m jx}
where R x is as in Lemma 3.1. From this one obtains the assertions
(iv) and (v) by a straightforward argument based on the fact
that Im J = !m P and Im R c Im pX. Note that (iii) and (v)
imply that I - K is Fredholm with index as indicated in (i). This
completes the first part of the proof.
For the second part of the proof of Theorem 2.1 we need
one more lemma which is concerned with the behaviour of separa-
ting projections under simple perturbations. It is convenient to
introduce the following notation. Let P and Q be projections of a
given Banach space X. We write P ~ Q if Im P c Im Q and
Ker P ~ Ker Q. Observe that P ~ Q if and only if P = PQ = QP. The
relation < defines a partial ordering on the set of all projec-
tions of X.
LEMMA 4.1. Let S be a linear operator acting in a
complex Banach space X, and let a,6 ~ ~, ~ < 6. Suppcse a - S and
B - S are exponentially dichotomous with separating projections
P(a) and P(6), respectively. Then P(a) ~ P(~). ALso P(a) = P(6)
if and only if p - S is exponentially dichotomous for all p in
the closed interval [a,B].
The result is practically obvious when S happens to be
a bounded operator on X. Indeed, in that case P(a) and P(B) are
the spectral projections corresponding to the parts a (S) and
aB(S) of d(S) lying in t~e open half planes Re h < a and Re h < B,
respectively. Hence P(a) = P(6) + P0' where P0 is the spectral
projection associated with the part ~ ,B(S) of ~(S) contained in
the open strip ~ < Re h < B.

Bart, Gohberg and Kaashoek 605
In the general, not necessarily bounded case, the situ-
ation is considerably more complicated due to the (possible)
presence of ~ in the extended spectrum of S. Again the (finite)
spectrum ~(S) of S splits into the three disjoint closed subsets
~ (S), ~(S) and ~ ,B(S) described above. But even in the seem-
ingly innocent case when ~ ,B(S~ is a compact subset of ~ (hence
a spectral set of S) it need not be true that P(~) is the sum of
P(~) and the spectral projection corresponding to ~ ,~(S). As an
example, let S be the infinitesimal generator of a strongly con-
tinuous group having the property that ~(S) = r (cf. [8], Section
23.16). For a a sufficiently large negative real number and ~ a
sufficiently large positive real number, we have that ~ - S and
S - ~ both generate exponentially decaying strongly continuous
semigroups. Hence ~ - S and B - S are exponentially dichotomous
with separating projections P(~) = 0 and P(B) = I, respectively.
Since ~(S) is empty in this case, it certainly is a compact
subset of the complex plane. Note that this example exhibits the
following interesting phenomenon: For the operator ~ - S the in-
finite spectral point ~ has to be counted as lying completely at
the left hand side of the imaginary axis. By shifting ~ - S to
B - S it changes its position (of course without actually moving
as finite spectral points would do) and goes completely to the
right hand side of the imaginary axis.
PROOF OF LEMMA 4.1. The projection P(B) is a bounded
linear operator commuting with B - S, hence with a - S. From the
integral representation for separating projections (cf. [5];[4],
Theorem 2.2) it is clear that P(B) also commutes with P(~). Put
P0 = P(~) - P(~)P(B). Then P0 is a projection commuting with S
and Im P0 = Im P(~) n Ker P(B). Let S O be the restriction of S
to X 0 = Im PO" Thus Sox = Sx for all x in the domain D(So) =
= D(S) n X 0 of S O . Since PO and S commute, S O is a linear opera-
tor acting in X 0.
Let S i be the restriction of S to Im P(~) considered as
a linear operator acting in X i = Im P(a). Then S I - ~ is the
infinitesimal generator of an exponentially decaying strongly

606 Bart, Gohberg and Kaashoek
continuous semigroup of bounded linear operators acting on X I.
The restriction P1 of P0 to X 1 is a projection of X 1 commuting
with S I. Also Im P1 = Im P0 = X0 and the restriction of S 1 to
Im PI coincides with S O . It follows that S O - ~ generates an
exponentially decaying strongly continuous semigroup too. But
then ~(S 0) lies in the open half plane Re ~ < ~ and (~ - S0 )-I is
bounded on the closed half plane Re ~ a ~.
In exactly the same way (using that X 0 c Ker P(B)) one
shows that ~(S 0) lies in the open half plane Re ~ > B and
(~ - S0 )-I is bounded on the closed half plane Re ~ ~ B. We con-
11 ~ , ~
clude that o(S0~ = 0 and (~ - S O ) is a bounded entlre functlon.
Hence (~ - S O ) is constant by Liouville's theorem. This implies
X 0 = (0), that is, P(a)= P(~)P(B) = P(B)P(~) or, equivalently,
P(a) ~ P(~).
Assume now that P(a) = P(B). With respect to the decom-
position X = Im P(a) ~ Ker P(a), we write
s 2
Then S I - ~ and ~ - S 2 both generate strongly continuous semi-
groups of negative exponential type. Taking ~ ~ p ~ 6, the same
is obviously true for S I - P and 0 - S 2. Thus P - S is exponen-
tially dichotomous for all P in [~,B].
In order to complete the proof we make the following
general observation. Suppose T is an exponentially dichotomous
operator of negative exponential type ~ (cf. [4] and [5]) and
with separating projection Q. Then for all complex E with
IRe ~I < -~, the operator T + a is again exponentially dichoto-
mous with separating projection Q. This is immediate from the
definitions. So, if for all p in [~,B], the operator P - S is
exponentially dichotomous, say with separating projection P(O),
then P(O) is locally constant on [~,~]. Since [~,B] is connected,
it follows that P(O) is constant on [~,B] and, in particular,
P(~) = P(~).

Bart, Gohberg and Kaashoek 607
PROOF OF THEOREM 2.1 (SECOND PART). It remains to
prove that if I - K is a Fredholm operator, then -iA x is exponen-
tially dichtomous. For this we shall use a perturbation argument.
Assume -iA is of exponential type ~ < 0 (cf. [4] and
[5]) and take ~ < s < -~. Then the operator c - iA is exponen-
tially dichotomous, its separating projection is the same as that
for -iA and
Putting
E(t;c - iA) = eStE(t;-iA), t ~ 0.
ka(t) = iCE(t;s - iA)B, t ~ 0,
we have ks(t) : eStk(t),- where as in the theorem k is given by
(2.1). One checks without difficulty that k s converges to k in
Li(~,L(~m)) when s tends to zero. Define K s on L~[O,~) by
Ks~ : f ks(t - s)~(s) d s, t a 0.
0
Then K s converges to K in L~[0,~) when s tends to zero. It
follows that for Isl sufficiently small the operator I - K is
Fredholm with the same index as I - K.
The Fourier transform k of k is analytic in the strip
llm 11 < -~. Fix 0 < h < -~. Then
lim ~(I) : 0.
IIm llsh
Hence the analytic function det(I - k(1)) has only a finite
m
number of zeros in the closed strip IIm 11 s h. So taking Isl
non-zero and sufficiently small, we have that det(I m - k(1)) ~ 0
for all real I.
Let s be a sufficiently small positive real number.
Then det(I m - ks(1)) = det(I m - k(1 - ie)) ~ 0 for all real I. By
the Basic Lemma of Section i, this implies that e - iA z is expo-

608 Bart, Gohberg and Kaashoek
nentially dichotomous. Its separating projection will be denoted
by PX(s). From the first part of the proof of Theorem 2.1 we know
that I - K is Fredholm and
S
dim Ker(i - K s ) = dim(Im P n Ker PX(s)) < ~,
codim Im(l - K s ) = codim(Im P + Ker Px(s)) < ~.
Since s was taken sufficiently small, the operator -s - iAx is
exponentially dichotomous too, while
dim Ker(l - K ) = dim(Im P n Ker P < ~,
-a
codim Im(! - K ) = codim(Im P + Ker PX(-s)) < ~.
-S
Here PX(-s) denotes the separating projection for -e - iA x
pX
By Lemma 4.1, we have P ~ (s). In particular
Ker PX(-s) n Ker PX(s), and so
dim Ker(l - K_a) ~ dim Ker(l - Ks) ,
codim Im(l - K_s) ~ codim Ker(l - Ks).
We may assume that ind(l - K_s) = ind(l - K) = ind(l - Ks). It
follows that the above inequalities are in fact equalities. But
then
Im P n Ker PX(-s) = Im P n Ker PX(a),
Im P + Ker PX(-s) = Im P + Ker PX(a).
Take x ~ Ker PX(-a). Then x can be written in the form x = Pu + v
with v ~ Ker PX(s) c Ker PX(-s). Hence 0 = PX(-a)x = PX(-s)Pu,
and so Pu ~ Im P n Ker PX(-a) = Im P n Ker PX(s). We conclude
that PX(s)x = 0. This proves that Ker PX(-s) = Ker PX(s). Also
pX(_s) ~ P Thus PX(-e) = PX(s). Applying (the second part

Bart, Gohberg and Kaashoek 609
of) Lemma 4.1, we see that 0 - iA x is exponentially dichotomous
for all P in the interval [-E,a]. In particular -iA x is exponen-
tially dichotomous. D
PROOF OF THEOREM 2.2. From Lemma 3.1 we know that the
operators I - K and J = P P: Im P § Im pX are matricially
coupled with coupling relation (3.&). Combining this with Theorem
2.& and [3], Theorem 2.1, we see that ! - K and jx are both
Fredholm. Also, given a generalized inverse (jx)+ of jx, one for
I - K can be defined by
(I - K) + = I - K x - uX(jX)+R x,
where K x, U x and R x are defined as in Lemma 3.1.
Let @ ~ Lm[0,~). Taking advantage of the special form
x P x
of the operators K , U and R , we obtain
[(T - K)+~](t) = ~(t) + f ~+(t,s)~(s~ds
f . ,
0
+
where the generalized resolvent y is given by
co
y+(t,s) = iOE(t-s;-iAX)B+ iCE(t;-iAX)(jX)+E(-s;-iAX)B.
In view of the identities (1.8) and (1.9), the second term in
the right hand side of this expression can be rewritten as
-iCE(t;-iAX)H+E(-s;-iAX)B or iCE(t;-iAX)(l - ~+)E(-s;-iAX)B,
where the projection ~+ of X onto Ker pX is defined by
~+ = z - P - (z - P 215215
5. THE RIEMANN-HILBERT BOUNDARY VALUE PROBLEM
In this section we deal with the Riemann-Hilbert boun-
dary value problem (on the real line)
(5.1) w(z)r = r -~ < z < ~,
the precise formulation of which reads as follows: Given an m x m

610 Bart, Gohberg and Kaashoek
matrix function W(h), -~ < I < ~, with continuous entries,
describe all pairs 9+, 9_ of ~m-valued functions such that (5.1)
is satisfied while, in addition, 9+ and 9_ are the Fourier trans-
forms of integrable Tm-valued functions with support in [0,~) and
(-~,0], respectively. For such a pair of functions, We have that
9+ (resp. 9_) is continuous on the closed upper (resp. lower)
half plane, analytic in the open upper (resp. lower) half plane
and vanishes at infinity.
The functions W that we consider are of the form
W(/) = I - k(1), where k admits a spectral exponential represen-
m
tation. In other words W(1) admits a realization as in (1.2).
This implies that W(1) is analytic in a strip around the real
axis.
As before, A stands for A - BC.
THEOREM 5.1. Consider the function W(I) : I m - k(1),
where k has the spectral exponential representation (2.1) and
(consequently)
(5.2) W(h) : I + C(l - A)-IB, -~ < I < ~.
m
Assume the operator -iA is exponentially dichotomous (or,
equivalently, det W(1) ~ 0 for all I e ~), and let P and P be
9 X
the separating projections for -iA and -IA , respectively. Then
the pair of functions 9+, 9_ is a solution of the Riemann-Hilbert
boundary value problem (5.1) if and only if there exists a
(unique) vector x in Im P n Ker P such that
5.3) r : C(l - A : S eilt(-iCE(t;-iA x) d t,
0
0
5.4) g_(h) : C(I - A)-lx : S eilt(-iCE(t;-iA)x)dt, h c ~.
Here E(';-iA) and E(-;-iA denote the bisemigroups generated by
-iA and -iA respectively.
h e JR,

Bart, Gohberg and Kaashoek 611
by
PROOF.
Take x in Im P n Ker P , and define ~+ and ~0_
~+(t) : -iCE(t;-iA
~_(t) : iCE(t;-iA)x.
Then ~+ and ~_ are integrable functions. Since x ~ Ker pX, it
follows from (1.8) that ~+ has its support in [0,~). In a similar
way one infers from x ~ Im P and (1.9) that ~_ has its support in
(-~,0]. We denote Fourier transforms of ~+ and ~_ by @+ and ~_,
respectively. The identities (5.3) and (5.4) are now clear from
the fact that for an exponentially dichotomous operator S the
resolvent has the following representation (see [4,5]):
(k -
S)-Ix
co
: ~ e-ktE(t;S)xdt,
--co
Re k = O.
Also, a straightforward computation using (5.2) shows that (5.1)
is statisfied. This settles the "if part" of the theorem.
In order to establish the "only if part", assume that
the pair r r is a solution of the Riemann-Hilbert boundary
value problem 5.1). Write ~+ and r in the form
r = ~ eilt~+(t) d t,
0
0
r = f eilt~_(t)dt,
where ~+ ~ L~[0,~) and ~_ ~ L~(-~,0]. A routine argument,
based on (5.1) and the fact that W(~) = I m - k(~), yields
~+(t) - f k(t - s)~+(S) d s : O, t a O.
0
Since k has the form (2.1), it follows from Theorem 2.1 that
there exists a unique x ~ Im P n Ker P such that
~+(t) = -iCE(t ;-IA " )x .
But then (5.3) is satisfied As

612 Bart, Gohberg and Kaashoek
~_(~) = W(h)r = C(h - A)-ix, the identity (5.4) holds too. D
REFERENCES
i. Bart, H., Gohberg, I. and Kaashoek, M.A.: Minimal
Factorization of Matrix and Operator Functions. Opera-
tor Theory: Advances and Applications, Vol.1,
Birkh~user Verlag, Basel etc., 1979.
2. Bart, H., Gohberg, I. and Kaashoek, M.A.: Wiener-Hopf
integral equations, Toeplitz matrices and linear sys-
tems, in: Toeplitz Centennial (ed. I. Gohberg), Opera-
tor Theory: Advances and Applications, Vol.4,
Birkh~user Verlag, Basel etc., 1982, p. 85-135.
. Bart, H., Gohberg, I. and Kaashoek, M.A.: The Coupling
method for solving integral equations, in: Topics in
operator Theory, Systems and Networks, The Rehovot
Workshop (ed. H. Dym, I. Gohberg), Operator Theory:
Advances and Applications, Vol.12, Birkh~user Verlag,
Basel etc., 1984, p, 39-73.
. Bart, H., Gohberg, I. and Kaashoek, M.A.: Exponential
dichotomous operators and inverse Fourier transforms,
Report 8511/ M, Econometric Institute, Erasmus Univer-
sity, Rotterdam, The Netherlands, 1985.
. Bart, H., Gohberg, I. and Xaashoek, M.A.: Wiener-Hopf
factorization, inverse Fourier transforms and exponen-
tially dichotomous operators, J. Functional Analysis,
to appear.
. Bart, H. and Kroon, L.G.: An indicator for Wiener-Hopf
integral equations with invertible analytic symbol.
Integral Equations and Operator Theory 6(1983), 1-20.
See also the addendum to this paper: Integral Equations
and Operator Theory 6(1983), 903-904.
. Gohberg, I. and Krein, M.G.: Systems of integral equa-
tions on a half line with kernels depending on the
difference of argument, Uspehi Mat. Nauk 13 (1958), no.
2 (80), p.3-72 [Russian]. Translated as: Amer. Math.
Soc. Trans. (2) 14 (1960), 217-287.
. Hille, E. and Phillips, R.S.: Functional analysis and
semigroups. Amer. Math. Soc., Providence R.I., 1957.
. Kailath, T.: Linear Systems. Prentice Hall Inc.,
Englewood Cliffs N.J., 1980.
10. Kalman, R.E.: Mathematical description of linear dynam-
ical systems, SIAM J. Control i (1963), 152-192.

Bart, Gohberg and Kaashoek 613
ii. Kalman, R.E., Falb, P. and Arbib, M.A.: Topics in
Mathematical System Theory. McGraw-Hill, New York etc.,
1960.
H. Bart,
Econometrisch Instituut
Erasmus Universiteit
Postbus 1738
3000 DR Rotterdam
The Netherlands
M.A. Kaashoek
Subfaculteit Wiskunde en
Informatica
Vrije Universiteit
Postbus 7161
1007 MC Amsterdam
The Netherlands
Submitted: May 6, 1985.
I. Gohberg
Dept. of Mathematical Sciences
The Raymond and Beverly Sackler
Faculty of Exact Sciences
Tel-Aviv University
Ramat-Aviv
Israel

Integral Equations
and Operator Theory
Vol. 8 (1985)
PRINCIPAL CURRENTS
Richard W. Carey and Joel D. Pincus
0378-620X/85/050614-2751.50+0.20/0
9 1985 Birkh~user Verlag, Basel
An operator theoretic analog of a local Chern class is
discussed for certain finitely generated operator algebras. This
gives rise to an index theory for the maximal ideals of the
algebra. The algebraic receiver for this index is the Grothendieck
group of the algebra completed at a maximal ideal.
INTRODUCTION
In [ 6 ] we reported the construction of a new index
theory for the maximal ideals of a finitely generated commutative
Banach algebra of operators on Hilbert space. The index was the
local degree of a holomorphic chain supported on the Taylor
spectrum of the generators. The specification of the chain
corresponds to the evaluation of certain weights determined by
composition series associated with the module action of the algebra.
The boundary of the holomorphic chain is a distinguished cycle
concentrated on the joint essential spectrum of the generators.
The index which we introduced via this association of
holomorphic chains to operator algebras differs from others
in that it does not vanish when the dimension of the Taylor
spectrum is less then the number of the given generators,and ill
that it is not constant on the Fredholm components. Our index
has jumps on the singular locus of the underlying spectrum. But,
because of this, has a natural transformation law under morphisms.
Here we give a synoptic account of the ramifications of
our idea of using analytic continuation to construct a local index
theory in terms of the multiplicity theory of ideals.
82 The analytic index
We begin by recalling the main ideas associated with
the so-called principal function of an operator. It has stability

Carey and Pincus 615
under perturbation by trace class operators, and has geometric
meaning. This object is associated with pairs of possibly non-
closed subspaces of Hilbert space which are operator ranges.
Recall that an operator range is a linear subspace that is the
range of some linear operator. The collection of all operator
ranges is the lattice L generated by the closed subspaces. An
ideal in L is a subset I of L such that if D'aL and D E I with
D' < D , then D'EI; furthermore, If both D and D' are in I then
so is D+D'. An ideal I is said to be invariant if whenver D is in
I then every D' unitarily equivalent to D is also in I.
If F is the ideal of finite dimensional subspaces of L
then two elements R and S of L are said to be congruent modulo F
provided that R+d = S+d' for some elements d and d' in F.
Suppose that R~S(modF). Then there are positive operators
(A,B,C,D) with R(/A) =R, R(/B)=S, R(/A+-C)=R(~-D-) where the ranges
R(/C) and R(/D) are in F. See[15] for references to the study Of
range spaces,
Consider now the perturbation problems
A~A+C and B~B+D.
The functions displayed above the arrows are the phase
shifts corresponding to the perturbations. Thus,
and
for smooth f.
trace [f(A+C)-f(A)] = /f'(1)~(1)dl
trace [f(B+D)-f(B)] = ff'(1):(~)dl
It is quite easy to see that if R =R(T) and S=R(T*) for
a Fredholm operator T, then Index T = T(O) -:(0).
Suppose that f, g and h are in L m , with h> 0 and all
=
the functions having compact support in [0,~]. The function f is
said to be h related to g at the origin provided that
If f(x)-g(x) dx I< /h(x) dx for s > O.
x + s = x+~
Recall that v is an approximate limit of the function f
at x provided that for every ~ > O, { y:]v-f(y)l

616 Carey and Pincus
approximate limits of h at g consists only of 0 and f and g are h
related, we will write f=g. This is an equivalence relation and
the set of equivalence classes [f] forms an abelian group K(L)
under the natural identification [f+f'] = [f] + [f'].
We have proved in [15] that if g and g' are represent-
atives of [f] with {g(O)} = w+(0,g) and {g'(0)}=m+(O,g'), then
g(O) = g'(O). If there is a g in [f] which is right approximately
continuous at the origin, then it follows that the value g(O)
does not depend upon the choices of g and h. We will denote this
value by f~(O), and say that 0 is in the extended Lebesgue set of
f.
Suppose that R and S are elements of the lattice L, and
that RES(modF). We proved the following result in ~5].
THEOREM I.i The equivalence class[~-T] is independent
of the implementing operators {A,B,C,D}. It depends only upon the
subspaces R and S.
Thus if {W,V,X,Y}is another choice of positive operators
n P
with R(~-) = R, and R(~ = S, R(/~ = R(/V-$-Y), V+V+X, W§
then [~-T ]=[D-p].
We will say that two subspaces R and S in L are F
comparable if Rs and the origin belongs to the extended
Lebesgue set of the class [~-~9
In this case the value (T~)~(O) will be denoted by
ind(R,S). We note that if T is a Fredholm operator then the
subspaces TH and T~H are F-comparable, and ind(TH,T~H) is the
usual Noether index.
stability result.
We have also been able to establish [15] the following
THEOREM 1.2 Suppose that T and K are bounded operators
on the Hilbert space H with R(K) in F. Suppose that TH and T~H are
F-comparable. Then [T+K]H and [T+K]~H are F-comparable and
ind([T+K]H,[T+K]~H) = ind(TH,T~H)
Now let Tz= T-zI and form the self adjoint operators
T ~T and T T with spectral resolutions E~ and F~ (respectively).
Z Z Z Z
If z is not in the essential spectrum of T, it is clear
that both spectral projections are constant finite rank operators

Carey and Pincus 617
for sufficiently small positive %. Thus, if p(T) is a polynomial
in T, we have
z z
J(p)(z) = trace [p(T)(E%-F%)] = p(z) Index(T-z).
Since we are interested in the case where z is in the
essential spectrum of T, we form another functional. Let
J (p)(z) = 2--~rtrace p(T) f (E%-F4) z z
~(1%1~) d%
where ~ is a real smooth compactly supported function normalized
2
so that ~ s@ (s) ds = I.
Of course J e(p)(z ) = J(p)(z) for e sufficiently small
if z is not in the ess~tial spectrum of T. But it is basic to
Our analysis that the hypothesis T~T-TT ~ is trace class implies
that there exists an Ll,real-valued, compactly supported function
gT,called the principal function of T, so that
1
Jg(p)(z) = -g2f P(~)~(I~-z[/~)gT(

618 Carey and Pincus
But there are circumstances in which it is important to
refine the definition so that we obtain a function which is defined
on a larger set. Thus, when the essential spectrum is curve like
it will be possible to define the principal function ~ the curve
as an average of the values on both sides of the curve. This even
makes sense when the principal function is weakly differentiable
i.e. when the distributional derivatives Sg /~x and 8g/~y are
measures. It is known ~4] that the principal function of an operator
T satisfies this condition provided that the spectral multiplicity
functions of Re T and Im T are Lebesgue summable.
In this case a theorem of Federer [i~ asserts that for
H I almost all b in R 2, and f a weakly differentiable function
lim ff(b+Ez)~(Izl)dL2z = 89 lim inf f(z) + (L 2) ap iim supf~,
E+0 + z~b z§
whenever ~ satisfies the conditions above. Thus, we see that if the
subspaces T H and T~H are F-comparable then 6 ~ (0) is equal to the
z z z
average of the lower and upper approximate limits of gT at z.
It therefore is reasonable to take this as a redefinition
of the principal function on the set of points z where the rotation
-ally symmetric regularizaNons of the principal function converge
Hausdorff one measure almost everywhere. Then ind(T~H,TzH ) = gT(z)
H 1 almost everywhere~
For an example, consider the unilateral shift. If T is
the shift and w is a complex number with modulus one, then
R(T-w)~/R(T-w) is a one dimensional vector space. Both ranges are
dense subspaces of H, but 89 =-gT(0) - by Theorem 1.1 - describes a
stable feature of their geometry, a new relative dimension.
Other examples of a simple nature are furnishad by
singular integral operators on smooth contours in the plane with
rational symbols that vanish. These operators actually have N(T)
and N(T ~ finite dimensional, but the difference of these dimensions
is not stable against finite rank perturbations. See [17].
82 Fundamental properties
The principal function of a single operator T is used to
define a current, called the principal current of the unital C*-
algebra corresponding to T. By definition the principal current

Carey and Pincus 619
for the C ~ algebra which corresponds to a T such that [T,T ~] is
in trace class is T 1 E21 gT
- 2~i -
basic properties:
The two-current defined by this relation has certain
With a suitable functional calculus
(i) ~T(~) = index f(T), when f ~ 0 on Oess(T).
(ii) ~T(fdh) = trace [h(T~,T),f(T~,T)]
(iii) Mass(T) < 89 II[T~,T]II trace norm
(iv) T~= ~#(T), where ~(T,T ~) is a "smooth" function of T and T ~
and T~ is the current formed from the operator ~(T#T~).
(v) TT=TT+ k for k in trace class.
The first of these relations remains the motivation for
the entire theory, ic led originally to the conjecture by the second
author that the principal function coincides with index T when z
z
is a Fredholm point. This conjecture was first established in full
generality in [12] .
It is fundamental to our approach that we consider
the functionals which appear in (ii) (or those that appear in higher
dimensional versions) not only as being represented by a global object
like a homology class, or some measure [13],but by an object defined
pointwise on the spectrum of the algebra.
Indeed, a variety of conditions are known which guarantee
even that the principal function of a single operator takes on only
integer values. This is so, for example, if T is subnormal. Integer
values are also assumed if T+T ~ and(T-T~)/2i have integrable
spectral multiplicity functions, [14].(We speak here of a determina-
tion up to a set of zero with respect to two dimensional Lebesgue
measure.)Thus, there is geometry associated with these point values.
Desideratu
We can rewrite (i) as an intersection of currents
index f(T) = [div (f) liT](1)
provided that f is analytic and non-vanishing on the essential
spectrum of T.
We put aside for the moment the general question of the
meaning of intersections with principal currents in the non-Fredholm
situation, and ask what the natural multidimensional extension of
this intersection result may be in the multidimensional Fredholm case.

620 Carey and Pincus
Suppose that we are given s commuting operators
=(TI,...,Ts) with joint essential spectrum X a closed 2k-1
dimensional smooth submanifold of CS.Then for holomorphic functions
= (fl ..... fk ) and h, the operators ~(~) and h(~) make sense
if they are understood in the sense of the Taylor functional
calculus, and we can ask if there are hypothesis which guarantee
the existence of a holomorphic (k,k) chain T in cS/x so that
x[h,E~(s = [div(i)(l T ](h)
where E~(~) denotes the Koszul complex of the tuple ~(T);
x[h,Em(~)] is the Lefshetz number of the endomorphism induced on
E~(~) by h(T), and div(~) is the current of integration which
corresponds to the zero set of ! counted according to multiplicity.
Thus, div(! ) = d(d c logll~N2a(ddClogIl~II2) k-l, where dC=i/4~(~-~),
see [ 4 ].
The Brown-Douglas-Fillmore-Kasparov theory is associated
with a 2k-i current ~ with support in X so that
x[h,E~(!) ] = ~(h 3log II!ll2A[dd c log ]lill2] k-1
The restriction of ~ to X defines an element of KI(X ).
But the C ~ algebra structure does not determine solutions Tof
dT = ~. Rather, it is our fundamental idea that the C ~ algebra
index theory is determined by analytic structure associated to
Commuting subalgebras generating the C ~ algebra. Now we define
our concept of local index theory. Recall that dual to the Chern
map ch~: K~(X)§ there is ch~:K~(X)+H~(X;Q) and an index
theorem is obtained by identifying ch~(~) = [~]. Thus~ if Fg~I(X)
then index(F)= ch~(~)ch~(~), corresponding to the pairing
H~(X;Q) OH~(X;Q) +Q under the Chern maps. Note that the boundary
-i
map~ takes ch~(~) to a class in H~(cS,x;Q). When ~ is holomorphic
with Bott symbol [B(!)] (see [ 3 ]) in ~I(X), ch~[B(!)] maps to
a bivariant class [div(~)] in H~(cS/x,(cS/x)/zero(~); Q]
We get
x(E~(~))=ch~(~)ch~[B(~)]=~-Ich~(~)'[div(~)]
:e([T]- [div(s
Here the e applied to a class of dimension zero of the
form ~gj hj[pj] , where [pj] is a zero cycle,i.e, a point, gives
~jhj gj.

Carey and Pincus 621
Note here that since we are treating [div(~)]as a
bivariant class it maps H~(cS/x) to H~(zero ~) and we think of [T]
as being in H~(cS/x), so that [T]'[div(~)] is a class of dimension
zero in H~(zero ~) having the form [gjhj[pj].
Thus we want to realize the restriction ofthe Ext class
ch~(~) to Bott symbols as an intersection with a holomorphic (k,k)
chain T in cS/x. If T and div(~) are holomorphic chains of
complementary dimensions k and s-k which meet in a finite number
of points {m} of the Taylor spectrum of ~, we write
e([T].[div(!)]) = ~ (T,div(~)) m
where (T,[div(~)) m depends only onT and div(~) in a neighborhood
of m.
The establishment of a local index theory then means
that we must find a way to calculate the pairing (T,(div(~)) m.
In this note we explain our results from [6] in these
terms rather completely for the case when the Taylor spectrum
has one dimensional structure, and we indicate how the general case
may be treated.
For maximal ideals which are 1-analytic (a concept which
we introduce below), we will define the operator theoretic local
Chern class ch~(~,m) by forming the unital Banach algebra generated
by T. Let A denote this algebra, and let A ^ denote the completion
-- m
of A with respect to the filtration{mn}, where bar denotes closure.
Under the generating condition it will turn out that
A^m is Noetherian~d we form the Grothendieck group K(A~) of
isomorphism classes of all finitely generated A ^ modules.
m
We view the Hilbert space H in two different ways:
For x in H and r in A, we consider the two actions (r,x)§
and (r,x)§ We let H~ denote the adj0int module associated
with the second action.Then we complete the Hilbert space regarded
as an A module by taking the filtration{mnH}. The resu]ting
completion is denoted by H^'m Similarly, we form H~, the completion
formed by taking the filtration{mnH~}
The local chern class ch~(~,m) is defined as the element
of K(A~) given by the difference [Him ] - [H ^] when m is 1-analytic.
m

622 Carey and Pincus
We note here that the condition of one-analyticity implies that
H ^ and H ^ are Noetherian A ^ modules.
m *m m
If a is an ideal of a ring R and E is an R-module,
denote by ~R[a,E] the Hilbert-Samuel multiplicity of a with
respect to E over R whenever it is defined. Recall [7] that
this is the coefficient in the so-called X-polynomial of E.
The multiplicity of the two modules above can be
^
written as limits. Take m to denote the closure of mA ^ in A ^. Then
m m
and
~A m^ [~,H~]= lim dimcmnH/mn+iH
UA~[~,H~%] : lim dimcmnH~/mn+iH ~
n §
If a is an ideal in A~, we will define its index by
setting I(a) = ~A~[a,H~] - ~A~[a,H~] , whenever the multiplicities
exist. Thus, the indexof a one-analytic maximal ideal m is obtained
by applying the multiplicity homomorphism ~Am[~,.] : K(A ) + Z,
to ch~(2,m) which gives
I(m) =UA^[~,H~ ] -~A^[~,H~]
m m
82 k-analyticity
We now give a definition which makes it possible to carry
out the program which we have outlined for the construction of the
local index. The definition which follows is merely a sufficient
condition which can certainly be weakened.
Let A be the Banach algebra generated by ~ = ~i ..... Ts}'
a commuting s-tuple of operators. Let (z) be the ideal in A generated
by T-z. We will say that (z) is k-analytic if k is the smallest
integer for which there is a surjective linear map ~ ={~i,~2 H
from C s onto C k such that the image of T-z in the Calkin algebra
is an invertible k-tuple in the sense of Taylor.
In other words, (z) is k-analytic if there are k
linear combinations of the generators T-z which form a Fredholm
k-tuple, but fewer than k such combinations are not Fredholm.
A condition such as this one guarantees that the
Taylor spectrum in a neighborhood of (z) consists of k -dimensional
analytic variety.

Carey and Pincus 623
result.
Results from [16 ] can be used to establish the following
THEOREM 4.1 Let (z) be !k-analytic maximal ideal of
C s
A. Then there is a neighborhood of (z) in which the Taylor
spectrum of T is an analytic subvariety of dimension at most k.
It is a consequence that the Taylor spectrum in a
neighborhood of a non-isolated one-analytic (z) consists of a finite
union of irreducible complex one-dimensional analytic varieties
{vj}.
THEOREM 4.2 ~(z) i__s on,analytic, then for all w
sufficiently near z, the module completions H2w). and H$(w) are
No etherian A~ . modules; fur____!therm______qor ~ th___!e multiplcities
~i~ dimc (w)~H)(w)n+lWl and ~i~ dimc (w)nH~/(w)n+iH~ exist.
This result seems to be the first realization that
Noetherian ring theory and index theory are related. We found it
by considering the standard definition of the Noether index of
a Fredholm operator, and rewriting it in the following way:
PROPOSITION 4.3 Let T be a Fredholm operator. Then
Index T = lim dim T~nH/T~n+IH - lim dim TnH/Tn+IH
Because the proof depends on a fundamental insight due
to Hilbert, we will give it here.
For n a positive integer,
Index T = Index T n+l - Index T n = (dim ker T n+l- dim ker T n)
- ( dim ker T ~n+l - dim ker T~n).
But Index T = Index TITn H because Index is invariant
1
under finite rank perturbations. Thus, with P the orthogonal
n
projection onto TnH, we have
Index T = dim ker T TnH - dim ker P T ~
n TnH
= (dim ker T n+l - dim ker T n) - dim TnH/Tn+IH.
as well as
Combining the two expressions for Index T, we get that
dim ker T ~n+l - dim ker T ~n = dim Tn/Tn+IH
dim ker T n+l - dim ker T n = dim T~nH/T~n+IH
Thus, Index T = dim T~nH/T~n+IH - dim TnH/Tn+IH for n=l'2 ....

624 Carey and Pincus
A classical theorem of Hilbert which underlies all of the
multiplicity theory of rings will now imply that both terms on
the right stabilize. Let X be a finitely generated graded module,
X = ~ ~ X k over the polynomial ring C[x 0 ..... Xr]. Then there is
a polynomial Px(t) of degree at most r with rational coefficients,
called the Hilbert polynomial of the module X, so that dim C X k
= Px(k) when k is sufficiently large.
To apply this result, we take X k = TkH/Tk+IH and define
the action of C[x0] as x 0 : h + Tk+IH § + Tk+2H , with a similar
construction for the adjoint term. Since r=0, the Hilbert polynomial
result implies that the dimensions stabilize.
It is remarkable that this result seems not to have been
previously observed; but we also have the following result which
shows that index theory is closely related to Noetherian ring
theory, and in particular to the multiplicity theory of ideals.
COROLLARY 4.4 Let T be a Fredholm operator, then
Z
H(z ) and H~(z) are Noetherian C[x O] modules and
Index %= length C[[x0~(H~(z) 8 C[[xo]] ) - length C[[XoI]](H(~ ) 8 C[[x0]]) "
For the case of the commuting s-tuple T we have not yet
introduced the analog of the principal function. Although the
local index of a single Fredholm operator coincides with the
classical Noether index, and thus with gT' this is a feature of
the geometry of the plane which does not carry to higher dimensions
where the spectrum can have singular loci imbedded in it.
We have defined Ch~(T,m) as a difference element in
the Grothendieck K(AI). Since this group is generated by the
^
quotients A~/p , where p is a prime ideal of Am, the local index
can be thought of as being the ~ement in K(Am) defined by the
difference [H ~m ^ ] - [H ~ ] , with the decomposition
^
[H.m] - [H i] = ~gp[A:/p]
The sum here is taken over the distinct primes. These will all
have depth zero or one, and the integer coefficients gj which
occur as the coefficients of primes of depth one will correspond
to the principal function.

Carey and Pincus 625
Under the assumption of one-analyticity, we can give a
separate formula for these "principal numbers".
Let Z denote the Taylor spectrum of ~. Let B = C(ZN U),
the continuous complex-valued function in a CS-neighborhood, U,
of a one-analytic maximal ideal m. Let P:A§ be the Gelfand-Mazur
map composed with restriction to U. We can take the closures of
A
the images p(m n) to form the algebra completion B ^ and a P which
m
extends P , and maps A^ to BA.
m m
Let E (resp. E~) be the annihilator in ^Am of H ^m (resp. H~)
the localization (A~)p~ of Am at the minimal prime
and form Pj
of E which corresponds to the i~reducible variety V. introduced
J
after theorem 4.1 above. In other words, pj is the minimal prime^
ideal of E whose image under the surjective map (induced by p)
^ ^
given by Am/E§ ) yields the prime ideal corresponding to
the sheet V and maximal ideal m. We note that this makes sense
J
only when the image is a minimal prime, i.e. V. may not appear
^ J
in the prime spectrum of B /P(Z).
m
Now we can use the associativity formula for the
multiplicity of ideals. See,for example,Nagata [7 ].
THEOREM 4.5 If (z) is one-analytic, then the principal
numbers are given by
gJ = lengthA(z)p~j[H~(z) ^ ^ 8 A(z)^ (A i z)p~j ]
- lengthA~z)pj[H(z)8 ^ A(z ^ ) (A)iz)p j ]"
The lengths which occur here are of course defined only
when the indicated primes exist. The lengths are set equal to
zero when the ideals are not well defined.
This theorem is the natural extension of corollary 4.4.
The proof, of course, is more than just an application of the
associativity law from algebra. The main point of the proof is
that we are able tO get rid of the quasi-nilpotent elements.
But the perhaps unexpected result is that the principal
numbers are unchanged under trace class perturbations of the
generators T which preserve commutativity.

626 Carey and Pincus
THEOREM 4.6 If L. = T.+K., where K. is in the trace class
-- 1 I 1 I
and if the commutator [Li,Lj] = O, then all of the numbers gi
remain unchanged when they are computed in terms of the perturbed
ideals.
We will see below that more can besaid if it is assumed
that the essential spectrum of the s-tuple ~ has some smoothness.
It is very important that we can now state transformation
lawsfor the local index under morphisms in terms of these principal
numbers.
For the simplest of these relations, consider a morphism
= (fl .... fr)- By the spectral mapping theorem the Taylor spectrum
of the r-tuple of operators ~(~) is the image set ~(~) in C r.
Suppose that the r-tuple ~(~) is one-analytic at y in ~(Z).
-I
Suppose also that ! (y) consists of one-analytic ideals.(The
analyticity assumption implies that this set is finite.)Suppose
also that ~(Z) is topologically unibranch at y.
THEOREM 4.7 The index of the maximal ideal which
corresponds to the new generators ~(~)- y is given by
I(!-y) =[{~mult f gj(x) }
J x:!(x~=y j
xEV. ]
where mult f.is the degree of the unramified covering defined by
x--3~--
restricting f to a small neighborhood o~f x in V..
- - ]
The point of such theorems as this is that there is no
restriction on the relation between r and s, and we are not merely
dealing with a global transformation law far homology classes-
which we could also construct- but with a transformation law for
the local index in terms of the principal numbers. The module
action of the algebra A on the Hilbert space has been broken up
into a purely geometric part corresponding to the multiplicities
associated to the map ~ on the varieties V. and lengths of the
]
composition series given by the gi"
For a very simple example, consider the pair ($2,S 3)
where S is the unilateral shift. The Taylor spectrum is the
patch {(Zl'Z2):
Z 3
1 -
2
z2 = O, z 3. < i, j=l,2}. Since the principal
function of the shift (its index) is -I inside the unit disc, we

Carey and Pincus 627
see that the theorem then implies that the local index of the
maximal ideal generated by (S2-zl,S3-z2) on this patch is -i
except at the origin where it takes the value -2. Indeed, if we
consider the map ! : C+C 2 given by z+(zkP,z kq) where p and q are
relatively prime and p < q,then it is clear that mult0~ = kp.
We will return to this example below, where we will again
compute the local index but in terms of a system of parameters
for the maximal ideal.
Now we turn to the computation of the local index
in terms of the transformation law. This is the analog of (i) in
82 we expressed that formula in terms of the residue theorem).
THEOREM 4.8 Suppose that f is a complex-valued analytic
function which does not vanish on Oess(~) , Then
Index f(T) = X{Xmult~ fj gj
~(x)=8
x in V.
]
82 System of parameters
An alternate description of the local index can be
given in terms of a system of parameters for the one-analytic
maximal ideal. We let (z) be a one-analytic ideal and let ~ belong
to the Grassmanian Gc(s-l,s ) so that:
a) the hyperplane corresponding to N intersects the
tangent cone to the Taylor spectrum at (z) only at the vertex point
z.
b) The operator ~1(~-z) is Fredholm.
The notation ~ refers here both to an element of the
Grassmanian and to the s-i linear combinations of the generators
with the indicated zero space. We prove that the ambiguity involved
here is harmless in the sense that different linear combinations of
the generators which vanish on the same subspace give the same
result in the next theorem. N(~-z) is the s-1 tuple of operators
formed by choosing a dual basis{Nl,-..,Ds_ I} for ~, and then forming
the operators Dl(~-z ) ..... Ds_l(~-z). The notation ~i(~-z) refers
J
to the operator D(T-z) obtained from any dual vector N to 9-, the
orthogonal complement to K relative to the standard inner product
in C s .

628 Carey and Pincus
THEOREM 5.1 If (z) is one-analytic, there is an open
set ~ in the Grassmanian Gc(S-l,s ) such that a) and b) above are
satisfied and such that if H is in ~ , then the local index of
the maximal ideal (z) is ~y_e_E by
. L
l(z) dim ker ~ (~-z)(l root space H (~-z)
I
- dim ker ~(~-z) ~ r~ root space H (~-z) ~
In this statement, we have used root space ~(~-z) to mean the
set of vectors in Hilbert space which are annihilated by some
positive power of all of the s-I operators ~(~-z). The first term
in I(z) above is the dimension of the space of these vectors that
I
lie also in the kernel of the single operator ~(~-z). Because of
b) this last space is finite dimensional.
We turn now to the example (S 2,S 3) treated in 82 above,
but now we wish to illustrate how we get I(O) for all ~ that do
not pass through the tangent cone.
We have seen that the Taylor spectrum is {(Zl,Z2):
3 2 = O, Izjl~ i}. This has a cusp singularity at the origin.
Z 1 ~ Z 2
The tangent cone at the origin is the zero set of the
2
polynomial z 2 , and this is the plane C z 1
Thus, to calculate the index by the formula in theorem
5.1, we can use ~($2,S 3) = aiS2+agS 3 as long as a2~ O. Then we
must have ~I($2 ~3. . ~2 . ~S 3
,~ ) = Dlb +D 2 w ere albl+a2b2 = 0 and O~Ib I ~Ib21.
The last condition is a result of the Fredholm assumption b).
Because ker(blS2+b2S3)is the zero vector, we have
i 2 3~
I(0) = - dim ker ~ (S ,S ) r] root space ~ ($2,$3) ~.
But, with % = - bl/b 2 , we have
i r span {l,e i0 } for Ibll > [b21
ker ~ ($2,$3)~ = I span {l,e i0 ~ %neinO } for 0 la21 . This
inequality is equivalent to ]bll < Ib21 . Thus~ine in0 is not in
the intersection. Hence, I(0) = -2.
The theorem above is the form in which we originally
announced our results in [6].

Carey and Pincus 629
There is yet another way in which we can express the
local index. It is the Euler characteristic of a Koszul complex
given in terms of the local parameters.
We regard the Hilbert space as an A module, and we form
the vector space H z, which is defined to be the localization of H
at the maximal ideal corresponding to z. The operator ~
!
induces an endomorphism [~'(~-z)] on Hz, and it is easy to give
another form to the last theorem:
THEOREM 5.2 I(z) is the Euler-Poincare characteristic
I
of the complex [H-(~-z)]
82 Holomorphic chains
O+ Hz +H z +0
One of the characterizations of a holomorphic one-chain
is as a locally finite sum ~ gj[Vj], where gj is an integer and
[Vj] indicates integration of two-forms over the regular points
of the irreducible variety V . It is a theorem of Lelong that
3
this defines a current.
The local degree of such a chain is defined to be
gjO(ll[Vj!i[ ; z), where 0(I [ [Vj] II; z) is the melong number of
[Vj] at z.
Recall that by definition the density of a current A
having measure coefficients is defined in terms of its associated
variation measure. For an open set Q in some fixed compact set
containing the support of A , this is defined to be
IIA(~)II ~ sup { I&(~) : ~Dk(~) with II~(x) II < I, all x ~ Q}
The density or Lelong number 0(I I ~ l];z) at the point
z in ~ , is lim @ II & II;r,z), the limit of the average mass of
r§
in a ball of radius r about z. When & = [Vj] it is known that
O(II[Vj]II;z ) = ~j(z), the multiplicity of the local ring of the
variety V. at z.
3 We have already seen that the Taylor spectrum of T in
a neighborhood of a non-isolated one-analytic (z) consists of a
finite union of irreducible complex one dimensional analytic
varieties V..
3
By the local principal current we shall mean a pair

630 Carey and Pincus
C s
{Uz,[Vz]}Consisting of a neighborhood U z of z and a holomorphic
one-chain [Vz] defined in U z so that UznZ consists of one-analytic
ideals, and [Vz] has the form ~ gj(z)[Vj]. Here gj(z) is the
principal number at z, and [Vj] indicates the current of integra-
tion of two forms over the regular points of V. with the induced
3
orientation.
The local principal current has local degree equal to
the maximal ideal index I(w) for w in U , and the local currents
z
piece together to give the globally defined principal current
THEOREM 7.1 If (z) and (w) are one analytic ideals
then the local currents [Vz] and [Vw] agree on the intersection
U :~ U .
z w
Using theorem 5.8 above we can we can sometimes evaluate
Index f(~) when f is a complex valued analytic function without a
zero on the joint essential spectrum of ~ and [Ti,Tj~] is compact.
But there are hypothesis on the joint essential spectrum which allow
us to conclude more.
According to a definition in Harvey [9], suppose that M
is a compact subset of a Riemannian manifold N. Suppose that there
is a closed subset S (the scar set ) of M with Hausdorff 2k-I
measure zero such that the set of points in M but not is S, M/S, is
a real 2k-I dimensional submanifold of N/S of class Clhaving finite
volume. Such sets are called scarred (2k-l) manifolds.
The assumption that the joint essential spectrum of
is contained in a scarred one-manifold has strong consequences.
The following theorem implies that all the maximal ideals of the
algebra A which are not in the joint essential spectrum are one-
analytic, and shows how they assemble globally.
THEOREM 7.2 Suppose that T is a commuting s-tuple o__~f
operators with [Ti,Tj~ ] compact for all i and j, and suppose that
th 9 .~oint essential spectrum of T is contained in a scarred one
manifold M. If z is not in the joint essential spectrum of ~,
then (z) i__~s one-analytic, and there is a holomorphic one-chain $
i__~n CS/oess(~) whose local degree a__tt z i__~s th e maximal ideal index
I(z). This chain $ has a unique compactly supported current extension
T to c s with finite mass. T is the unique compactly supported

Carey and Pincus 631
A~
rectifiable (i,i) current so that aT(y) = Index f(~) when f i__ss
analytic and not vanishing on ~ (~).
- - ess
Under the additional assumption that the commutators of
the C ~ algebra generated by ~ are in trace class, we can evaluate
the boundary of T.
THEOREM 6.3 I__f_f [Ti,Tj~] is in the trace class for all
i and j, then ~T(fdh) = trace [ h(~,~), f(T,~)] where f(~,T~)
and h(~,~ ~) are operators defined for test functions f and h by
the Weyl functional calculus.
Because we are assuming that the two dimensional Hausdorff
measure of the joint essential spectrum of ~ is zero, we can use
the Hartogs-Rosenthal theorem to conclude that the continuous
functions on the joint essential spectrum of ~ can be approximated
by rational functions having poles in the complement of Gess(~).
It follows then from the last part of Theorem 6.2, that
~T determines the Kl(Oess(~homology class which corresponds to the
tuple T.
The boundary current here is a real MC cycle, where we
recall that a one dimensional real MC cycle on a Stein manifold
is a closed one-current ~ for which ~(~) = 0 whenever ~ is a
holomorphic one-form.
In [8] Harvey and Lawson eonslder the construction of
holomorphic chains which span real MC cycles supported on real
2k-i dimensional submanifolds of C s. This work was extended by
Harvey to the scarred case in [9].
The crux of the cited construction is the introduction
of a certain line bundle defined in terms of a function which is
shown (after some difficulty) to be rational using results of
Hadamard. We proceed directly to construct this function in terms
of Lefshetz numbers associated with the z-parameterized Koszul
complexes introduced above. In the one chain case we identify the
cocycles which determine the line bundle as a quotient of character-
istic polynomials of endomorphisms which act on the Koszul complex
of Theorem 5.2 above. Our local construction contrasts with the
Harvey-Lawson use of global data associated with the boundary cycle.

632 Carey and Pincus
A version of this construction holds in all dimensions.
While we do not know the invariance properties of the
local index beyond those described in Theorem 4.6 in general, the
additional assumptions of Theorem 6.2 imply stability under compact
perturbations for the principal current T.
THEOREM 6.4 Under the assumptions of Theorem 6.2, the
local index I(z) is stable under compact perturbations of the
generators which preserve commutativit[. The chain T is obtained
from the collection of operators TI+KI,...,Ts+K s if K. is compact
for all j, and these new generators commute.
- - J - -
Although the results above may seem somewhat special,
they are in fact just the relevant ones for certain geometric
applications. There is a canonical relationship between algebraic
geometry and operator theory which emerges from our considerations
and which may be fruitful for the further study of the singular
locus of analytic varieties.
THEOREM 6.5 Let W be a complex one-dimensional analytic
variety supported in an open se____~t of C s Let x be a point of W,
and let B~(x) be the ball of radius 6 about x __in C s. Let ds denote
arc length measure on F= W n ~B6(x), and let H2(ds) be the closure
i__n_n L2(F,ds) of the polynomials i__n_n (z I .... Zs). Then the subnormal
operators T defined to be multiplication by the coordinate
J
functions zj on th____ee Hardy space H2(ds), have commutators [Tj,Ti~ ]
in trace class. There is a principal current for the algebra
generated by these T. and the local index I(~) i__~s equal to the
J
negative of the multiplicity of the local ring of W for ~ in the
intersection of W and the ball.
We pose the question of connecting the multidimensional
versions of this theorem with the study of more subtle invariants
of singularities.
82 Remarks
We will not give a full development of the multidimensional
theory here. Instead we will merely indicate how the local index can
be obtained. A full description will be given elsewhere9
First, let us remark that the theory which we have

Carey and Pincus 633
introduced has the study of pseudo-differential operators which are
microlocally like Toeplitz operators as one of its applications.
Accordingly, in our limited space here, we will treat
only one case, that of the so-called crypto-integral algebras.
These algebras are generated by pseudo-differential operators of
order zero with scalar symbols which are asymptotically homogeneous
and which have compact support See [13].
The local index theory is most easily developed in terms
of the Koszul complex associated with a system of parameters,and
we will discuss the case where assumptions are made on ~ss(T).
We willassume that: T is a commuting s-tuple
of operators which generate a crypto-integral C~-algebra of
dimension two, and suppose that the joint essential spectrum i__ss
contained in a C 1 smooth scarred three-manifold
Form the real MC cycle defined by setting
~(fldf2~ df 3 ~ df4) = -2~ 2 trace [fl(~),f2(~),f3(~),f4(~) ]
where [',',','] denotes the completely antisymmetric form with
four entries.
Assume that ~ is a flat current. Of course this needs
further discussion, and will be analyzed in a forthcoming account.
Suppose also that z is a Fredholm point for T. Then the set of
projections ~ for which ~$(z) is not in HI(M ) -- is an open dense set
in the complex Gc(s-2,s). For such a ~ , let the pair
of operators ~ (~-z),obtained as before through the choice of a
dual basis, be denoted by (V1(z),V2(z)).
The Koszul complex defined by this commuting pair of
operators is
where the boundary maps are
and
d 2 d 1
0 + H § H @ H § H § 0
d2(x) = -V1 (z)x ~ V2 (z)x
d2(xSy ) = Vl(z)x + VZ(z)y
We will let H k = ker dk/range dk+l, and suppose for
the following theorem that H satisfies the conditions above.

634 Carey and Pincus
THEOREM 7.1 Consider the endomorphisms induced on the homology of
the Koszul complex by the s-2 operators ~(~-~). There is a holomorphic
two chain $ in cS/o ,~. whose local degree is equal to
- - ess~i ) -- __
I(z) = dim fl root space H(~-~)i H 0 + dim N root space~(~-~)IH 1
- dim l~ root space H (~-~) H
This chain has a unique compactly 2 supported current
extension T to C s with finite mass. T is the unique compactly
supported rectifiable (2,2) current so that ~T = ~. ibove'
the range hyperplane associated to ~ intersects the tangent cone
to the Taylor spectrum only at the vertex z.
Theorem 5.2 above was obtained by writing the dimensions
of the intersection of the corresponding root spaces as dimensions
of homology modules of a localized Koszul complex. We can rewrite
theorem 7.1 in the same way. We again view the Hilbert space as a
module which we localize at the maximal ideal which corresponds to
z. Then the pair of operators {V1(z),V2(z)} induces commuting
endomorphisms {[V1(z)] [V2(z)]} on H . The Koszul complex E(Hz,[V])
' Z
is defined to be the localization of the complex of the preceeding
page obtained by replacing H by H z, and by replacing the boundary
maps by the ones formed from the commuting endomorphisms of H just
z
described.
9 s
THEOREM 7.2 The Euler-Polncare characteristic of the Koszul complex
E(H [V]) is I(z).
When the Taylor spectrum has higher dimensionality
we proceed similarly. If we have a k-analytic maximal ideal, we
form parameters which correspond to projections in the Grassmanian
Gc(s-k,s), and the local index can be defined to be the Euler
characteristic of the corresponding multistep localized Koszul
complex. There is a corresponding Grothendieck group construction.
Just as in Theorem 4.8 above, we can express the
Noether index of operators formed from the generators in terms of
the Functional calculus by taking the intersections of divisors
and principal currents.
Suppose,for example, that F = (FI,F2):Cz+C2~ , 0 ~ F(M)

Carey and P 635
for holomorphic F . Then we can consider the Bott operator
]
I
-F2(2)" FI(I)~ ]
on H@H. The Noether index of this Fredholm operator is equal to
the current intersection T {] [div([)](1), where [div(~)] is the
so-called Poincare-Martinelli divisor of F. See [2].
Our main idea from [6] was that there is a holomorphic
chain representation of the Ext class with operator theoretic
meaning for the local degree. The study of holomorphic chains can
be expressed sheaf theoretically, and we will exploit this language
elsewhere. But we note here that our attention has been called, at
a recent conference organized by INCREST, to a preprint by M. Putinar
[18], and an announcement by R. Levi [19] where sheaf theoretic
language is used to study a generalized index theory associated
with the Euler-PQincare characteristics of Koszul complexes
constructed from s-tuples of operators. There are interesting
relations between this work and ours; but neither of these
cited authors introduces the concept of a local index, and they
do not make contact with the multiplicity theory of ideals nor with
the relationships between traces of commutators and boundary
currents. Yet their work has connections with ours and may be
looked upon (from our viewpoint) as being a step towards the
kind of localization which we have described above as the con-
struction of operator-theoretic local Chern classes.
Our interpretation of local Chern classes- and especially
the relationships between principal numbers and the local Euler
obstruction in higher dimensional situations will be treated elsewhere
in terms of the Grothendieck local cohomology. See [20].(The local
Euler obstruction was introduced by MacPherson in [21 ]as a numerical
invariant associated to singular points of algebraic spaces, and
used to construct Chern classes in homology for algebraic spaces.)
See [22] and [23]. We envisage the fully developed local index theory
of our operator algebras as providing relationships between ~ and
m
these geometric objects as well as relationships between integrals
of curvature forms in the neighborhood of singularities and appropriate
traces involving multicommutators.

636 Carey and Pincus
Of course the restriction of the theory to the situation
where the Taylor spectrum has analytic structure involves too much
loss of generality. There is the beginning of a more general
formulation for situations in which there are no underlying analytic
varieties.
We describe such an example now. Suppose that N is a
bounded normal operator and that Y is a bounded self-adjoint
operator such that the commutator [N,Y] is in the trace class. There
is no fully satisfactory notion in the literature,of which we are
aware, for the spectrum of the non-commutative algebra generated
by such a pair of operators. The joint essential spectrum is some
subset of R 3, and when this is a smooth curve one can talk in terms
of finding a spanning surface. But there are many ways to span-
even minimally.
Despite this we have a result which is the protQtype
for more general ones involving greater numbers of operators.
Using the Weyl calculus, it is possible to define a real closed
one-current A concentrated on the joint essential spectrum of
the pair of operators N and Y (without any smoothness assumptions
on the joint essential spectrum) by setting
A(RdS) = 2~i trace [R(N,Y),S(N,Y)].
THEOREM 7.2 A is a flat current having the form d[G] for [G]
rectifiable current having the form [G] = (H2 I_ E^ g~) where ~ i__ss
an orientation function, ~: Z§ (unit simple two vectors in R 3
tangent t__oo E, a bounded two dimensional rectifiable set.
The role of the principal function here is played by
the function g, which need now not be an integer, and the
rectifiable set E plays the role of a new spectrum.
It seems quite remarkable that the trace assumption on
the commutator [N,Y] implies so much geometric structure. But it
is also the case that the current [G] is constructed from local
data expressed in terms of symbols.
In [I0] this theorem is used to study self-adjoint
(and symmetric) Toeplitz operators on multiply connected plane
domains. The symbol of these operators has no smoothness beyond
essential boundedness, and the joint essential spectrum can be

Carey and Pincus 637
dense in Z. Nevertheless, the principal function is integer valued
H 2 almost everywhere on Z. A certain multiplication operator plays
the role of N in this example; and the joint essential spectrum of
the Toeplitz operator and N is computed as the H 2 I_ Z essential
discontinuity set of the principal function. We have thus construced
an integral varifold which spans the essential spectrum and whose
support is understood as a spectrum for the algebra.
It should be noted that the Euler characteristic of
a Koszul complex built directly from the operators N and Y in the
preceeding theorem - by passing to the Calkin algebra- v&nishes.
See [ 27] or [28]. This is also the case in the situation of
Theorem 7.2 above, where the number of generators is greater than
the "intrinsic" dimension of the Taylor spectrum. This circumstance
may perhaps make the distinction between the underlying ideas of
the original principal function approach as contrasted with its
variant in [13] more pointed. In the single operator case the
integrability of the principal function was perhaps the main fact.
Now rectifiability of the principal function seems to be of cardinal
importance in relation to an associated slicing and intersection
theory. We hope for a theory in which geometric singularities
are related to the local structure of the underlying operator algebras.
The trace can be associated with a type II factor.
For, in fundamental work, M. Breuer [25],[26] has created an index
theory in this context. We will close our remarks by giving an
example of a principal current in the yon Neumann algebra setting.
It is known that the C e algebra generated by the translations on
L 2 (R +) can be represented faithfully in a type II factor. In
[24] the authors showed how the principal function of a Toeplitz
operator whose symbol is a finite sum of exponentials coincides
with the Lagrange mean motion of the symbol. Now we wish to study
the more special Fredholm version of this result in which the
frequencies in the exponential are rational and we have Bohr mean
motion.
Let { Tj } be the commutative s-tuple of Wiener-Hopf
operators with the symbols exp(-2~iljt) _ where I. is a rational
J

638 Carey and Pincus
number expressed in lowest terms as %j = pj/qj, and where we take
0 < %1 < %2 < "'"

Carey and Pincus 639
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
............ On boundaries of complex analytic varietiies
II, Annals of Math. 106 (1977) 213-238
Harvey, F.R., Holomorphic chains and their boundaries,
Proc. Sympos. Pure Math., 3 0:I (1977) 309-382, Amer.
Math. Soc., Providence, Rhode Island
Pincus, J.D. and Xia, J. Self-adjoint and symmetric
Toeplitz operators on multiply connected plane domains,
Jour. Functional Analysis 59 (1984) 397-444
Federer, H., Geometric Measure Theory, Grundlehren
Math. Wiss. 153, Springer Verlag (1969) Berlin and
New York
Carey, R.W. and Pincus, J.D., An invariant for certain
operator algebras, Proc. National Acad. of Sciences
71 (1974) 1952-1956
Helton, J.W. and Howe, R.e., Integrql operators,
commutators, traces, index and homology, Lecture Notes
in Math. 345 (1973) 141-209
Carey, R.W. and Pincus, J.D., Pincipal functions, index
theory, geometric measure theory, and function algebras,
Integral Eq. and Operator Theory 2 (1979) 441-483
Carey, R.W., and Pincus, J.D., Index theory for operator
ranges, preprint (1984)
Markoe, A., Analytic families of differential complexes,
Jour. Functional Analysis 9 (1972) 181-188
Prossdorf, S., On the stability of the index of a one
dimensional singular integral operator with vanishing
symbol, Prob. Math. Analysis, Leningrad State Univ.
(1966) 70-79 Translation by Consultants Bureau, New York
Putinar, M., Base change and the Fredholm index,
Increst preprint No. 33 (1984)
Levi, R.N., Cohomology of collections of essentially
commuting operators, Functional Analysis and its
applications, 17 (1984) 229-230, Translation by Plenum
Press, New York
Hartshorne, R., Local cohomo~gy , Lect. Notes in Math.,
41 (1967) Springer Verlag, Berlin and New York
MacPherson, R., Chern classes for singular varieties,
Annals of Math i00 (1974) 423-432
Gonzalez-Sprinberg, G., L'obstruction locale d'Euler
et le theoreme de Mac-Pherson, Seminare E.N.S. (1978-79)
7-32 in ASTERISQUE 82-83 , Soc. Math. de France
Langevin, R., Courbures au voisanage d'une singularitie
algebraique isolee, Seminare E.N.S. (1978-79)33-43
in ASTERISQUE 82-83, Soc. Math. de France
Carey, R.W., and Pincus , J.D., Mean motion, Principal
functions, and the zeros of Dirichlet series, Integral
Equations and Operator Theory 2 (1979) 484-502

640 Carey and Pincus
[25]
[26]
[27]
[28]
Breuer, M., Fredholm theories in von Neumann algebras,l
Mth. Annalen 178 (1968) 243-254
Breuer, M., Fredholm theoried in yon Neumann algebras,II
Math. Annalen 180 (1969) 313-325
Curto, R. Fredholm and invertible n-tuples of operators.
The deformation problem, Trans. Amer. Math. Soc. 266
(1981) 129-159; and Fredholm and invertible tuples of
linear operators, Dissertation ,SUNY, Stony Brook,1978
Vasilescu,F.H.,On pairs of commutifig operators,Studia
Math. , 62 (1978) 203-207
Mathematics Department Mathematics Department
University of Kentucky StateUniversity of New York
Lexington, Kentucky 40506 Stony Brook, New York 11794
This research was partially supported by grants from the
National Science Foundation.
Submitted: January 7, 1985

Integral Equations
and Operator Theory
Vol. 8 (1985)
ON THE SYMBOL HOMOMORPHISM
OF A CERTAIN FRECHET ALGEBRA
OF SINGULAR INTEGRAL OPERATORS
0378-620X/85/050641-0951.50+0.20/0
9 1985 Birkh~user Verlag, Basel
Heinz-0tto Cordes and Elmar Sehrohe
We prove the surjeotivity of the symbol map of the
Frechet algebra obtained by completing an algebra of convolution
a~d multiplication operators in the topology generated by all
z . n . ,
L -Sobolev norms. The proof is based on an R -verslon of Egorov s
theorem valid for non-homogeneous principal symbols, discussed in
[5] , [6] . We use the hyperbolic equation ~u/~T:ilDl~u , 0

642 Cordes and Schrohe
rem on commutative C -algebras that the symbol map is surjective:
im ~ = A ker ~ = K(H ) In [4],[8] also a Frechet algebra A
s ~ s "
was investigated, obtained by completing A 0 above in the locally
convex topology generated by all Sobolev norms {II. II : s E N }
s
This algebra A also possesses a symbol homomorphism o:A + C(~) ,
governing the Fredholm properties of its operators.Moreover, A
may be remarkable by its property of being a ~ -subalgebra of
i(H ) , for any fixed s It shares one of the most important pro-
s ,
perties of C -subalgebras :If A 6 A is invertible in [(Hs) , then
the inverse is in A (ef.Gramsch [I0] ).
oo
On the other hand it seems an open problem whether the
symbol homomorphism o still is surjective,as for A s ,sE~. In that
respect a first result was given in [4] : ira o at least contains
a space of functions in C(~) which are infinitely differentiable
in every x-direction at points (x,~) with finite x .
In the present paper we will show that no condition
at all is required. In other words, also the symbol of the algebra
A maps onto C(~) The proof consists of a refinement of the
methods developed in [ 4] and [ 8] , replacing the translation ope-
iDz i< D) ~t
rators e used there by the one-parameter family e ,
with some fixed 0 < D < I . The important point is that the cha-
racteristic flow corresponding to this family vanishes at infinity
so that no differentiability of the symbol at all is required.
A very similar Freehet algebra on a compact manifold without
boundary was investigated by Atiyah and Singer [ I] Our present
method is also applicable for that algebra,as will be discussed in
a separate publication [II]
wing notations.
i. DEFINITIONS AND PRELIMINARY RESULTS.
1 9 1 . DEFINITION: For x E Nn , s @ 9 we use the follo-
(a) (x) : (l+IxI2) I/2 , ~x = (2~)-n/2dXl...dXn
(b) f^= Ff denotes the Fourier transform of a
tribution of H = {fES' : IIfll = ~ If^($)(~)Si2d $ < ~} denotes the
s s ~ n

Cordes and Schrohe 643
L2-Sobolev space of order s over ~n (a Hilbert space)
(c) L =L(H ) is the algebra of bounded operators
S S
H s§ ' Ks=Ks(Hs ) is the ideal of compact operators in Ls
(d) For f 6 L~(2 n) define the (Fourier-)multipliers
f(M) and f(D) by f(M)u(x) = f(x)u(x) , f(D) = F-If(M)F ,on H 0
Here f(D) 6 L ,for all s ,and llf(D) ll = sup{llf(D)ull :IIutT

644 Cordes and Schrohe
have A-SAA s - A E K t , for all s,t E ~ .
,
(e) A /K is a commutative C -algebra with unit and
8 S
thus isometrically isomorphic to some space C(~) of continuous
functions over a compact Hausdorff space ~, by the Gelfand-Naimark
theorem. We call the function a 6 C(~) corresponding to A+K the
S
the symbol of A , denoted by o(A) or OA By (b) all the A s
are isometrically isomorphic, and the 'symbol spaces' 9 for diffe-
rent s may be identified. By (d) this can be done such that the
'symbol map' A + ~ for the generators is independent of s
(f) (Calderon interpolation theorem) If s

Cordes and Schrohe 645
PROOF: Use prop.l.6 and induction.
An important tool in the theory of ~do's is the result
below,which is an immediate consequence of the Calderon-Vaillan-
court theorem (cf.[2]),and calculus of ~do's ([3],[6],for example)
1.9.THEOREM: The pseudo-differential operator A=Op a
with symbol in CB~(N n) is in L for every s. Moreover, for every
s
fixed s we have an estimate
I'AII < c sup {ID~D~a(x,~)I : x,~ E ~n IeI IBI~
S -- S ' ' ~S } '
with constants c s ' Ys depending on s and n , but not on a .
2.RESULTS AND PROOFS.
The following theorem gives the main result of this article.
2.1. THEOREM: If a E C(N) , then there exists an
operator A ~ A with ~ = a .
We will break up the proof of thm.2.1 into a series of
lemmata. In the remainder of this section we keep a real number
, 0 < ~ < i given fixed.
2.2. DEFINITION: For A E A 0 ,t 6 N define the operator
A t = e-iA-~tA eiA-nt ,
with the unitary operators e iA-qt = Op (e i(~)nt) of every H s .
2.3. LEMMA: Let A be an operator of the algebra A 0
(a) We have A t E A , for all real t .
(b) The map t + A t is differentiable from N to As '
in the operator norm of L s , for every s.
(c) The commutators, below, are in every L , and
s
d/dt A t = i[At,A-n] = i[A,A-n] t .
(d) We have
OAt(X,~) = OA(X,~) , x,~ e N .
PROOF: Clearly it is sufficient to give the proof for
the generators of A 0. We get At= A for A = ~(D) , ~ = ~ , sj ,
so that the statement is trivial for the Fourier multipliers. On
n
the other hand, for the multipliers we must invoke the N -vers!on

646 Cordes and Schrohe
of Egorov's theorem discussed in [5] (for a discussion in full
detail we refer to [6] (to appear)). We Only require the result
for the simplest case of the l system du/dt = int
with the solution operator e This equation satisfies all
assumptions of thm.4.1 in [5] (or better,thms.12.1 and 12.2)
In particular it is important that we do not require the hamilto-
nian system defining the characteristic flow to be homogeneneous
in $ ,as most other authors do. That system here is of the form
x" = (

Cordes and Schrohe 647
radicand R we get the estimate (2+t2) -I _< R ~ (2+t 2) = nt 2
First of all, R!l+2ulvIIwI+u21wI2 qt 2 . Furthermore, if IvI>ultI, then we
get R>u2+(Ivl-ultl)2=l-2ulvIItI+u2t2>l+t2-~(l-u2)-~-lt2u2=(l-E)_
+u2(E-(e-l-l)t 2) , with arbitrary 0

648 Cordes and Schrohe
(b) The commutator [A-n,B A] (first considered as an
operator in L(Hq,H_n) ) is (extends to) an operator in A 0. We have
[A-q,BA] = -iA+iB A , hence [A-q,B A] has symbol zero .
PROOF: (a) is an immediate consequence of lemma 2.5.
For (b) we first assume A E A 0 again. One gets
t ~ t
-A = I ~ d/dt(e-tAt)dt =- I e- Atdt_i I e- [A-I,At]dt =_B+[A-I,B] ,
0 0 0
where the fact may be used that A and B are @do's in ~c 0 (ef.[5]).
Then, for a general A E A 0 we again do the approximation in the
proof of lemma 2.5, observing that -iA+iB A converges in A 0 . Hence
[A-q,BA]U converges for u E H q , and the limit is (-iA+iBA)U .
Hence BA(dOm A -q) C dom A -n, so that [A-I,BA ] : dom A -q + H 0 is
well defined as a commutator of a bounded and an unbounded opera-
tor. Moreover, that commutator even is bounded and equal to
-iA+iB A ,as stated. Since we have OA=OB it then is trivial that
the symbol of the commutator vanishes.
2.8.LEMMA: Let a E C(N), and let k=l,2,...
be arbitrary. Pick any operator P E A 0 with symbol a , and define
A = Bk(p) . Then o A = a . Moreover, for j=0,...,k-i we have
Aj = ad]A-n(A) = i](I-B-I)JBk(p) E A 0 ,and OA. = 0 ,
3
with the operator B=BA=B(A) of 2.6, and B-l(Br(Q))written for
Br-I(Q)
The proof is an evident consequence of Lemma 2.7.
2.9.LEMMA: Given any a C C(N) It is possible
to find an operator A E A 0 ,j=0,1,..., with symbol a such that
Aj = ad]A-q(A) 6 A 0 , and OA.= 0 , j=l,2, ....
]
PROOF: By lemma 2.8, for k=l,2,..., we get a set
of operators A k , Alk ,..., Akk corresponding to the operators
A , Aj , of the lemma. Thus the proof of 2.9 is reduced to the
construction of an operator A from the operators A k , all having
symbol a , in a way that extends the properties of lemma 2.8 to
all j=8,i,2, .... The way to achieve this is described in detail
in [4] ,p.162-164.

Cordes and Schrohe 649
PROOF OF THEOREM 2.1. Let {Aj} be the sequence
of 2.9. We have ~ = a ,and a(adkA-~(A))=0 . Hence all operators
adkA-l(A)A k and Akad~-l(A) have their symbol vanishing over 9 .
This means that they are compact. Now lemma 1.8 implies that
A E A
1.
2.
3.
4.
5.
6.
7.
8.
9.
I0.
II.
REFERENCES
Atiyah, M., and Singer, I.: The index of elliptic ope-
rators, Ann.of Math.66 (1968) 484-530.
Calderon, A.P., and Vaillancourt, R.: On the bounded-
hess of pseudodifferential operators, JJ ~th. Soe. Japan
23 (1971), 274-378.
Cordes, H.O.: A global parametrix for pseudodifferen-
tial operators over Nn, preprint SFB 72 Math.lnst. Uni-
versitaet Bonn (1976).
Cordes, H.0.: Elliptic pseudodifferential operators,
an abstract theory, Springer Lecture Notes in Math.
voi.756, Berlin Heidelberg New York 1979.
Cordes, H.0.: A version of Egorov's theorem for
systems of hyperbolic pseudodifferential equations,
J.of Functional Analysis 48 (1982) 285-300.
Cordes, H.0.:Techniques of pseudodifferential opera-
tors, to appear.
Cordes, H.O. and Herman, E.: Gelfand theory of pseudo-
differential operators, Amer. J.Math.90 (1968) 681-717,
Cordes, H.0. and Williams, D.: An algebra of pseudo-
differential operators with nonsmooth symbol, Pac. J.
Math. 70 (1977) 101-116.
Gohberg, I.C.: On the theory of multi-dimensional
singular integrals, Soviet Math.l ~1960) 960.
Gramsch, B.: Relative Inversion in der Stoerungstheo-
rie von Operatoren und ~-algebren, Math. Annalen 269
(1984) 27-71.
Schrohe, E.: The symbol of an algebra of pseudodiffe-
rential operators, to appear.
H. 0. Cordes,
Department of Mathematics,
University of California,
Berkeley,Ca. 94720
U.S.A.
Submitted: May 13, 1985
Elmar Schrohe
PB Mathematik
Johannes-Gutenberg Univ.
Saarstr.21
6500 Mainz
West Germany

Integral Equations
and Operator Theory
Vol. 8 (1985)
0378-620X/85/050650-2451.50+0.20/0
9 1985 Birkh~user Verlag, Basel
WIENER-HOPF OPERATORS AND GENERALIZED ANALYTIC FUNCTIONS
Raul E. Curto*, Paul S. Muhly*, and Jingbo Xia
An almost periodic generalization of H +C is defined
and analyzed9 Applications of this analysis are made to the type
II index theory of Wiener-Hopf operators with almost periodic
symbols9
w INTRODUCTION
In [5], Coburn, Douglas, Schaeffer, and Singer developed
an index theory9 relative to a certain II~ factor, for Wiener-
Hopf or Toeplitz operators with continuous almost periodic sym-
bols. Their results generalize the well-known theory of Toeplitz
operators on the circle whose symbols lie in C(~). Our objec-
tive in this note is to find an analogue of H +C for almost
periodic functions and to use it to extend the theory of Coburn,
Douglas, Schaeffer, and Singer in much the same way that Douglas
[6] extended the theory of Toeplitz operators with continuous sym-
bols to cover Toeplitz operators with symbols in H +C. As one
might expect, a completely smooth transference of the theory for
Toeplitz operators with symbols in H +C to the almost periodic
setting is not possible. The obstacles involve some interesting
problems in the value distribution theory of analytic almost peri-
odic functions.
Let ~B denote the Bohr compactification of ' ~ and
let CAP~R) denote the algebra of continuous almost periodic
functions on ~. Then CAP(R) is a commutative C*-algebra9
under the sup-norm and pointwise operations, whose maximal ideal
space is ~B If AAP(~) denotes CAP(E) N H~(~) where H~(~)
is the classical Hardy space of boundary-values of functions that
Supported in part by the National Science Foundation.

Curto, Muhly and Xia 651
are bounded and analytic in the upper half-plane, then AAP(~)
may be viewed as a function algebra on ~B; in fact, it is a
Dirichlet algebra there, i.e., the real parts of the functions in
AAP(~) are uniformly dense in the real-valued functions in
CAP(~). Moreover, Haar measure m on ~B is multiplicative on
AAP(~), and so, for each p, 0< p~ ~, one may build an abstract
Hardy space, HP(m), based on m and AAP(~). The real line
may be viewed as a one parameter subgroup {et~t~
of ~B and
the action of ~ on ~B determined by this subgroup is ergodic
with respect to m. It results from this that for each function
6 H~(m), there is a null set E in ~B, that depends on ~,
such that for x not in E, the function of t, ~Cx+et) ,
belongs to H~(~). For each such x, then, we may extend the
resulting function to the upper half-plane using the Poisson ker-
I Im(z)
nel: ~x(Z) = ~]R~(x§ where Pz(t) = ~ It_z[2. As a
function of z, with x ~ E fixed, ~x(Z) is nice and smooth,
for Im(z) ~ 0, but as a function of x, ~x(Z) need not be
smooth in any sense. 0ur replacement for H ~ is an algebra H
consisting of functions ~ 6 H~(m) such that ~x(Z) is continu-
ous in x and z, or what comes to the same thing,
H ~ {~ E H~(m)I ~* Piy(X) is continuous on ~Bx (0,~)] where
* Piy(X) = ~(x+et)Piy(t)dt" The key feature of H that we
will want to exploit is that we may view functions in H as ele-
ments of both H~(m) and H~(~). Our analogue of H~+C, then,
is H+ CAP(~). When represented on ~, H+ CAP(~) is norm-closed
in L~(~) and when represented on ~B it is norm-closed in
L~(m).
The main result of Section 2 is an invertibility cri-
terion for functions in H+ CAP(~). It is a direct analogue of
the familiar criterion for H +C. In Section 3, we develop a
type II index theory for Wiener-Hopf operators with symbols in
H+ CAP~). Our main theorem here provides a sufficient condition
for a Wiener-Hopf operator to be Fredholm, but unfortunately we
are unable to decide if the condition is necessary. We conclude
by exploring the difficulties that arise in one approach one
might take to find a necessary condition. Here is where contact

652 Curto, Muhly and Xia
with the value distribution theory of analytic almost periodic
functions is made.
In closing this introduction, we note that in addition
to [5], we were motivated by the papers of Carey and Pincus, [3]
and [4], where detailed investigations are made of Wiener-Hopf
and related operators with almost periodic symbols.
w THE FuNcTION ALGEBRA H+ CAP(E)
We begin by recalling how to view ~B as an inverse
limit of tori. Regard the discrete reals ~d' which is the dual
of ~B as a vector space over ~, the rational numbers, and
let A be a basis for ~d over ~, consisting of positive num-
bers. Let I = [(y,n) [ y is a finite subset of A and n is a pos-
itive integer] and direct I by the relation (y,n) < (~,m) if
and only if y ~ ~ and n ~ m. For (y,n) 6 I, we write
rmlYl m2Y2+ mkYk ]
n~!Z= ~-~.~T n! "'" +--~! m i E Z], where y = [yi,Y2,..-,yk ].
Then for (y,n)~< (o,m), is a subgroup of m! " It follows
that we may view ~d as the direct limit li~ n~! ~. If T(y,n)
I
denotes the dual of n-~!~, then T(y,n) is a torus of dimension
equal to the cardinality of y. Moreover, for (y,n) < (o,m),
there is a natural projection from T(o,m) onto T(y,n), and
this yields the representation of ~B as lim T(y,n).
Corresponding to this representation of ~B we obtain
the representation of CAP(E) as lim CAP(y,n) where CAP(y,n)
is the set of all functions in CAP(E) whose Bohr-Fourier coef-
ficients are supported in n~!~. The maximal ideal space of
CAP(y,n) is T(y,n). We may also view AAP(~) as lim AAP(y,n)
where AAP(y,n) = CAP(y,n) a H~(~). However, we want to refine
this limit representation of AAP(~). For each fixed y,
AAP(y,n) is naturally isomorphic to AAP(y,m) for any two posi-
tive integers n and m. So, for the sake of discussion, we
shall consider AAP(y,I) with y = [yl,Y2,...,yk]. A function
f(t)
Z ~(mlYl+---+mkYk)exp~i(mlYl+m2Y2+'''+mkYk)t] in CAP(y,i)
is identified with the function ~ on ~k whose Fourier series

Curto, Muhly and Xia 653
ml m2... mk
is ~ ~(mlYl+'.-+mkYk)Zl z 2 z k Bohr called ~ the spatial
extension of f and proved that the correspondence f ~ ~ ~ is
an algebraic *-isomorphism between CAP(y,I) and c(~k). Under
this representation, then, AAP(y,I) is identified with the al-
gebra A(P) ~ C(~ k) where P = [(ml,''',mk) 6 Z k ] mlY I+ --. +
mkY k 0] and A(P) = If 6 c(~k) [ ~(ml,..j,m k) = 0,(ml,...,m k)
P]. Now P totally orders Z k and (~K,p) may be viewed as
a dimension group in the sense of [7]. Therefore, as is well
known (cf. [7]), we may realize (~k,p) as the inductive limit
of a sequence [(zk,P(s where for each ~, P(~) is lin-
k
early isomorphic to ~+ via a linear transformation in SL(k,Z).
This, in turn, shows that A(P) is the inductive limit of a se-
quence of function algebras on Tk each term of which is iso-
morphic to the polydisc algebra on Tk. The usefulness of this
observation for our purposes is that the functions in each term
of the sequence have smooth extensions into the polydisc. For
technical reasons, it is useful to us to have a slightly differ-
ent representation of A(P) as an inductive limit of algebras
with this property.
For each j, I~ j ~ k, let [~]s (resp.
j ~ )
[~s163
be an increasing sequence (resp. decreasing sequence) of positive
irrationalk numbers such that lim~ ~J yjs = .= lJm~ ~,J and set
P~ = jNl[(ml,-.-,m
k ) = E zk [ mlYl + ...+mjc~+ ..-+mkY k m 0, c js =
~J and ~]. Then it is clear that P~ is isomorphic via a
k
linear transformation in SL(k,Z), to a subsemigroup of Z+. We
now define the subsemigroup P](n) of m+ to be [l(mlYl+-..
+mkYk) [ (ml,''.,m k) s P~] and we set AAP(P~(n)) = [f E CAP(m) [
~(a) = 0, ~ ~ P~(n)]. Then it is clear that AAP(y,n) =
li9 AAP(P~(n)).
Viewing AAP(P~(n)) as a subset of c(~B), we denote
the weak-* closure of AAP(P~(n)) in L~(m) by H~(P~(n)).
From the discussion above, each algebra H (P (n)) is isomorphic
to a subalgebra of H~(~ k) where k = card(y) and for y ~ ~,
n ~ m, and ~ ~ k, H (P (n)) ~ H~(P (m)). The algebra we are

654 Curto, Muhly and Xia
seeking is H, which is defined to be the norm closure of
U H~(P (n)) in L (m), i.e., in an obvious sense,
y,n,~
H = Ig~ H~(P~(n)). As defined, H is a subalgebra of L~(m), in
fact, a subalgebra of H~(m); we want to show that H may be
viewed also as a subalgebra of H~(~).
To this end, recall that we view ~ as acting on ~B
via the dense one-parameter subgroup ~et]tE ~
of ~B that con-
sists of those characters of ~d that are continuous whe~ ~ is
given the usual topology. A function ~ 6 C QR B) restricts to
[et]tE ~ yielding an almost periodic function on ~; i.e., as a
function of t, ~(e t) is almost periodic. And conversely, each
almost periodic function on ~ may be viewed as ~(e t) for a
unique ~ E c(~B). Given a function ~ E LP(m), I~ p~ ~, and a
function f E LI(~), we define ~, f by the formula ~* f(x) =
~(x+et)f(t)dt. Then ~* f E LP(m) and [I~* fIILP(m ) ~
l[~l[ Lp (m)II flJ LI(~ ) 9
For a real number ~, we write [~] for the
character of ~B that X determines; so [X](x) is the value
of the character at x s ~B. The Fourier series of ~* f is
~(k)~(~) [k] where kE~ ~(~)[~] is the Fourier series of
and ~ is the ordinary Fourier transform of f. In particular,
Piy( I y is the Poisson kernel then
if f(t) = t) = ~ t2+y 2
2.1) ~,P. ~ Z $(X)e-YlxlExl.
ly kE~
The thing that prevents this series from converging on ~B is
A
that the support of ~ may have finite cluster points.
LEMMA 2.2. Let ~ E H~(P~(n)). Then the series (2.1)
converges uniformly and absolutely for x E~B and y > O. More-
over, there i8 a C~-function h on ~k (0,~), where
iwlt iwkt
k = card y, such that ~* Piy(et) = h(e ,''',e ;y) for
certain real numbers Wl,-'',w k.
Proof. It suffices to consider P~(1) and, in view of
the preceding discussion, it suffices to prove that if mlY I+ ---
+ mku k E P~(1), then mlYl + ... +mku k ~ 6([ml[+[m2I+.-.+imk[)

Curto, Muhly and Xia 655
0, then + 9 9 9
for some 6 > 0. Fix j, i~ j ~ k. If mj mlY I
+ mkYk = (mlYl +---+mj~j+--.+mkYk)+ mj(yj-~) ~ mj(yj-~), while
J J" j
if m. < 0 then mlY I+ -.. + mkY k ~ (yj-~)mj where ~s and ~
j
are the numbers used to define PY(1). If we take
6 = infj min[yj-a~,~j ~-Yj]/k, then the desired inequality is sat-
isfied.
COROLLARY 2.3. If ~ s H, then for y > O, ~* P.
my
E C(]RB). Moreover, if F is defined on the upper half-plane by
the formula F(t+iy) = ~o* Piy(et)" then F is bounded and analytic
there, and IIFIIH~(:]R ) = IlcgliH~(m). Thus the map ~ ~ F is an
isometric isDmorphism from H onto a norm-closed subalgebra of
H~(~) consisting of functions F such that for every y > O,
F(t+iy), as a function of t, lies in AAP(~).
Proof. The first assertion is true for ~ E H~(P~(n)),
by the lemma, and it is clear that the property is preserved when
taking uniform limits. Therefore ~* P. 6 C(]R B) for all y > 0
my
and ~ E H. The fact that F is bounded and analytic on the
upper half-plane is clear. The only thing that needs checking is
the equality of I[F[IH~(~ ) and II~II
H~(m)
. By [12], for each
E H, lim ~* P. = ~
y~0+ my
a.e. (m). Therefore IIqolI~
lim II~0, Piyll ~ = lira IIF(-+iy)ll~ ~ IIFII~. on the other hand,
y~0+ y~0+
IF(t+iy)I = I~* Piy(t)I ~ II~* PiyiI~ ~ II~oli~ 9
IIH| m)"
Therefore liFil
Henceforth we view functions in H either as functions
-- II
on IN B or as functions on I~ and we will not distinguish be-
tween these two points of view notationally. Likewise, we will
view H+ CAP(I~) as H+ C(I~B).
We note in passing that when viewed as a space of func-
tions on ]R B, H ~ ~ s H=(m) I ~* P. E C(]R B) for all y > 0],
~Y
while on JR, H ~ [~0 s H~(]R) I ~* P-
~y
s AAP(~) for all y > 0]. It
would be very interesting to determine if equality holds in
either case.
The following lemma is due to Rudin [4].

656 Curto, Muhly and Xia
LEMMA 2.4. Let
Banach space X and let
A on X such that
a) AX m Y and AZ ~ Z
b) sup AlllA <
c) for each y E Y and
such that IIAY - Yll ~ r
Then Y+ Z is closed.
Y and Z be closed subspaces of a
be a family of bounded linear maps
for all A s ~;
and
c > O, there is a A 6
We wish to apply this lemma in the context where X is
the closure of H+ CAP(R) in L~(~) or in L~(m), Y = CAP(~),
Z = H, and ~ = lAy]y>0 where Ay~ = ~* Piy' ~ s L~(~) or
L| Since A CAP(~) ~ CAP(~), and A H ~ AAP(~) ~ H, it is
Y Y
easy to see that condition a) of Lemma 2.4 is satisfied. Also,
condition b) is satisfied since each A is a contraction on
Y
both L~(~) and L~(m). Finally, condition c) is satisfied be-
cause translation is continuous on CAP(~) and [Piy]y>0 is an
approximate identity for LI(~). We conclude, therefore, that
H+ CAP(~) is closed in X and so when viewed either as a sub-
space of L~(~) or of L~(m), H+ CAP(R) is closed. This leads
us to
THEOREM 2.5. When viewed either as a subspace of
L~(~) or L~(m), H+ CAP(~) is a norm closed subalgebra.
Proof. We need only prove that H+ CAP(F) is an alge-
bra. For this, it is convenient to work on ~B, and then it
suffices to show that if ~ s H~(P~(n)), for some y, n, and ~,
and if w is a positive real number, then [-w]~ E H+ CAP(~).
(Recall: [-w] is the character of ~B determined by -w.) We
may suppose that w = N(Yl+Y2+...+yk) for some positive integer
N, where y = (yl,...,yk). For if w is not of this form, then
we may write w = N(Yl+..-+y k) - w 0 where w 0 z 0 and
l
[w0]H~(P~(n)) ~ H~(P~e(nl)) for some y1 ~ Y, n I ~ n, and an
appropriate Za. It follows that [-w]~ E H+ CAP(~) if and only
if [-N(Yl+Y2+..-+yk)]~ lies in H+ CAP(~). But by definition
of P~(n), it is easy to see that [-N(Yl+Y2+...+yk)]~ differs
by a trigonometric polynomial from a function in H~(P Y (n))

Curto, Muhly and Xia 657
where ~I > ~" This completes the proof.
THEOREM 2.6. For ~,~ E H+ CAP(~),
~=~+LI(~ *Piy)(* * Piy)-(~)* Piyll~ = o;
i.e., the Poisson kernel is asymptotically multiplicative on
+ CAP(~).
Proof. Since P. is multiplicative on H and since
my
polynomials are dense in CAP(~), it suffices to verify the
assertion of the theorem when ~ = [-w], for any w > 0, and
any
E H~(P~(n)). Using equation (2.1), a simple calculation
shows that (~, Piy)(r Piy)
~(k) [e-Y(W+k)-e -y(w-k)] [k-w]
O~k~w
+ (~)* Piy' where the sum is, in fact, finite. The result fol-
lows immediately.
We come now to the statement of the main theorem of
this section. Its proof is somewhat involved and will be spread
through a series of lemmas. Each will be of interest in its own
right, and the route is scenic.
THEOREM 2.7. View H+ CAP(~) as a subalgebra of
L~(~) and let ~ s H+ CAP(E). Then ~ is invertible in
H+ CAP(E) if and only if there is an r > 0 and 6 > 0 such
that
(2.8) inf[I~, Piy(t)] I 0 < y < r, t s ~ 6.
The necessity of this condition follows easily from
Theorem 2.6. So we may suppose that (2.8) is satisfied and then
show that ~ has an inverse in H+ CAP(~). For 6 as in (2.8),
we can find w > 0 and ~0 E H~(P~(n)), for suitable Y, n,
and s such that ll~(t)-e-Wt~0(t)II~ < -- 88 Then ~(t)
e-iwt~0(t) is invertible in L~(~) and its inverse ~-i has
norm at most 4__ 36" Consequently, I[l-~II~ ~ ~. so, if we can
show that ~-i E H+ CAP(~), then ~ will be invertible in
H But to do this, we need only show that ~I lies
+ CAP(E). in
H+ CAP(~). This is where the lemmas begin.
Recall that the mean motion of a nonvanishing almost
periodic function ~ is defined to be the limit

658 Curto, Muhly and Xia
tim 1 [Arg ~(~)-Arg ~(~)]
(~_~).~ ~-~
whenever the limit exists. If ~ is invertible in CAP(R),
then the limit exists and if the limit is w, then one may
write
~(t) = e i~t exp(~(t))
for an almost periodic ~. (See [9] for a comprehensive treat-
ment of the theory of mean motions of almost periodic functions.)
LEMMA 2.9. If ~ E AAP(~) is invertible in CAP(R),
then the mean motion of ~ is nonnegative and the following
assertions are equivalent:
i) ~ is invertible in AAP(~);
ii) ~ = exp ~ for some ~ E AAP(~); and
iii) the mean motion of ~ is O.
Proof. Let w be the mean motion of ~. Then since
w belongs to the spectrum of ~ [I0, Chapter VI], it follows
that w ~ 0. Clearly ii) implies i) and iii). That i) implies
ii) follows from the fact that the maximal ideal space of AAP(~)
is homeomorphic to the cone over ~B ([2]) and therefore is con-
tractible. On general principles, then, every invertible element
of AAP(~) has a logarithm in AAP(~). Suppose, now, that w =0.
As was noted before the statement of the lemma, we may write
= exp 9 where 9 E CAP(~). We must show 9 6 AAP(~). If
is a polynomial, then 9 is an absolutely convergent Bohr-
Fourier series, and we may write 9 = 91+~ 2 where 91,92 E
AAP(~). But then, on the one hand~ exp @2 ~ AAP(~) (unless 92
is constant), while on the other, exp 92 = ~ exp(-@l s AAP(~).
This shows that ~ E AAP(~). In the general case, we approximate
by a sequence of polynomials that are invertible in AAP(~),
[~n]n:l , and write ~n = exp ~n' ~n s AAP(~). Then
Ill- exp(gn-~)II =
II~-~nIIII~-III~, which implies that there is a
sequence of integers [s I such that I It- ~n + 2nis o.
it follows that 9 s AAP(~), completing the proof.
It is instructive to make explicit the geometric con-
tent of Lemma 2.9. The maximal ideal space of AAP(~) is the

Curto, Muhly and Xia 659
so-called "big disc", ~B [0,~]/~B ~]. If ~ s AAP(~) is
viewed as a function on ~B, and if y > 0 is finite, then the
A
value of the Gelfand transform ~ of ~ at (x,y) is @(x,y) =
~* Piy(X) while the value of @ at [~B ~]] is ~B ~dm.
(See [ii] for an expanded discussion of this.) If ~ is invert-
A
ible in CAP(E) then ~ is bounded away from 0 in some neigh-
borhood of the boundary ~B ~0] of the big disc and the mean
A
motion "counts" the zeros of ~ in the interior of the big disc.
The mean motion of ~ vanishes precisely when there are no zeros
of $ in the interior of the big disc, i.e., precisely when
is invertible in AAP(~).
LEMMA 2.10. Let ~ s AAP(P~(n)) and suppose ~ is
invertible in CAP(JR). Then ~ may be written as ~ = exp ~,
for some ~ E AAP(P~(n)), if and only if the mean motion of
is zero.
Proof. The necessity is trivial. So suppose the mean
motion of ~ is 0, and apply Lemma 2.9 to write ~ = exp
with r E AAP(~). Without loss of generality, we may assume that
~oB ~dm = 0. Then ~II~* PiyiI~ = o. it sufffces to show that
for some y > 0, the spectrum of ~* Piy lies in P~(n). Since
convolving with the Poisson kernel is a homomorphism of AAP(~)
and of AAP(P~(n)), we find that ~* P.y = exp(@ I'PlY) for all
y > 0 Choosing y so that Ill-exp(@l*Piy)II~
9 ~, we find,
upon expanding the logarithm in a power series, that @, P. =
my
log[l-(l-exp(@, Ply)) ] = log[l-(l-~* Ply)] lies in AAP(P~(n)).
This completes the proof.
LEMMA 2.11. Let ~ 6 H~(P~(n)) and suppose that there
are positive numbers, r and 6, such that
inf[l~, Piy(t)l t t E~, 0 < y < r] 6
and that for each y E (0, r), ~, P. has mean motion zero~
IV
Then ~ is invertible in H~(P~(n))[
Proof. For each y > 0, ~* P. E AAP(~) and so we
my
may view ~ as extended to the interior of the big disc,
~B (0,~]/~B ~]. Our hypothesis says that this extension is

660 Curto, Muhly and Xia
is bounded away from 0 by $ in the region ~Bx (0,r) and
that it has no zeros in ~B (0 ~]/~R B x [~]. We would like to
conclude that the reciprocal of the extension represents the in-
verse of ~ in H~(P~(n)).~ To this end, fix Y0 E (0,r). Then
, P. E AAP(P~(n)). Our hypotheses imply that ~, P. is in-
lY 0 ~ lY 0
vertible in CAP(~) and has mean motion zero. Therefore by Lem-
ma 2.10, there is a ~Y0 ( AAP(P~(n))~ such that ~, P.my0 =
exp ~Y0" For Y > Y0' we have ~* Ply = exp(@y 0* Pi(y-y0)) by
the multiplicativity of convolving with the Poisson kernel and by
the fact that the Poisson kernel forms a convolution semigroup.
So, while @Y0 is not quite unique, it is possible to find a
family [~Y]y>0 in AAP(P~(n)) such that ~* Piy = exp(~y) for
all y > 0 and such that @YI* P'iy2 = @yl+y 2 for yl,y 2 > 0.
It follows from this quite easily that the_family [exp(-@y)]y>O
converges in the weak-, topology on H~(P~(n)), as y w 0+, to
an inverse of ~.
COROLLARY 2.12. Let ~ s H satisfy the conditions of
Lemma 2.11. Then ~ is invertible in H.
Proof. Choose u ~, n, and a
Y
~0 E H (P~(n)) such
that II~-~oll ~ m w~' where ~6 is as in the statement of Lemma
2.11. Then I~ 0* Piy(t)I ~ ~6 for y 6 (0,r) and ~* Piy and
* Ply have the same mean motion, namely zero, when y is suf-
~0
-I
ficiently small. By Lemma 2.11, ~0 has an inverse DO in
H~(P~(n)) and of course
-i
so ~0 ' and hence ~,
II~0111 ~ ~6 9 Therefore,
is invertible in H.
Ill-~lll ~ ! 3,
COROLLARY 2.13. Let K be any one of the spaces
y
AAP(P~(n)), H (Ps or H, and let ~ E K satisfy
inf[I~, Piy(t)I I t (~, y > 0] ~ 6,
for some 6 > O. Then ~ is invertible in K.
Proof. We need only check that ~* P. has mean
my
motion zero for each y > 0. However, the map Y * ~* PiY from
(0,~) to CAP(~) is continuous and lim ~* P. is the constant
y~ ly
function whose value is -~ B ~dm, which can't be zero by our
-m

Curto, Muhly and Xia 661
hypotheses. It follows that the mean motion of ~* P. is zero
my
for each y ~ 0.
We know that a function in the disc algebra A(~) or
the algebra H~(T) is invertible if and only if its harmonic ex-
tension to the open unit disc is bounded away from zero. This
fact is, of course, an immediate consequence of the definitions
of these algebras. An analogous fact is true for AAP(~) be-
cause ~B [0,~]/~B {~] is its maximal ideal space. Corollary
2.13 says that the algebras AAP(P~(n)), H~(P~(n)), and H
share this property with AAP(~). However, the proof is rather
roundabout.
The following lemma is the analogue for H of
the fact that if ~ E H~(~) and if (the harmonic extension of)
satisfies [~(z)[ ~ 6 > 0 for r < Iz[ < I, then ~ = ~i~2
where ~I is invertible in H~(~) and ~2 is a finite Blaschke
product. It is the last step needed to prove Theorem 2.7.
LEMMA 2.14. If ~ E H~(P~(n)) and if there are posi-
tive numbers 6 and r such that
inf{]~, Ply(t) [ It Em, o < y < r] 6,
then ~ = ~i~2 where
vergent; and
motion.
a) ~I E H and ~I is invertible in H;
b) ~2 E AAP(~) and ~2 is invertible in CAP(~);
c) the Bohr-Fourier series of ~2 is absolutely con-
d) the spectrum of ~2 is bounded above by its mean
Proof of Theorem 2.7
If ~0+@0 E H+ CAP(~) satisfies the condition of the
theorem, then with a slight decrease in 6 we can replace ~0+@0
with ~+~ where ~ belongs to some H~(P~(n))_ and 40 is a
trigonometric polynomial. But then, for a suitable w E ~, y~,
n l , and ~, eiWt(~ct)+~(t)) lies in
4
H~(P~, (n ~ )) and satis-
fies the estimate of Lemma 2.14. The conclusion of the lemma
implies that eiWt(~(t)+@(t)) is invertible in H+ CAP(~) and
so, therefore, ~+@ is invertib!e in H+ CAP(~). Finally, then,

662 Curto, Muhly and Xia
we conclude that ~0+@0 is invertible in H+ CAP(~).
Proof of Lemma 2.14
Fix Y0 E (0,r). Then ~* P. E AAP(P~(n)) and Lemma
mY0
iwlt iwkt
2.2 implies that ~ , P. (t) = h(e ,..-,e ) for certain
mY0
real numbers Wl,...,w k and a C~-function h on ~k. On the
other hand, the hypothesis that 9" P. is invertible in
IY 0
CAP(~) implies that we may write 9" P. = e iwt exp @ where
mY0
E CAP(~) and w k 0 is the mean motion of ~* Ply 0. Evident-
~k
ly, ~ may also be viewed in terms of a C -function on
and, in particular, we see that @ has an absolutely convergent
Bohr-Fourier series. ~rite @ = @i+r where @i E AAP(~) and
@2 has negative spectrum. Since e iwt exp(@2(t)) =
iwt
)(t) belongs to AAP(~), e exp(r
exp(-@l(t))( 9. Piy 0
may be written as
= ~ akeY0keikt.
Z akeikt where Z ..[akl < ~. Let
0mk~w 0~kmw
Then ~2 E AAP(~) and Z [akle y0~
92(t)
e wyO Z [axl
O~k~w
< ~. On the other hand, it is an easy calculation
to show that for y E [0,Y0) ,
-w(y-YO) iwt
92, Piy(t) = e e ~ exp[@2* Pi(Y0-y )(t)].
Therefore, ~2 is invertible in CA~(~) and each ~2" Piy'
y E [0,Y0) , lies in AAP(~). Now define
Since
[9 * Piy(t)
~l(t+iy ) = 1~ 2. Piy(t ) , t ( m,
~exp[r Pi(y-y O)(t)]'
0 ~ Y ~ Y 0
t E JR, y0 ~ y < ~.
92, FiYo(t ) ; e iwt exp(~2(t)) , it is easy to see that ~o 1
is bounded and analytic on the open upper half-plane and %Ol
-i
boundary values ~2 Since 92 is invertible in CAP(JR),
has
we

Curto, Muhly and Xia 663
have ~i = ~+~, where ~ E H and ~ E CAP(~). But
~, Piy(t ) = (q~@~l) * Piy(t ) _ ~ * Piy(t ) = 91(t+iy ) _ (~ * Piy)(t)
and so ~ has nonnegative spectrum; i.e., ~ E AAP(~). Thus we
-I
see that 91 = ~2 E H. By definition, there is a positive $i
such that I(~ I) * Piy (t)I = 191 (t+iy)I ~ ~i for all t 6m
and y 6 (0,Y0] ; also, by definition, for each y E (0,Y0] , the
mean motion of ~I* Piy is zero. Thus, by Corollary 2.12, ~I
is invertible in H. This completes the proof of the lemma.
We note in passing that we are unable to decide if, in
Lemma 2.14, the function 91 is invertible in H~(P~(n)), al-
though we feel that it ought to be.
w WIENER-HOPF OPERATORS
For 9 E L~(~), we denote by W the Wiener-Hopf (or
Toeplitz) operator on H2(~) defined by the formula
W 9 = PM I H2(~) where M is the operator on L2(~) of multi-
plication by ~, P is the orthogonal projection of L2(~) onto
H2(~), and the vertical bar denotes restriction. If ~(t)= e iXt,
for some X E~, we will write W~ for W 9. If ~ : L2(~) ~
L2(~) denotes the Fourier-Plancherel transform, then we have
W~ ~-I = T~ where TX is the compression of translation (to
the right) by ~ to L2([0,~)); i.e., TX= I[0,~)Lx I L2([0,~))
where I[0 ~ ) denotes the indicator function of [0,~) and
(L~)(x) = ~(x-X). The norm-closed algebra generated by
[L I ~ E CAP(~)] will be denoted by ~. It is a C*-algebra and
is generated by [Wx I ~ E~]. The norm-closed algebra generated
by ~W i 9 E H+ CAP(~)] will be denoted by ~. Of course,
is not a C*-algebra. In [5] and [15] it is discussed how to
represent ~ in a certain Ii factor ~ through a certain
representation p. It is shown that for ~ 6 CAP(~), p(W ) is
Fredholm relative to ~ if and only if ~ is invertible in
CAP(~) and, in this case, the Breuer-Fredholm index of p(W )
is the negative of the mean motion of ~. Our objective is to
extend these results, as far as we can, to the algebra ~.
Let ~ denote the commutator ideal of ~. Then a
moment's reflection reveals that ~ is the norm-closed ideal

664 Curto, Muhly and Xia
generated by the operators of the form W~W_x- W_xW~ =
= W_~W~(WkW_~-l) where k ~ 0 and ~ E H. Moreover, [W +K I
6 H+ CAP(~), K s ~] is dense in ~, and IIw~+~li ~ [I~II~-
LEMMA 3.1. The map ~ W +~ is an isometric isomorph-
ism from H§ CAP(~) onto ~/~. In particular, ~ = [W +K I
E H+ CAP(~), K E ~].
Proof. We need only show that IIW~+KI[ ~ I[~[[~ for any
K E ~; and, in fact, we may assume that
K = ~j kH W jk(WxjW-xj-I)W~jk' where ~jk and ~jk~ have the form
e-ikt@(t), k > 0, @ E H. For such a K, ~N > 0 such that
KW k = 0, if k ~ N. On the other hand, for any Wiener-Hopf
operator W~. and any k > 0 we have W~. = W~W@Wk,~ and conse-
quently k88 ) = (W,g ,n)H2(~ ) . Since l[~[[~ = Ilw~l[,
as is well known, given r > 0, we may choose unit vectors
and ~ in H2(~) such that [I~][~ ~ I(w~g,~)l + ~. But from what
was just noted, [(W g,~)I = liml((W +K)Wkg,Wk~) I ~ HW +KI[. This
completes the proof.
In [5], the von Neumann algebra in which ~ is repre-
sented is essentially the group-measure von Neumann algebra ob-
tained from letting the discrete reals ~d act on ~. The Hil-
bert space is L2([0,~)) | L2(~ d) and the von Neumann algebra is
generated by the operators (1[0,~) | I)(M | I)(1[0,~ ) | I),
s L~(~), and (I[0,~)|174174 , k s where
TX ~ denotes translation by k on L2(~d ). For our purposes, it
is more congenial to take the Fourier transform of this algbra.
So, our Hilbert space will be ~ = H2(~) | L2(m) (recall m is
Haar measure on ~B) and for ~ 6 L~(~ B) we define ~ =~ I~
2 2 ~
where ~ = P| I and M is defined on L (~) | L (m) "="
L2(~x~B,p x m) (~ is Lebesgue measure on ~) by the formula
(~)(t,x) = ~(x+et)~(t,x) , ~ E L2(~) | L2(m). The von Neumann
algebra ~ generated by [W~I ~ ( c(~B)] is unitarily equiva-
lent, via the Fourier transform, to the yon Neumann algebra de-
scribed above that was first used by Coburn, Douglas, Schaeffer,
and Singer; for the proof of this see [15, Section 3]. The alge-
bra ~ is a II~ factor and we denote its canonical trace by 7.

Curto, Muhly and Xia 665
For T = ~ ~ W E ~, write p(T) = S ~ ~ Then as is shown
j k ~jk j k ~jk
in [5] and [15], 9 extends to a faithful *-representation of
onto ~ which is defined to be the C*-algebra generated by
[W% I ~ E c(~B)]. Our objective is to extend p to an isometric
representation of 9 in ~.
To this end, we denote the operator of convolving with
the Poisson kernel on H2(m) and on ~ by Ply" Thus on H2~R),
(Piy[)(x) = f~ [(x-t)Piy(t)dt , while on ~, (Piy~)(t,x) =
f~ ~(x-s,X+es)Piy(S)ds. Then on ~, Ply E ~, and in Lemma 2.1
of [15] it is shown that p may be extended to a faithful repre-
sentation Pl of the C*-algebra ~i generated by ~ and
[Piy ] 9 onto the C*-algebra ~i generated by ~ and
Y
[Piy]y9 in ~.
LEMMA 3.2. If ~ E CAP(m), then [Piy]~9 asymptoti-
cally commutes with W and with %; i.e., lim~P. W-W P II
y~0 ly ~ ~ iy
= 0 and
limIIP. W -U P ~ = 0
y~0 ly ~ ~ iy "
Proof. It suffices to check the case when ~ is a
pure exponential. But then, the result is clear since on H2(m),
PiyWk = e-[k]YwkPiy, while on ~, PiyW[k] = e-lk[Yw[k]Piy.
LEMMA 3.3. The map which sends an operator of the form
Z Hwr
j k jk
to Z ~ ~@ ,
j k jk
~ ~ H+CAP(m)
@~k
extends to an isometric
representation, denoted 92 , of onto a subalgebra
of ~.
Proof. Lemma 3.2 implies that ~01im[IPiyW~-W'k~*zi~ )PiYI~I
N Y
= 0 for all ~ E CAP(m) and likewise for W . On the other
hand, for @ E H, we have PiyW@ = W(@,Piy)Piy and a similar
equation holds for ~@. Now fix a finite family [~jk] g
H+ CAP(m), let S = Z H W@ and let T = Z H ~ . For y 9 0,
j k jk j k @jk
let Sy denote the same combination as S, but with @jk re-
placed by ~jk* Piy' and define Ty similarly. By our initial
observations, we may write
Pi S*SPi = P" S~ S 2 Pi + A
y y my y y y y

666 Curto, Muhly and Xia
and
PiyT*TPiy = P. T~ T 2 P. + B
zy y y my y
where Ayll = ByII = 0. since P s~ S~ P. E ~i and
iy ~y ~y my
Pl (PiyS*2yS2yPiy) = P'myT~yT~Ly P'my (cf. [15]), we see that IISIl 2 =
Hs*sII = s*sP. IL mr- llP. T*TP. LI = L,IT*Tli = llTl 2. This
y~0 my ly ly ly
completes the proof.
Let ~ be the norm-closed ideal in ~ generated by
the trace class elements of ~; i.e., ~ is the closure of
IT E ~I ~((T*T) I/2) < ~]. Then ~ is the ideal of generalized
compact operators in ~. In [5], it was shown that p-l(~N p(~))
is the commutator ideal of ~. Our next objective is to obtain a
similar result for $ and P2"
LEMMA 3.4. r = p~l(~n p2(~)).
Proof. Recall that { is the norm-closed ideal gener-
ated by operators of the form W_xW~(WxW_k-I) where ~ ~ 0 and
E H. Since P2(WkW_k-I) = ~[x]W~x]-I is a finite rank projec-
tion in ~, it is clear that ~ m p~l(~ p2(Z)) " Now by Lemma
3.1, every element of 9 may be written as W$+K where K E {.
So, to prove the reverse inclusion, we need only show that if
P2(W~) E R, then ~ = 0. Define [Ut]ts ~ on ~ by the formula
(Ut~)(s,x) = ~(s-t,x) and set ~t(B) = utlBut , B s m. Then
[~t]tE~ is a o-weakly continuous, one-parameter group of *-
automorphisms of 9 that carries ~ into ~, because [Ut]tE ~
~. Moreover, for A s ~, the map t * ~t(A) from ~ to ~
is norm continuous. [To see this, let r > 0 be given and write
e
A = AI+A 2 where A I is trace class and A 2 E ~ with IIA2II < ~.
Then since [~t]t~ ~ is ~-weakly continuous and ~ is the dual
of LI(~,~) (the completion of the trace class operators in the
trace norm), it follows that it is possible to choose ~ 9 0 so
r Vt It] < ~ and that when this is done,
that II~t(AI)-AI[II < y ,
e e
then II~t(A)-AII ~ IIaI(AI)-AIILI + Ilat(A2)-A211 < y+ 2y = r So, if
A s ~, and f s LI(~), then ~ ~t(A)f(t)dt s ~. If

Curto, Muhly and Xia 667
~@ : P2(W@) E ~, then since ~ ~t(~r = W@*Piy' as an
easy calculation shows, we see that W$,p. s ~ for all y > 0.
ly
Since, however, @* Ply E CAP(~) we conclude from [5] or [15]
that @* Ply = 0 for all y > 0. This implies that @ : 0 and
completes the proof.
We come now to the main theorem of this section, a suf-
ficient condition for P2(W@) to be Fredholm in ~,
s H+ CAP(~), together with a calculation of the index.
THEOREM 3.5. If @ E H+ CAP(~) is invertible in
H+ CAP(~), then P2(W~) is Fredholm in ~. Moreover, there is
an r > 0 such that ~, P. is invertible in CAP(~) for all
ly
y E (O,r) and the index of P2(W~) is minus the mean motion of
~* Ply for any y E (O,r).
Proof. By Lemmas 3.1 and 3.4, P2(W~) is Fredholm in
for every invertible ~ E H+ CAP(~). To calculate the index
of P2(W@), we may assume that r = e-iWt~(t) where w ~ 0
y
and ~ E H (Pz(n)) for suitable y, n, and s By Theorem 2.7
and Lemma 2.14, we may write ~ = ~i~2 where ~I is invertible
in H and ~2 s AAP(~) is invertible in CAP(~). Since ~I is
invertible in H, P2~W i) is invertible on ~ with inverse
P2(W _i ). Also, since P2(W@ ) = P2 (W -iwt )P2(W ]) we conclude
~1 e ~2 "
that the index of P2(W~) is the index of P2 (W -iwt )" Since
e ~2
~2 E CAP(~), the index of P2 (W -iwt ) = p(W -lwt ) is minus
e ~2 e ~2
the mean motion of e-imt~2 by [5]. To see that this mean mo-
tion coincides with that of ~* Piy for each y E (0,r), for a
certain r > 0, simply use Theorem 2.7, the invertibility of ~I'
and Theorem 2.6. This completes the proof.
Conspicuously absent from the statement of Theorem 3.5
is a necessary condition that P2(W@) be Fredholm. One would
expect, of course, that if P2(W~) is Fredholm in ~, then
is invertible in H+ CAP(~). However, while this appears plaus-
ible at first glance, we are unable to find a proof. In fact,
upon deeper reflection, we are not at all certain that it is

668 Curto, Muhly and Xia
true. In the following discussion, we shall indicate briefly how
far we are able to go in proving the assertion that ~ is in-
vertible in H+ CAP(R) whenever Pi(W~) is Fredholm and we
shall identify the obstruction that prevents us from completing
the proof. First, we recall the Fredholm theory of Toeplitz
operators with symbols in H~+C. Let ~ s H ~ and assume that
the Toeplitz operator on Hi(~) determined by ~, T~, is Fred-
holm. To prove that ~ is invertible in H~+C, one first shows
that ~ is invertible in L ~, and then, using the fact that
dim(H 2@~H 2) = dim(H 2GT H 2) is finite, one shows that ~ = ~
where ~ is a finite Blaschke product and @ is invertible in
H . The first step can be carried out in our setting, the second
is where the difficulties lie.
LEMMA 3.6. Let ~ E H and suppose that Pi(W~) is
Fredholm in ~. Then ~ is invert~ble in L~(m); i.e., there
is a 6 > 0 such that l~I ~ 6 a.e.
Proof. If W~ = Pi(W~) is Fredholm in ~, then so is
I~ ~ If 6 is the distance from 0 to the spectrum
12 = (~)*~ .
of ~ + ~ in ~/~, then for each ~ s [0,6), W - ~I is
I 12 I 12
also Fredholm. Consequently, for each ~ E [0,6), there is a
finite projection E E ~ and a positive number ~(~) such that
i i2
- ~I ~ ~(~)(I-E ) Now choose a sequence {~n]::! of pos-
~B
itive continuous functions on such that ~n(x)dm(x) = i
for all n and limI@Ii,~n = I@I 2 a.e.m. (The choice of a
sequence with this property is possible since I$I 2 has only
countably many nonzero Bohr-Fourier coefficients and so may be
viewed as a function on a separable quotient of ~B.) For fixed
n, the ergodic theorem implies that lim i ~T l$12(x_et)~t(0+et)dt
T~ ~ -T
= ~]RBl@li(x-Y)~n(Y)dm = I@l 2. ~n(X) a.e. (m) on the one hand,
while on the other, lim ~-~ l T T at(g I 2)~n(0+et)dt = ~ "
It follows that ~ = ~ - ~! ~ ~(~)(I-E~) where

Curto, Muhly and Xia 669
I T
E no is defined to be t(E~ Tho operators
Ena are not necessarily projections, but since each ~n is posi-
tive, E n~ ~ 0 and Fatou's lemma implies that ~(E~) ~ ~(Ea);
i.e., each E n is trace class. Thus we have shown that
is Fredholm for each n and for each ~ s [0,6).
Since I~12" ~n is continuous on ~B, we conclude from [5] that
I~12. ~n(X) ~ ~ for all x s all n, and all ~ s [0,6).
Since !imI~I2* ~n = I~I2 a.e. m, we ultimately find that
n4~
I~I 2 m 5 a.e. m, and this completes the proof.
COROLLARY 3.7. If ~ s H is such that P2(W@) is
Fredholm in ~, then P2(W~) is bounded below on ~.
Proof. By Lemma 3.6, there is a 5 > 0 such that
I~I a 6 a.e. m. But by definition of ~@ = P2(W~), W@ is a
direct integral of Wiener-Hopf operators with symbols ~(x+et)
in H~(~). Since almost everyone of the integrands is bounded
below by 5, the same is true of W~.
Passing now to the second step, and the obstruction to
finding a converse to Theorem 3.5, we need to investigate how a
function @ in H+ CAP(~) can fail to be invertible there. For
our purposes, it suffices to assume that ~ is in H. Let
Y = {y > 0 I ~* P- vanishes somewhere on ~]. In view of Theo-
ly
rem 2.6, there are three possibilities to consider:
3.8) Y A (0,r) is not dense in (0,r), for any r > 0,
and there is a sequence ~Yn]n= I with Yn %0 such
that each @, Piy n has zeros on ~;
3.9) Y ~ (0,r) is dense in (0,r), for some r > 0; and
3.10) Y ~ (0,r) is empty for some r > 0 and there is a
sequence
{Yn]~=l~ with Yn ~ 0 such that
infn tE~inf It* PiYnI(t) = 0.
In analogy with what happens on the disc, (3.8) and ~.9) corre-
spond to the case where the inner factor of @ contains an infi-
nite Blaschke product while (3.10) corresponds to the case where

670 Curto, Muhly and Xia
the inner factor of $ is the product of a finite Blaschke prod-
uct and a singular inner function. Our objective is to show,
among
other things, that if $ E H~(P~(n)) is such that P2(W~)
is Fredholm, then (3.8) cannot happen. Our analysis employs
classic tools in the value distribution theory of analytic almost
periodic functions.
THEOREM 3.11. Let ~ be a function in H and view
as a function on ~B. Then inI~I ~ Ll(m). If y ~ O, and if
89 is defined to be ~BlnI~* PiyIdm, then ~ is convex.
For each YO 9 0 such that ~(yo ) has a derivative, ~, PiYo,
viewed now as a function on ~, has a mean motion and its value
is -~(y0 )._ Moreover, the functions ~* Ply are free of zeros
on ~ for all y in the interval (a,~) if and only if ~ is
linear on (~,~).
Proof. The only new aspect of this theorem is the
integrability of InI@ I for ~ s H. The rest follows easily
from Theorems 7 and 8 of [9], once one notes that when y 9 0,
so ~, Ply s AAP(~), ~]R BlnI~* PiY Idm = T~limlf~- InI~* PiY I(t)dt
[I]. To show that inI$ I s Ll(m) for ~ E H, it suffices to
assume that @ s H=(P~(n)) for some y, s and n. In this
case, @ may be viewed as a function on a certain torus ~k,
and to be contained in H~(~k). The integral, then, of InI@I
over ~B coincides with the integral of inI@ I over ~k. Since
this is well known to be finite [14], the proof is complete.
Alternatively, the Integrability of inI~ I for ~ E H~(P~(n))
follows from a corollary on page 9 of [8] and the fact that
has a first nonvanishing Bohr-Fourier coefficient.
We note in passing that as a result of Theorem 3.11,
every function @ E H has an inner-outer factorization analogous
to the one for functions on the unit disc. It would be interest-
ing to know whether these factors belong to H. Presumably they
don't.
The function ~@ is commonly called the Jensen func-
tion of ~. At first glance, one might expect the possibilities
for it to be more or less arbitrary. However, this is not

Curto, Muhly and Xia 671
entirely so, as the following corollary illustrates.
y
COROLLARY 3.12. If @ s H (P~(n)), with Jensen func-
tion ~, then in each interval (~,~), a ~ O, there are only
finitely many subintervals on which 89 is linear. Moreover, on
any one of these, -~' lies in P~(n).
Y
Proof. The first assertion is essentially contained in
Theorem 22 of [9]. The second follows from the fact that the
spectrum of $ has only finitely many points in any interval of
the form [0,r), r < ~, therefore the greatest lower bound of
the spectrum of $ belongs to the spectrum of @. The same is
true for $* Piy' for any y > 0, and so the result follows
from Theorem 3.11 and Lemma 4 on page 271 of [i0].
LEMMA 3.13. Let $ E H and let E be the projection
onto ~ @ P2(W@)~. Also, for y > O, let Ey be the projection
onto ~ @ P2(W~.Piy)~. Then ~(Ey) ~ T(E).
Proof. Since the operator P. on ~ is Hermitian
ly
and has zero kernel, it suffices to prove P. E ~ ~ E~. This
my y
implies that E is equivalent in ~ to a subprojection of E,
Y
and so T(E ) ~ T(E). But the containment relation follows from
N
the fact that for ~ s Ey~ and ~ s F~g, (Piy~,W@~) =
(g,W( Piyn) = 0
@,ply)
With the preliminaries attended to, we come now to our
final result relating the value distribution theory of $ to the
Fredholmness of P2(Wo).
THEOREM 3.14. Let ~ (H~(P~(n)) and suppose P2(W0)
is Fredholm in ~. Then ~ has only finitely many intervals of
linearity in (0, ~) and, in particular, ~ does not satisfy
(3.8). If ~ takes on only countably many values on (O,r),
for some r 9 O, then ~ takes on only finitely many values in
(0,r) and so, if ~ is not invertible in H+ CAP(~), ~ satis-
fies (3.10).
Proof. Since P2(W$) is Fredholm, the projection E
of ~ onto ~ O P2(W~)~ is finite. Therefore, by Lemma 3.13,
for each y 9 0, the projection E onto ~ ~ P2(W],p~ )~ is
y
ly
also finite with ~(Ey) ~ ~(E). Suppose y belongs to an

672 Curto, Muhly and Xia
interval of linearity of 89 Then by Theorem 8 of [9], @* Piy
is invertible in CAP(m). Consequently, P2(W@,Piy) is Fredholm
with index equal to minus the mean motion of 4' Piy" By Theorem
3.11, then, the index of P2(W~,p ) is ~(y). But since
iy
@*Piy is analytic, the index of P2(W@,Piy) is ~(Ey) and we
conclude that ~(y) ~ T(E) < ~ by Lemma 3.13. Since ~ is
convex by Theorem 3.11, ~(y) exists for all but countably many
values of y and is increasing. Also, by Corollary 3.12, @~(y)
E P~(n) for y in any interval of linearity. Since P~(n) has
only finitely many points in any finite interval, we conclude
that 89 can have only finitely many intervals of linearity. Of
course the fact that (3.8) is ruled out for @ now follows from
Theorem 3.11, as do the other assertions of the theorem.
i.
2o
3.
.
.
.
7.
.
9.
I0.
REFERENCES
R. Arens, The boundary integral of logI~ I for gener-
alized analytic functions, Trans. Amer. Math. Soc. 86
(1957), 57-69.
and I. Singer, Generalized analytic func-
tions, Trans. Amer. Math. Soc. 81(1957), 379-393.
R. Carey and J. Pincus, Mosaics, principal functions,
and mean motion on yon Neumann algebras, Acta Math.
138(1977), 153-218.
and , Mean motion, principal func-
tions, and zeros of Dirichlet series, Integral Equa-
tions and Operator Theory 2(1979), 484-502.
L. Coburn, R.G. Douglas, D. Schaeffer, and I. Singer,
C*-algebras of operators on a half-space, II. Index
theory, Inst. Hautes ~tudes, Sci. Publ. Math. No. 40
(1971), 64-81.
R.G. Douglas, Toeplitz and Wiener-Hopf operators in
H~+C, Bull. Amer. Math. Soc. 74(1968), 895-899.
E. Effros, Dimensions and C*-alsebras, CBMS Regional
Conference Series in Mathematics, Amer. Math. Soc.,
Providence, R.I., 1981.
H. Helson, Ana!yticity on compact abelian groups, Alge-
bras in Analysis, Academic Press, N.Y., 1975.
B. Jessen and H. Tornehave, Mean motions and zeros of
almost periodic functions, Acta Math. 77(1945), 137-279.
B. Ja. Levin, Distribution of Zeros of Entire Functions~

Curto, Muhly and Xia 673
ii.
12.
13.
14.
15.
Translations of Mathematical Monographs, Vol. 5, Ameri-
can Mathematical Society, Providence, R.!., 1964.
P. Muhly, Function algebras and flows II, Arkiv f~r
Mat. 11(1973), 203-213.
, Function algebras and flows III, Math. Z.
136(1974), 253-260.
W. Rudin, Function Theory i_~n Polydiscs, Benjamin, New
York, 1969.
, Spaces of type H~+C, Ann. Inst. Fourier
(Grenoble) 25(1975), 99-125.
J. Xia, Wiener-Hopf operators with piecewise continuous
almost periodic symbols, to appear.
Department 0f Mathsmatics
I01 MacLean Hall
The University of Iowa
lowa City, Iowa 52242
Submitted: January 2, 1985

Integral Equations
and Operator Theory
Vol. 8 (1985)
BASE CHANGE AND THE FREDHOLM INDEX
Mihai Ptrtinar
INTRODUCTION
0378-620X/85/050674-1951.50+0.20/0
9 1985 Birkh~user Verlag, Basel
The present paper deals with a relationship between 2.L.Taylor's functional cal-
culus and the multidimensional Fredholm index, both referring to commutative n-tuples
of operators. We develop an enlarged frame, with the purpose of making the Fredholm
objects more flexibile under analytic transformations. In particular two base change
results, Theorem 3.1 and Theorem 4.2, are obtained in this paper.
Let us recall for the beginning some terminology and facts from multidimen-
sional spectral theory. Let H be a complex Hilbert space and let T = (T1,... ,T n) be a
commutative n-tuple of linear bounded operators on H. 3.L.Taylor defined in [19] the
joint spectrum Sp(T,H) of the n-tuple T on H as the subset of those points ), r C n, with
the property that the Koszul's complex K(T- X, H) is not exact. In other words,
X I~ Sp(T, H) iff the condition
(1) Hq(T- X, H) = O, qZO,
is fulfilled, where we have denoted by H.(T- , H) the homology spaces of the complex
K(T - ~, H). This joint spectrum has many of the expected properties, when compared
with the spectrum of a single linear operator, among which we recall only Taylor's theo-
rem [20, Theorem 4.3] about the existence of a continuous functional calculus f ~+ I(T),
with analytic functions f defined in neighbourhoods of Sp(T,H).
Motivated by Taylor's ideas, several authors developed (in convergent direc-
tions) the Fredholm theory for commutative n-tuples of operators. Thus it is unanimously
accepted that the point X e C n doesn't belong to the essential joint spectrum SPe(T, H)
if the condition
(2) hq=dim Hq(T-)t,H)< ~, q_> O,
holds true. Then the Fredholm index of T is the integer

Putinar 675
n
ind(T)= ~. (-l)qh .
q=0 q
These definitions extend the corresponding notions for a single linear operator
and the expected stability results hold (see [22, Chap.HI, w 7]). However, in the multi-
dimensiona! case new phenomena appear (see for instance [6]) and the aim of this paper
is to investigate one of them.
More precisely, the computation of the index, or of some finer invariants as the
numbers h of the m-tuple f(T), where f:U + C m is an analytic map defined on an
q
open neighbourhood U of Sp(T, H), is the motivation of this paper. It turns out that,
instead of working on C m with the above usual definitions it is more convenient to lift
the objects to U, to investigate there a Tor-intersection condition and to compute the
corresponding intersection number between the analytic fibre f-l(0) and the n-tuple T.
Let us explain this procedure in more detail.
The Fredholm condition (2) can be interpreted as follows. Let L(H) denote the
algebra of bounded linear operators on H, with its two sided closed ideal K(H) of com-
pact operators. Then condition (2) is equivalent by [5, w 3] with
(2)' Hq(T - X, L(H)IK(H)) = O, q >_ O,
where the operators T i -X i act by left multiplication on the CaIkin algebra L(H)/K(H).
For another interpretation consider O(C n) the algebra of holomorphic functions
on C n and let m X be its maximal ideal associated to k :
m X : {f c o(cn)[ fG1) = 0}.
The quotient o(cn)-module o(cn)/m k admits a canonical free resolution given by the
Koszul complex associated to the system z -X, where z = (Zl,... ,z n) stands for the co-
ordinate n-tuple in C n :
K.(z - ~, o(cn)) --* o(cn)Imx ~ 0.
Note that the quotient algebra o(cn)/mx is canonically isomorphic with C. Then
regarding the n-tuple T as the topological representation
0 : o(cn) -* L(H)/K(H)
which associates T i to z i, the condition (2)' can be reformulated independently of the
coordinates and of the Koszul resolution, as follows:
(2)" Tot O(cn) (o(cn)/mk, L(H)/K(H)) = O, q > O.
q

676 Putinar
This equivalent definition for the Fredholm property of the n-tuple T - X can be
easily extended to more general objects, a fact which will turn out to be extremely use-
ful.
Thus~ let F be an analytic coherent sheaf on C n, which admits a finite globally
free resolution with finite type O-modules, like the sheaf O/m k above. For explana-
tions concerning such sheaves, see w I. Throughout this paper O denotes the sheaf of
holomorphic functions on C n, Then the vanishing condition
(3) Tor O(cn) (F(Cn), L(H)IK(H)) = 0, q > 0,
q
will be interpreted as a generalized Fredholm "incidence" relation between the sheaf F
and the n-tuple T. A natural K-theoretic difference construction will provide under the
assumption (3) an integer
which generalizes the Fredholm index.
ind O(cn) (F, L(H)/K(H)) ~ Z,
Let us illustrate these notions with two examples.
EXAMPLE 1. Consider the n-tuple T and an integer k, I < k < n. Let us denote
by / the ideal of the sheaf O generated by Zl,... ,z k. Then the sheaf F = 0/I, which is
supported on the complex (n-k)-dimensional linear subspace {z[ = z 2 ..... z k = 0} of
C n, has a finite free resolution given by the partial Koszul complex K.(Zl,... ,Zk; O).
In this case condition (3) is equivalent with the Fredholmness of the k-tuple (TI,... ,T k)
and
ind O(cn) (O/(Zl, .... Zk) , L(H)/K(H)) = ind (T 1 .... ,Tk).
Remark that the n-tuple T is Fredholm if (3) holds, but the n-dimensional index ind(T)
vanishes when k < n.
EXAMPLE 2. Let f be a non-constant analytic function of C n and let F be the
f
quotient sheaf O/(f). It has a finite free resolution, 0 -+ O --* O --+ F --~ 0, and it
is supported on the hypersurface { f = 0}. Then (3) means exactly that the operator f(T)
is Fredholm, and in this case
ind O(cn) (O/(f), L(H)/K(H)) = ind I(T).
Other examples will be presented in w 3 and w 5,
The present paper is an amplified version of an earlier paper, with the same
title, circulated as INCREST preprint (April, 1981). Meanwhile two recent complemen-

Putinar 677
tary papers dealing with Fredholm theory for n-tuples have appeared. R.Carey and a.
Pincus have announced in [4] an interesting analytical method of checking9 in our termi-
nology, condition (3) and of computing the corresponding index for some special classes
of n-tuples of operators and certain coherent sheaves with support of complex codimen-
sion 1. Secondly, R.Levi announced in [14] a K-theoretic framework for multioperato-
rial Fredholm theory. The reader is referred to the final part of Section 4 for more
details.
Stein spaces.
The present paper has five sections.
The first section contains some preliminaries on analytic coherent sheaves on
In the second section we define the notion of a Fredholm sheaf with respect to
a Stein algebra representation into a Banach algebra with a distinguished two-sided
closed ideal. Some stability results for Fredholm sheaves and for the associated index
under geometrical and algebraic operations are then presented.
In the third section the notions and the results of w 2 are applied to the case of
commutative n-tuples of operators on Banach spaces. The main result is Theorem 3.1.
The fourth section is expository and contains another base change result, Theo-
rem 4.2, which refers finally also to commutative n-tuples of operators.
In the fifth section a non-operatorial example is presented, which illustrates in-
tuitively the general construction of the index.
w 1. PRELIMINARIES
A class of analytic sheaves, which is stable under some algebraic and geometric
transformations, is described. The reader is referred to the recent book [13] for an
excellent introduction to analytic geometry.
Let X be a Stein space with structure sheaf O X. We denote by FR(X) the class
of analytic coherent sheaves F on X, which admit a finite globally free resolution with
finite type Ox-modules:
(4) 0--+ L n--+ Ln_ 1--+ ... --~ L 0 -+ F-+ 0~
m.
where L i = OXI for some non-negative integers mp 0 < i < n.
In view of the equivalence between Stein modules and coherent sheaves on X
established in [Ill, an element F r FR(X) is completely determined by the Fr~chet
O(X)-module F(X). Similarly, the exact sequence (4) can be derived from the free O(X)-
-resolution
(4)' 0--+ L n -+ Ln_ l--~ ...-+ L 0-+ F(X)-+ 0,

678 Putinar
O(X)mi.
where L i = Thus we shall use, depending on the context, one of these two
equivalent descriptions of the same object.
For the sake of completeness we sketch the proof of the following.
LEMMA I.I. Let 0-+ FI-+ F2-+ F3-+ 0 be a short exact sequence of
analytic coherent sheaves on a Stein space X. If two of the sheaves F l, F 2, F 3 belong to
FR(X), then the third also belongs to FR(X).
PROOF. Let M i denote the finite type O(X)-module Fi(X). If two of the modules
M I' M2' M3' which are related by a short exact sequence
0 -+ MI--+ M2-+ M3-+ 0,
belong to FR(X), then the third admits by [3, Proposition III.6.3] a finite resolution with
finite type projective modules.
Because each projective module is a direct summand of a free module, the proj-
ective resolution can be modified up to a special form:
-+ -+ -+ L -+ 0,
(5) 0 -+ Pn Ln-I "'" o
where Pn is projective and Lj, 0 < j < n-l, are finite type free O(X)-modules. Then by
comparing the three resolutions (of M 1, M 2 and M 3) through the short exact sequence,
one gets that Pn is stably free. Therefore the complex (5) is equivalent to a free resolu-
tion.
Ox/mx r FR(X).
We describe now some examples of elements in FR(X).
LEMMA 1.2.[11, Proposition 4.6]. Let X be a Stein manifold and Zet xr X. Then
We have already remarked in the introduction that this lemma is valid when X =
= C n, because of the existence of a canonical resolution of Ox/m x. We mention also
the following example. Let Z be a locally complete intersection in a contractible Stein
manifold. Then the structure sheaf 0 Z = OXfl(Z) belongs to FR(X).
LEMMA 1.3. Let f " X --~ Y be a flat morphism of Stein spaces and let F be a
coherent sheaf on Y. Then f~F c FR(X) whenever F ~ FR(Y).
Recall that the morphism f : X --~ Y is said to be flat if the Oy,f(x)-module
OX, x is flat for any point x r X.
The proof of Lemma 1.3 is immediate because a flat morphism preserves by
pull-back the exactness of a resolution.

Putinar 679
LEMMA 1.4. If F and G belong to FR(X), then F(~G has the same property,
provided that F is fiat over G.
PROOF. Let L' and L." be finite free resolution of F, respectively G. Then the
simple complex associated to the bicomplex Lr(~L~ ' is a finite free resolution for the
sheaf F(~)G.
w 2. THE MAIN NOTIONS
In this section a generalized Fredholm theory for coherent sheaves of FR(X) and
relative to a Banach algebra representation of O(X) is described.
Let A be an unita[, not necessarily commutative, Banach algebra and let I be a
two-sided closed ideal of A. The basic facts concerning the K-theory of Banach algebras
can be found in [20], where the commutativity is inessential. Thus the boundary opera-
tor
8 : KI(A/I) -+ K0(I)
in the long exact sequence of K-theory for the pair (A,I) can be interpreted as a genera-
lized index. We recall the classical situation A = L(H), I = K(H), where H is a Hilbert
space. Then the group K0(K(H)) coincides with Z and 8 is, after this identification, the
Fredholm index map.
DEFINITION 2.1. Let X be a Stein space, let F c FR(X) and let p : O(X) -+ A be
a unital representation of the Stein algebra O(X) into a Banach algebra A with a distin-
guished two-sided closed ideal I. The sheaf F is said to be Fredholm relative to p and to
I, if
(6) TorO(X)(F(X) A/I) = 0, q_> 0.
q
Formula (6) should be compared with relation (3) in the introduction.
Let us assume (6) holds and let L be a finite free resolution of F(X) with finite
type O(X)-modules. Then the complex of free (A/I)-modules L (~O(x)A/I is exact, hence
homotopicalty trivial, that is, if the boundary operator d of L has degree -1, then there
exists an operator c of degree +l on L.(~)O(x)A/I , such that (d~l) c + e (d~)l) = id.. The
A/I - linear map
is invertible, where
d(~lA/I + e : Le(~O(x)A/I --~ Lo(~O(x)A/!
L e =~L2p and L ~
= nL2p+l 9

680 Putinar
The free O(X)-modules L e and L ~ have the same rank, and, after choosing bases
in these two O(X)-modules, the above map gives an element [d~IA/I + el in the group
KI(A/I). We set
(7) indO(X)~, A/I) = a [d~)lA/I + el,
where ~ : K I(A/I) --~ KD(1) is the generalized index.
PROPOSITION 2.2. The definition of the index is independent of the choice of
the resolution L., the homotopy ~ and the basis in L. .
PROOF. If e' is another homotopy of the complex L.~)O(x)A/I between the null
morphism and the identity, then for each t ~ [0,I], te + (I - t)e' remains a good homoto-
py, thus the classes [d~)IA/I + e] and [d(~IA/I + e'] coincide.
Changing the bases in the components of the complex L. the element
[d~)IA/I + e] becomes [u(~)IA/I] [d~IA/I + e] [V~IA/I] , where ucAuto(x)(L e) and v~
c AUto(x)(Lo). If we denote by ~ : A -~ A/I the natural projection, then [ux IA/I] =
= Ir~[u~)IA], therefore ~ [u~)IA/I] = 0 and analogously 8 [v~)IA/I] = 0. Then the additi-
vity of ~ implies the invariance of (7) with respect to the choice of bases.
Let L'. and L'.' be two finite resolutions of F(X) with finite type free O(X)-modu-
les. There is a morphism of complexes of O(X)-modules h : L'. -+ L'.' which is a quasi-
isomorphism (i.e. it induces isomorphisms in homology). Let L. be the cone of h, that is
the complex with components Lp = L'-~b~L p'~ '' p-l and boundaries
dp : (-1)PhP
The complex of free O(X)-modules L is exact by [3, Corollary 1.6.6], hence it is hom~-
topically trivial via a homotopy e and consequently
indO(X)(L, A/I):= a [df~lA/I + e~lA/I] = ~ ~,[d~l A + e~l A] = 0.
Thus we have reduced the proof of Proposition 2.2 to the following lemma.
LEMMA 2.3. Let 0 -+ L': -+ L. -+ L'. -+ 0 be a short exact sequence of finite
complexes of finite type free O(X)-modules. If the three complexes are exact over A/I,
then
indO(X)(L: ' , A/I) + indO(X)(L '. , A/I) = indO(X)(L., A/I).
PROOF. The boundary operator d of L. is of the form

Putinar 681
d
where (-l)Ph : L' -+ L" is a morphism of complexes.
P P P
Consider the complex of continuous functions C([O,l], L.(~O(x)A) with the
boundary operator
d t= ~1 A, re[0, 1].
d'
By [18, Theorem 2.1] the complex C([0,1], L I~O(x)A/I) is exact, and being a
complex of free C([0,1], A/D-modules, there is a homotopy e t which depends continu-
ously on t and which trivializes it. Then the elements d0Q1A/I + e 0 and dl~)lA/I + e l
define the same class in KI(A/I), and
indO(X)(L., A/I) ; ~ [dll~)lA/I + e l] - ~[d0~)IA/I § e 0] =
; indO(X)(L'.~L'o', A/I) ; indO(X)(L'o , A/I) + indO(X)(u ,. ' A/I), q.e.d.
COROLLARY 2.4. Let 0 -+ F' -+ F -+ F" -+ 0 be a short exact sequence of
Ox-modules of FR(X). If two of the sheaves F', F and F" are Fredholm relative to P and
I, then the third has the same property and
indO(X)(F, A/I) = indO(X)(F', A/I) + indO(X)(F", A/I).
PROOF. The first assertion follows from a long exact sequence of tor's, while
the second assertion was proved in Lemma 2.3.
THEOREM 2.5. Let f : X + Y be a flat morphism of Stein spaces and let P :
: O(X) + A be an unital representation, where A is a Banach algebra with a distin-
guished two-sided closed ideal I.
If F ~ FR(Y) is Fredholm relative to po f* and I, then f~ F is Fredholm relative
to P and I, and then
(8) indO(X)(f*F, A/I) = indO(Y)(F, A/I).
PROOF. Let us assume F r FR(Y) is Fredholm relative to p e f* and I, and let L.
denote a finite free resolution of F(Y) by finite type free O(Y)-modules. Then I*F
FR(X) by Lemma 1.3 and L.~)O(y)O(X) is a resolution of (f*F)(X) by free O(X)-modu-
les. The sheaf F is Fredholm relative to p and I iff the complex L.~)O(y)A/I :
: (L.(~O(y)O(X))~)O(x)A/I is exact, hence iff f*F is Fredholm.

682 Putinar
The complexes which appear in the computation of the two sides of (8) are iso-
morphic) so that the proof is complete.
w 3. AN APPLICATION TO COMMUTATIVE N-TUPLES OF OPERATORS
Let E be a complex Banach space and let T = (TI,... ,Tn! be a commutative n-
-tuple of linear bounded operators on E. First of all we explain the definitions of the
previous section in the particular case of the representation
p:O(C n) -+ L(E), p( ~ a z (~)= ~ a(T c(,
~>0 ~>0
and relative to the ideal K(E) of compact operators.
Let X ~ C n be fixed. Then the sheaf O/m~, which belongs to FR(Cn), is Fred-
holm relative to p and K(E), iff the Koszul's complex K(T-X, L(E)/K(E)) is exact,
because K.(z- X) O) provides a finite free resolution of O/m I. This definition of a
Fredholm n-tuple on a Banach space is similar with (2) t. However, it is not known
whether the condition (2) is equivalent with (2)' on an arbitrary Banach space.
As for the abstract index defined in the preceding section) we shall replace the
morphism 8 with the usual Fredholm index map ind : KI(L(E)/K(E)) -+ Z) in order to
obtain numerical results. It is important to notice that this morphism of groups still
factorizes through Coker(K 1 ~ (E)) -+ K 1 (L(E)/K(E))).
In order to compute the index of T-l, let e be a homotopy which splits the
exact complex K(T - X ) L(E)/K(E))) that is de + r = id. It turns out that) denoting by e
a lifting for r with coefficients in L(E)) the operator
is Fredholm, and consequently
d+e:p~)->0K2p(T-X,E) -+ ~ K2p+I(T- X, E)
ind O( C n )(O/m), L(E)/K(E)) = ind (d + e).
An induction argument, e.g. ris, Proposition 2.6]) shows then that
ind (d + e) = p~0 (-I)p dim Hp(T - )., E).
This integer will be by definition the index of the n-tuple T - X) and it will be denoted
ind(T - ).).
The invariance theorems under small norm or compact perturbations of this
multidimensional index are proved for instance in [15) w 2].

Putinar 683
Adopting the weaker definition (2) for n-tuples on Banach spaces and defining
the index by the same Euler characteristic all the stability results remain true, except
for the invariance of the index under compact perturbations which is still an open
question (see [10] and [21] ior details).
The joint essential spectrum is by definition the set:
5Pe(T,E) = {X r cnl T - ~ is not Fredholm}.
In view of the above interpretation of Fredholmness, the joint essential spectrum
coincides with Taylor's joint spectrum with coefficients in the quotient algebra
L(E)/K(E):
SPe(T,E) = Sp(T,L(E)/K(E)).
This equality implies, as in the Hilbert space case, some properties of the joint essential
spectrum, like for instance the spectral mapping property.
The Fredholm index vanishes on the unbounded connected component of the
open set on\ 5Pe(T,E) , so that Fredholm domains of T, corresponding to a non-vanish-
ing index, contain only 0-dimensional and n-dimensional analytic subsets'
Let U be a connected Stein open neighbourhood of Sp(T,E) and let f : U -* C m
be an m-tuple of non-constant analytic functions defined on U. Then Taylor's functional
calculus produces an m-tuple of commuting operators f(T)= (fl(T),...,fm(T)), with
joint spectra related to those of T by the spectral mapping relations:
We distinguish three cases:
5p(f(T),E) = f(Sp(T,E)), 5Pe(f(T),E) = f(SPe(T,E)).
l) m < n. Then f(SPe(T,E)C)= Sp(f(T),E), where K c denotes the union of the
compact K and of the bounded connected components of cn\K. In particular f(X)r
c SPe(f(T),E) whenever )~ I~ SPe(f(T),E) but ind(T - X) ~ 0.
One might say that in this case the Fredholm behaviour of the m-tuple f(T) can
be read on the analytic fibres of the map f. Indeed, let Us assume in order to apply the
general scheme of w 2, that f is a flat map. If 1J r cm\ SPe(f(T),E), then by Theorem
2.5, i*(O/mlj) e FR(U) is Fredholm relative to T and
ind (f(T) - IJ) = indO(U)(f*(O/m ),L(E)/K(E)).
The analytic sheaf f*(O/m ) is concentrated on the analytic fibre f-l(lJ), which, as we
tJ
already remarked, is disjoint of SPe(T,E)C.
2) m > n. Then the set f(U) has no interior points, hence Sp(f(T),E) does not
contain open sets. Therefore ind (f(T) - I J) = 0 for any 1J e (:m\ SPe(f(T),E).

684 Putinar
3) m = n. This is the case when numerical relations between the indices of T and
I(T) are expected to hold. We prove further on such a formula in a particular but
generic case.
THEOREM 3.1. Let T be a commutative n-tuple of linear bounded operators on
a Banaeh space E and let f : U --~ C be an analytic flat map with finite fibre at 0 c C n,
where U is a Stein open neighbourhood of Sp(T,E).
and
If the n-tuple f(T) is Fredholm, then the n-tuples T - ~, f(~) = O, are Fredholm
(9) ind f(T) = fO,~=0 "~k(f) ind(T - k),
where v)(f) denotes the multiplicity of f at
PROOF. We are under the hypotheses of Theorem 2.5, with F = Ocm/m 0 and
the functional calculus representation p : O(U) --+ L(E) associated to T. Hence the O U-
-module f*F is Fredholm relative to p, and
ind f(T) = indO(U)(f*F, L(E)/K(E)).
Thus we have to analyse the right term of this equality.
The sheaf f*F is concentrated on the zeroes set of f. More precisely, at the
level of global sections, there are isomorphisms
(f*F)(U) = O(U)/(fl,...,fm ) = X) ~ O~/fem O.
f( : 0
The multiplicity v~(f) is by definition the integer dim O),/m (O)/f*m 0) and it is
finite by the assumption on finite fiber at 0.
By induction on N = ~vk(f) =/(f'F), the length of the Ou-modules f'F, we shall
prove the equality
(I0) indO(U)(f*f, L(E)/K(E)) : I(XI: 0 V)(f) indO(U)(o/m~,L(E)/K(E)),
and the Fredholmness of the n-tuples T - )., f(k) = 0.
If N = l, then f*F = O/ink and (10) is true. Moreover we claim that the sheaf
O/m is Fredholm relative to p iff the n-tuple T- lJ is Fredholm, 1J e U. Indeed
IJ
because Koszul's complex K.(z - p, O U) provides a finite free resolution of Ou/m l~ and
because p is a functional calculus representation for T, both conditions are equivalent
with the exactness of the complex K.(T - H, L(E)/K(E)).
Let us assume relation (10) holds for N = I(f*F)-I. There exists an exact se-
quence of Ou-modules 0-+ G -+ f*F-+ Ou/mk0--~ 0, where f(),0 ) = 0. By the
additivity of the length, l(G)=/(f'F)-1 and by Lemma l.l, G r FR(U). Then (10)
follows from Corollary 2.# and the n-tuple T - k 0 is Fredholm by the above remark.

Putinar 685
The proof is complete after noticing that
indO(U)(o/mz, L(E)/K(E)) = ind(T - k).
Let us illustrate the theorem with some examples of analytic flat and finite
maps. With the above notations the n-tuple T is Fredholm whenever f(T) is Fredholm, if
the analytic map f : U -+ C n, f(0) = 0, belongs to one of the following classes."
- f is one to one,
- f(z) = (Pl(Zl),... ,Pn(Zn)) with PI,... 'Pn non-constant polynomials,
- f(z)= (Sl,.~ denoting by Sj the elementary symmetric polynomials in
Zl,...,z n. In each of the three cases the multiplicity of f at its zeroes can be easily
computed.
R E M A R K S 3.2. a) Contrary to the case 1) above, in formula (9) only points
contained in non-vanishing index Fredholm domains of T occur.
b) The theorem still holds true when U is allowed to be an arbitrary open neigh-
bourhood of Sp(T,E), but then the flatness assumption muCt be replaced with the follow-
ing condition
TorO(Cn)(o(cn)/m0 ,
(1 1) q O(U)) = 0, c~l.
This means that the comple x K.(f, O(U)) is exact in positive dimensions.
In this case the proof consists in passing to the envelope of holomorphy 0 of U,
and to make use there of coordinateless arguments, in the spirit of [16]. Namely, let
: 0 -+ C n be the analytic extension of f. Then the flatness of f is replaced by (11),
while the finiteness assumption which is not necessarily valid for f, is corrected in
formula (9) by the vanishing relations
indO(U)(oo/m l, L(E)IK(E)) = O, ~ c U\ U.
These are in turn consequences of the inclusions of the coordinateless spectra in the
generating part U of U:
a(O ,L(E)/ K(E)) u a(U ,E)CU,
see [16] for details. Then the proof of Theorem 3~ can be adapted to this more general
situation~
w 4. BASE CHANGE FOR FREDHOLM COMPLEXES
A central role in the last section was played by the family of complexes
K.(T - X, E)) parametrized on X r C n. In order to analyse separately the behaviour of
the homology spaces Hq(T- X)E) under analytic transformations we shall reduce ;this
problem to a finite dimensional one and there we shall use the methods of analytic

686 Putinar
geometry,
Let us recall for the beginning some terminology. A complex (L, d) of Banach
spaces and linear operators ~ dq
9 .. "-* Lq+ 1 Lq Lq_l--'* ...
is said after G.Segal [17] to be Fredholm if the homology spaces are finite dimensional,
dim Hq(L) < % and if every boundary operator dq is direct, that is Ker(dq) and Im(dq)
are closed complemented subspaces in Lq, respectively in Lq_ 1"
With the notations of the preceding section, the n-tuple T is Fredholm iff the
complex K (T,E) is Fredholm in the above sense.
Let X be an analytic space with structure sheaf O X. The (topological free) O X-
-module associated to the presheaf U -~ Ox(U)QF , where F is a Banach space will be
A
denoted by OxQF. An analytically parametrized complex of Banach spaces is by
definition a sequence (Lq)qr z of Banach spaces and of elements dq r r(x, Ox(~L(L q,
Lq_l)) , such that dq o dq+ 1 = 0, q oZ. The image of dq through the natural projection
F(X, Ox~)L(Lq, Lq_ 1) -'* OX,x(~L(Lq, Lq_ 1)/mx~)L(L q, Lq_ 1) =L(Lq, Lq_ 1)
is a linear bounded operator dq(X)cL(Lq, Lq_l). Thus one gets for every x r X a
complex (L. , d.(x)) of Banach spaces whose boundaries depend on x. This complex will
be denoted in short by L(x) and its homology spaces by Hq(L.(x)). On the other hand for
an analytically parametrized complex there is a corresponding complex of sheaves
(OxQL" , d) which will be denoted by L , and its homology sheaves by Hq(L ). When X
is a complex manifold, the above definition of an analytically parametrized complex
means nothing more than that the operators dq(X) depend analytically in the norm
topology.
Finally, by a right bounded complex L. we mean a complex with vanishing nega-
tive terms, Lq -- 0 for q < O.
There are many stability results for analytically parametrize d complexes of
Banach spaces ([18, Lemma 2.2], [21], [22, Chap.III]). As a by-product of them one
obtains rather directly by a descending procedure the following result (see also [l/#,
Proposition l]).
PROPOSITION 4.1. Let L be a right bounded complex of Banach spaces, analy-
tically parametrized on a reduced analytic space Y, and let N be a non-negative inte-
ger. Suppose that the complexes L(y) are Fredholm for y c Y. Then there exists locally
on Y a right bounded complex P of finite type free Oy-modules and a morphism of
complexes

Putinar 687
which induces the isomorphisms
for 0 < q < N and locally on y c Y.
qb : P. --+ L
H CP) ~- Hq(L), Hq(P(y)) -~Hq(L(y))
This proposition reduces many statements of the infinite dimensional analytic
Fredholm theory to the finite dimensional case. In order to illustrate this principle we
present without proof the following infinite dimensional version of Grauert's continuity
theorem [2, Theorem 3.3.#].
THEOREM 4.2. Let L. be a right bounded complex of Banach spaces, analyti-
cally parametrized on a reduced analytic space Y and let q be a non-negative integer.
Suppose that the complexes L.(y) are Fredholm for every y ~ Y. Then the fol-
lowing assertions are equivalent:
a) The function y -~ dim Hq(L.(y)) /s locally constant on Y.
b) The families of subspaces {Ira dq+l(y)} and {Ker dq(y)} form analytic Banach
subbundles of the trivial bundle Y x Lq.
c) If f : X -+ Y is a morphism of analytic spaces, then the natural map
is an isomorphism.
f*Hq(L) ...+ Hq(f* L)
The proof of the theorem reduces the statement, via Proposition 4.1, to the
finite dimensional case and then uses Theorem 3.3.4 from [2]. We point out only the
importance of the splitting assumption in the definition of a Fredholm complex, in order
to reduce the property c) to the finite dimensional case.
As a continuation of the preceding section we derive some applications of
Theorem 4.2 to multioperatorial Fredholm theory.
Let T be a commutative n-tuple of linear bounded operators on a Banach space
E and let Dc::C n be a Fredholm domain of T. Then K.(T-I, E) is an analytically
parametrized family of Fredholm complexes on 1 e D. Let q be a fixed non-negative
integer.
Then Proposition 4.1 implies that the iumping points of the function
form a thin analytic subset S of D.
-+ dimHq(T- X,E) , teD,
Applying Theorem #.2 to the complex K.(T- ~,E), one gets that the homology
spaces Hq(T - t,E) form in a natural way an analytic vector bundle on D\S. Indeed, let

688 Putinar
X be the simple point X and let f: X--* D\S be the inclusion map. Then
Hq(L.)/mxHq(L.)~. Hq(L.(X)) by c), hence Hq(L.) is a locally free OD\ S - module, where
L = K.(z - T, ODt~E).
A
Let Tq denote the homology sheaf Hq(K.(z- T,O x E)), defined on C n. Then,
again by Proposition #.1, Tq is a coherent sheaf on the open set U = cn\SPe(T,E). This
fact was used by R.Levi [14] in order to define the following abstract index
n
r(T) := y. (-l) i IT i lu],
i=0
as an element of the Grothendieck group K0(U) of analytic coherent sheaves on U.
Levi's index K(T) contains much more information than the usual Fredholm index.
Notice that the set which carries K(T),
n
X := Sp(T,E)\SPe(T,E) = U Supp(Ti l U)
i=0
is an analytic subset of U. The element K(T) has an integer multiplicity in every point
x r X, which is up to a sign the local index computed in some particular cases by
R.Carey and J.Pincus in terms of traces of commutators. This local index can be
expressed as follows:
ix T' X
xcV i=0
where V runs over the irreducible components of the analytic set X, and the multiplicity
of the sheaf T i along V is defined by the formula
where y is a generic point of V.
mv(T i) =length- (T. ~)~ O.. ))
UV,y I,y Uy v,y
Let us interpret from this point of view the indices relative to coherent sheaves
defined in Section 2. Let F c FR(C n) be a Fredholm sheaf relative to the representa-
tion in the algebra L(E) associated to the n-tuple T and relative to the ideal of compact
operators. We assume in addition that the set Supp(F) is disjoint of SPe(T,E); this
assumption follows from the Fredholm condition of F, when for example this sheaf is a
locally complete intersection ideal of Ocn.
Because the joint spectrum of T is bounded, the analytic set Sp(T,E)~
]7 Supp(F)= XNSupp(F) doesn't contain components of dimension greater than zero,
hence it is a finite subset of U. For any pair (i,j) of integers, the sheaf Tort~ (F, Ti[U) is
supported on XNSupp(F). Then Definition 2.1 of the index of T relative to F gives rise
to the equality

Putinar 689
indO(Cn)(F, L(E)IK(E)) =..~ (-l)i+Jdim TorU(F, Ti)(U),
i,]
which in turn can be interpreted as follows:
C n
ind O{ )(F, L(E)/K(E)) : [F]" K(T).
Concluding, the index of the n-tuple T relative to the sheaf F and to the ideal
of compact operators represents the Tor-intersection multiplicity of F against the class
K(T).
w A FINAL EXAMPLE
We exemplify the general notions of w by representations of Stein algebras
into commutative C*-algebras.
Let X be a Stein manifold of complex dimension n and let x be a fixed point of
X. Consider a pair (A, B) of compact CW-complexes, BCA, and let ~:A -+ X be a
continuous map. We investigate the Fredholmness of the sheaf O/m x relative to the
representation
p : O(X)-~ C(A), p(f) = ~*(i) = fo~, f ~ O(X),
and relative to the ideal I = V(B) = {g r C(A) [ glB = 0}. We have denoted as usually by
C(K) the commutative C* - algebra of continuous functions on the compact K.
By rll, Proposition 4.6] the simple point x is a complete intersection in X,
hence there exists a n-tuple f = (fl'" " " 'fn ) of holomorphic functions on X, such that the
augmented Koszul complex
is exact.
K.(f, O X) -+ Ox/m x --~ 0
Thus the sheaf Ox/m x is Fredholm relative to p and I iff the complex:
K.(f o r C(A)/I)
is exact. But the C*-algebra C(A)II is naturally isomorphic with C(B), therefore we
have proved the following fact:
A) The sheaf Oxlm x is Fredholm relative to 0 and I = V(B) iff x ~ ~(B).
In order to compute the index we identify the groups K I(C(A)/I) with KI(B) and
K0(1) with K0(A, B). Then the boundary operator 8 : KI(C(A)/I) --+ K0(1) becomes the
usual coboundary 8 : K I(B) --* K0(A,B)in topological K-theory, see [20].
By its definition in w 2, the index indO(X)(O/mx , C(B)) is the class in K0(A, B) of
the complex of trivial vector bundles K.(f o~, A which is exact on B, in short

690 Putinar
indO(X)(O/mx , C(B)) = X(K(f o ~, A x C)),
where X : Cn(A, B) § K0(A, B) stands for the generalized Euler-characteristic map [l,
w 2.6].
Let U be an open neighbourhood of x in X, such that U is diffeomorphic equi-
valent by f with a ball B(0, 6) of radius 6 > 0. We may assume that the n-tuple f - X =
= (fl-~l''"'in-Xn) is still a regular sequence in O X for every XcB(0,6). Let e be
chosen such that 0 < 6 < 6. Then ~(5)~ V = r where V = f-lg(0, r
Let i : ({(A), {(B)) -~ (X, X\V) denote the inclusion map and let
: KO(u, U\V) --* KO(A, B) be the composition of the morphisms
i* ~*
: K0(U, U\V) = K0(X, X\V) --+ K0(#(A), {(B)) --* K0(A,B),
where the first one is the excision isomorphism [1, w 2.#]. Then for any X r B(0, ~) the
sequence f - ;k is still regular, hence
indO(X)(O/my, C(B)) = ~ f(y), OxC)), y r V.
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
positive generator of this group, for every y r V, [1, p.llS]. Concluding we can state
the following assertion.
B) The function x-+ indO(X)(O/mx , C(B))c K0(A, B) is locally constant on
X\~(B) and equals r * (Tx) , where T x stands for the positive generator of the group
K0(X,X\ {x}) = 7, inherited from the complex structure of X.
Notice that indO(X)(O/mx , C(B)) = 0 if x ~ #(A). Indeed, in this case the complex
K.(foq~, Axe) is exact. Therefore the index vanishes identically when X\~(B) is a
connected subset of X.
Let us conclude with an example of non-vanishing index. Let X = C, A =
: {Izl < 1} and B= {Izl = 1}, If(~: A-~ C is a continuous map, then the sheaf Olin 0
is by A) Fredholm relative to the representation p = @* and the ideal I = V(B) iff (~
doesn't vahish on B', The group K0(A, B) coincide with Z, and in this case assertion B)
has the following numerical interpretation:
indO(C)(O/m0 , C(B)) = deg(~ I B).
Notice the formal analogy with the classical theory of Toeplitz operators.
Acknowledgment. I would like to thank Dan Yoiculescu for very useful convers-
ations about this material.

Putinar 691
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
REFERENCES
ATIYAH, M.F. : K-theory, Benjamin, New-York, Amsterdam, 1967.
BANICA, C.; STANASILA, O. : Methodes alg~briques dans la th~orie globale des
espaces complexes, Gauthier-Villars, Paris, 1977.
BASS, H. : Algebraic K-theory, Benjamin, New-York, Amsterdam, 1968.
CAREY, T.; PINCUS, 3. : Operator theory and boundaries of complex curves,
preprint, 1982.
CURTO, R.E. : Fredholm and invertible tuples of operators. The deformation
problem, Trans. Amer. Math. Soc. 266 (1981), 129-159.
CURTO, R.E.; MUHLY, P. : C~-algebras of multiplication operators on
Bergman spaces, preprint, 1983.
CURTO, R.E.; SALINAS, N. : Generalized Bergman kernels and the Cowen-
-Douglas theory, preprint, 1982.
DOUGLAS, R.G.; VOICULESCU, D. : On the smoothness of sphere extensions,
J. Operator Theory 6 (1981), I03-II I.
FAINSTEIN, A.S.; SHULMAN, V.S. : On Fredholm complexes of Banach spaces
(Russian), Funct. Analysis and Appl. 14 (1980), 87-88.
FAINSTEIN, A.S.; SHULMAN, V.S. : Stability of the index of a short Fredholm
complex of Banach spaces under small perturbation in the non-compactness
measure (Russian), in "Spectral theory of operators", 4, Baku, 1982.
FORSTER, O. : Zur Theorie des Steinschen Algebren und Moduln, Math. Z. 97
(1967), 376-405.
KATO, T. : Perturbation theory for linear operators, Springer, Berlin-Heidelberg-
-New York, 1963.
KAUP, L.; KAUP, B. : Holomorphic functions of several variables. An introduc-
tion on the fundamental theory, Walter de Gruyter ed., Berlin, 1983.
LEVY, R. : Cohomological invariants for essentially commuting systems of
operators (Russian), Funct. Analysis and AppL, 17 (1983), 79-80.
PUTINAR, M. : Some invariants for semi-Fredholm systems of essentially
commuting operators, J. Operator Theory 8 0982), 65-90.
PUTINAR, M. : Uniqueness of Taylor's functional calculus, Proc. Amer. Math.
Soc. 89 (1993), 647-650.
SEGAL, G. : Fredholm complexes, Quart J. Math. Oxford Set. 21 (1970), 385-
-402.
TAYLOR, 3.L. : A joint spectrum of several commuting operators, J. Functional
Analysis 6 (1970), 172-191.
TAYLOR, 3.L. : Analytic lunctional calculus for several commuting operators,
Acta Math. 125 (1970), 1-38.
TAYLOR, 3.L. : Banach algebras and topology, in "Algebras in Analysis",
Academic Press, 1975, pp.I18-186.

6 9 2 Putinar
21.
22.
VASILESCU, F.-H. : Stability of the index of a comlex of Banach spaces, J.
Operator Theory I (1979), Ig7-205.
VASILESCU, F.-H. : Analytic functional calculus and spectral decomposition,
Ed. Academiei and D.Reidel Co., Bucharest and Dordrecht, 1982.
Department of Mathematics, INCREST
Bd. P~cii 220, 79622 Bucharest
Romania
Submitted: November 28, 1984

Integral Equations
and Operator Theory
Vol. 8 (1985)
TRANSVERSALITY AND THE INDEX THEOREM I)
N. Teleman
INTRODUCTION
0378-620X/85/050693-2751.50+0.20/0
9 1985 Birkh~user Verlag, Basel
In this paper, we give a new proof of the index theorem
based on transversality.
The topological index of elliptic operators involved in
the Atiyah-Singer [3'] ,[14] and Sullivan-Teleman [18],[19],[20]
index theorems, have two main ingredients : the Chern character of
the symbol of the operator and the L-character (the Hirzebruch
[10] polynomials in the Pontryagin classes) of the base manifold.
The expression for the L-character, in terms of Pontrygin classes,
is complicated.
The Thom construction [21] provides a very elegant,
simple, geometric realization of the L-character of homology
manifolds; the rational Pontryagin classes appear to be an alge-
braic by-product of the L-character. The Thom construction -
based on transversality - shows that the L-character is built-in
naturally, geometrically.
It is spontaneous then to use the Thom construction
for a proof of the index theorem.
We are going to show here that by realizing this idea,
I)
Research supported by the N.S.F. Grant N ~ MCS 8102758. Paper
revised while the author was visiting I.H.E.S., France.

694 Teleman
the signature operator with coefficients in an arbitrary aspheri-
cal complex vector bundle ~ ~plits-up in a product of two
operators : a signature operator with constant coefficients over
the open submanifold (of the base manifold) consisting of the
sapport of the bundle ~ , and a signature operator over an even
dimensional sphere; the bundle ~ is called aspherical if it is
the pull-back of some bundle over a sphere. The Serre theorem [15]
states that any complex vector bundle is, essentially, over %he
rationals, an aspherical bundle.
The operator splitting above allows us to prove the in-
dex formula for such a signature operator by direct eomputation~
on the base manifold of the operator. This computation involves
the following pieces of information :
i) Hirzebruch-Thom formula
2) Bott-isomorphism
3) Hodge theory
4) Excision invariance of the analytical index
5) Multiplicativity property of the index
6) The index formula for spheres
The proof of the index theorem we give here provides a
proof of the index theorem for Lipschitz and topological manifolds
[18] ,[19],[20] independent of the Atiyah-Singer index theorem for
smooth manifolds. Our method allows also to recapture the index
theorem for signature operators on admissible pseudomanifolds,re-
sult due to J. Cheeger [7].
The presentation of the background material for this
paper is presented in such a manner as to allow a possible axioma-
tization of the index theory. For the reading of this paper, fami-
liarity with [6],[18],[19] and [20] is necessary.
The author thanks Jeff Cheeger for useful discussions
about pseudomanifolds.
2. Basic notations; recall of basic results. The aim of
this section is to introduce the basic notations and recall basic
results which are necessary for the rest of the paper. Secondarily,
by doing so, we might be able to produce an axiomatization of the

Teleman 695
index theory.
Let HT (i.e. Hodge and Transversality) denote one of
the categories : Diff, PL or Lip (of Lipschitz manifolds). In
w we explain,, brielfly, how our considerations extend to the
category of admissible pseudomanifolds introduced by J. Cheeger
[6].
Any object in HT satisfies the following properties
(A.I)-(A.12). The notations and definitions here are as in [19].
(A.I) Any object in HT is a finite CW-complex. The
geometrical dimension is the dimension of the object.
(A.2) Any object in HT has a Lebesque measure
induced by a Riemannian metric.
(A.3) Let C(X) denote the Banach algebra of all
real continuous functions on X . For any pair (X,~) as in (A.I),
(A.2), the sequence of separable real Hilbert spaces,
{Ri}o

696 Teleman
is a bounded operator which decreases supports, i.e.
Supp Af(~) c Supp ~ (The support of ~ , Supp ~, can be defined
because ~r-- is a C(X)-module)
tries :
A.6. The Hilbert spaces ~r are related by isome-_
,r : r m-r
§ ~ , m = dim X
(derived from the Riemannian metric);
These are C (K) -homomorphisms, and
,m-r ,r r (m-r)
o = (-i)
(A. 7) The operator r , with domain :
r
~ = , ~m-r
d
Br = (_l)m(r+l)+l dm-r ,
is the formal adjoint to d (Stokes' formula).
(A.8) The pairing :
~r x ~m-r § m
~,a + (~'*S)r ' (d~ = 0 ,d~ = O)
where ( ' )r denotes the scalar product on er , is non-degene-
rate (Poinear~ duality pairing).
r ~r
(A.9) Let W 1 c be the subspace
r r
~i = nd n nB
endowed with the norm II II
The inclusion
2=
Ill II 2+lld ll 2 II II 2

Teleman 697
(w~,II Ill ) + (~r, II If)
is compact. (Relliah lemma).
(A. IO) The spheres S r , O < r < ~ , belong to the
category HT , and H~(S r) = H*(Sr,~) .
Any morphism :
f : xm+ S m-r
in HT is homotopic to a transverse regular mapping
: xm+s m-r
to a point Po . This means that there exists an object
(vr,u v) 6 HT of dimension r , and an embedding
F : V r Bm-r § m
(~m-r is the unit disc in ~m-r) such that
V r ]Bm-r ~? F ~Bm-r
v r "f
{Po }
= _ ~-r)+
is a bundle map (S m-r is thought as S m-r Bm-r U m.r
Bm-r
being the upper (lower) hemispheres, and
~(X

698 Teleman
in HT
(Xl,U I) , (X2,~ 2)
, their product (X 1 X2,Ul x u2 )
(i) ~r(x I X2,~ 1 ~2 ) = @ ~P(Xl,P I) ~ gq(x2,~ 2) ,
p+q=r
(ii)
(iii)
(@ denotes the Hilbert tensor product)
d(~ @ ~) = d~@ ~ + (-i) deg ~ @ dG ,
,(~ @ ~) = (_l)deg ~. deg~(,~) @ (,~)
belongs to HT , and :
(in (iii) , X 1 and X 2 are even dimensional).
(A.12) The KUnneth formula holds for H*
w Review of Signature Operators.
3.1. Hodge Theory. (A.I)- (A.8) provide Hodge theory. As cus-
tomary, the space of harmonic forms of degree r on an object X
is defined :
is an isomorphism
r d~ = O Be = O}
Hr(X) = {~I~ 6 W 1 , , ..
Theorem 3.1. For an~ ob/ect X g HT
(i) the~a~ural Hodge homomor~hism.
~r r
: H + ~r(x)
(ii) the operator * induces an isomorphism :
*: Hr(x) ~ HdimX-r(x )
(iii) the spaces d~-i , d~+l , Hr
and mutually orthogonal in ~r ; in addition
Proof. See w [19].
~r = H r @ d~-i @ ~+i
are closed

Teleman 699
3.2. Signature operator.
The operator
+ ~ D = d+~ = D + -
D-( WI§ O , 8 D ,
+
are defined in the usual way, see [3'] and [19]; the operator D
is called signature operator.
Theorem 3.2. The signature o~erator
+ +
D : W ! § W O
is a Fredholm operator, and
Index D + = Sig X ,
where Sig X denotes the signature of the Poincar~ pairing on the
middle eohomology of X (we suppose here dim X ~ 0 mod. 4).
Proof. For the smooth case, see Atiyah-Singer [3']; for the
Lipschitz case, the Telemam [19].
Let ~ § X be any continuous complex vector bundle.
We suppose that the bundle ~ has an A(X)-structure, i.e. a
vector subspace A(X,~) of the space of continuous sections in
~,C(X,~) , is prescribed, and A(X,~) is a finitely generated
projective A(X)-module, such that :
A(X,~I 8 ~2 ) = A(X,~I) 8 A(X,~ 2)
A(X,~) = A(X) ~...8 A(X) (N times)
where N denotes the product bundle of rank N
The operators :
D : W (X,~) +W (X ~)
D~
= D + D[
are defined as in w [3'] and ~6 [19]; D + (X,~) is called signature
operator with values in
+
Theorem 3.3. D~ is a Fredholm operator.
The Proof of Theorem 3.3. invokes in addition to
(A.I) - (A.8), the Rellich lemma (A.9).

700 Teleman
3.3. The excision theorem.
(A.4) -(A.7) and (A.9) provide the necessary structure
to carry over all considerations from w167 [19] into the
category HT. As a consequence we have the :
a common o~en part U
Theorem 3.4. (Excision)
Let (M,~) ,~ = 1,2 , be two HT-Spaczs which have
i I i 2
M 1 ~ U , M 2
Let W c U be a closed set, and let
§
@ -~ M ~ = 1,2 ,
be complex bundles of the same rank. Suppose that :
(i) PllU = ~21u
(ii) the following bundle isomorphisms
A21
qlu 2Ju
A89
81 I g ' e21u
~1 (M I "-W) ~21 (M2"-W)
811 (M I ~W) 021 (M2"~W)
are given such that the diagram

Teleman 701
be commutative. Then
(i) (3.3)
(ii)
(3.4)
~11(u ~ w)
e11lu ~ w)
A21
!
A21
~21 (U~W)
821 (U~W)
Index D + - Index D~2 + - Index D +
~I = Index D81 82 ;
if 81 and e 2 are trivial bundles, then
Index D~I - Index D~2 = (Rank ~i ) Sig M 1 -
- (Rank ~2 ) Sig M 2
As a corollary of this theorem we get :
Theorem 3.5. For any complex vector 5undle ~ § X 2m
in the category HT , Index D~ + does not depend on the chosen
measure on X .
It is clear that
+
Index D~
and that
+
+ Index D~
induces a homomorphism.
K~ +
Any element u s K~ is a formal difference
u = [~i ] - [~2 ] of vector bundles, where ~2 may be chosen to
be trivial (this will be tacitly used, later, in some occasions).
By definition, the (virtual) rank of u is :
Rank u = Rank ~I - Rank {2
If ~ is a bundle, let I{I denote the product bundle
of the same rank as { on the same base space.
depends only on [~] 6 K~
We can rewrite the formula (3.4.) for ~1,~2,eI,82
replaced by I~ll, [[21, 1811, 1821 , where the bundle isomor-
phisms A21,A 89 are each the identity; (3.4.) becomes

702 Teleman
(3.4 ') Index D + +
. 1611- Index DI~21 =(Rank I~I) Sig M I -
the
-(Rank I~21) Si s M 2
If we substract (3.4) and (3.4') side by side, we get
Corollary 3.6. For ~1,~2 bundles as in theorem 3.4 (ii), we
have :
+ +
(3.4") Index D~ = Index
1-1~11 D 2-1 21
of virtual rank zero).
(Notice that ~i-[(iI , i = 1,2 , are vector bundles
w The Thom's construction; L-classes.
4.1. For this exposition, see R. Thom [21]
Theorem 4.1. (Serre [15]). Let X be any finite CW-complex.
Let g 6 Hr(sr,~) be a generator. Then the mapping
~r(x) | ~ ~ Hr(X,~)
If] | p/q , p/q f*(g) ,
(for f : X § S r a continuous map) is an isomorphism for
2r > dim X+ 1 . Here ~r(x) stands for the cohomotopy group of
x in dimension r , i.e.
~r(x) = {homotopy classes of maps x + S r} ;
this is an abelian group.
4.2. Suppose now that X m E HT , dim X = m is even. Take any
continuous map :
f : X m § X r , m+1 < 2r , m-r ~ 0 (mod 4)
By (A. IO) , f is homotopic to a map ~ which is transverse regu-
lar to a point p 6 S r . Let F and V be as in the statement
of (A.IO) :
vm-r ~r~F X m

Teleman 703
We associate with f the signature of V m-r , or
Sigvm-r,when m-r ~ 0 mod 4 ; by definition, Sig(V m-r) is the
signature of the Poincar4 pairing (A.8.) on the middle L 2-
cohomology of V m-r . We require :
(A. IO') Sig V m-r is independent of the transverse
regular mapping belonging to the homotopy class of f , and
r (X) + ~ ,
[f] § Sig V
is a homomorphisms. []
By definition, (see Thom [21], and Goreski-MacPherson
[8]) the homology L-class Lr(X m) is that rational homology class
Lr(Xm) 6 Hr(Xm,~)
with the property that :
(4.1) f*(g) (Lr(Xm)) = Sig V m-r ,
where g 6 Hr(sr,~) is a generator. We set :
L~(X m) = ~ Lm_4i(xm) 6 H4,+e(xm,~ ) ,
i=O
E ~ dim xm(mod. 4)
As X m has a Poincar4 duality, relative to the ratio-
nal singular cohomology ~) , the cohomology class
L4i(x m) 6 H4i(xm,~)
may be defined by the formula
(4.2.)
X m
L* (X m)
(L4i(x m) [J f*(g)) [X m] = Sig V m-4i ,
denoting the fundamental class of X m and
L*(X m) = Z L4ilX m)
i=o
is the Hirzebruch L-character of x m
*)This is not the case for pseudomanifolds.

704 Teleman
The formulas (4.1), (4.2) show that L4i(x m) and
Lm_4i(xm) are Poincar4 dual :
(4.3) Lm_4i(xm) = L4i(x m) n [x m]
This construction allows us to define Lm_4i(xm) f
resp. L4i(xm) in dimensions above, resp. below, the middle dimen-
sion of X m . The other classes are defined by taking the direct
product of X m by S m , see [8] and [12].
Proposition 4.2.
(i) L~ m) = 1
L4n(x 4n) X 4n = Sig X 4n (Hirzebruch signature
formula)
L4n(x Y) = Z L4P(x) x L4q(y)
p+q=n
L*(S r) = 1
(ii) L*(X4n)([x4n]) = Sig X 4n (Hirzebruch signature
o
formula)
L4n(X 4n) = [X 4n]
Li(X x y) = Z L (X) L (Y)
p+q=i P q
L,(S 2r) = [S ,2r]
Proof. The two groups of relations (i), (ii) are related by
Poincar4 duality. For the proof of the multiplicative property of
L, , see w Appendix, given here for the benefit of the reader.
4.3. The even dimensional rational cohomology of any finite com-
plex is generated, over ~ , by the Chern character of all complex
vector bundles over it. We would like to use this fact in order
to reformulate the defining formula for the L-classes.
To do this, we have to suppose first that X m is even
dimensional, m = 2s .
Let
f: X2s §
S2s

Teleman 705
be a continuous map, and let { § S 2s be a complex vector
bundle over S 2Z-4i . As the Chern character is natural, and
K(S "25-4i) ~ Z (Bott isomorphism), the defining formulas (49
and (4.2) for L,(X) and L*(X) become :
Theorem 4.29 (Hirzebruch-Thom formula)
(i) Ch(f*{) (L2s163 = Sig v4i.Ch~([S2Z-4i])
or, for Poincar~ duality spaces :
(ii) (L4i(x 2s [] Ch(f~)) [X 2s = Sig V 4i
9 Ch6([$2s
A virtual vector bundle over x of the form f*~ , where
f : X § S 2r
is a continuous map, and ~ 6 K~ 2r) , will be called aspherical.
Notice that the Chern character of an aspherical ele-
ment f*~ is of the form
where
f*~ , and
of dimension
Ch(f*~) = n + U2r ,
n 6 H~ is the virtual dimension of the bundle
U2r 6 H2r(x,~)
Proposition 4.3. Let X m be a finite CW complex
m . Then for any cohomology classes
a o 6 H~ , a r 6 Hr(xm,~) ,
r = even, 2r > m+l , there exists one and only one
~ 6 K~ m) | ~ having the properties :
(i) ~ is aspherical
(ii) Ch n = a + a
o r
Proof. 1) Existence. The cohomology class a r is aspherical.
Serre's theorem 4.1. implies that there exists a continuous map
f : X m § S r
such that f*(gr ) = a r , where gr 6 Hr(sr,~) is a generator.

706 Teleman
As the Chern character (on S r) is an isomorphism, there exists
an element ~ 6 K~ r) | ~ such that
Theorem 5.1. (Index Theorem for signature operators in
HT). Let X 2s 6 HT and let ~ + x 2s be a complex vector bundle
over x 2~ . Then :
(5.1)
Ch ~ = gr
Let e = I | a 6 K~ r) | ~ , where I is the trivial
o
bundle of rank I on S r . Then q = f*(e+~) has the desired pro-
perties.
isomorphism. []
2) Uniqueness. The Chern character (on X m) is an
Let ~ denote the external tensor product of bundles;
the same symbol will be used to denote external products of
K-theory classes.
property that
Let 82r 6 K~ 2r) | ~ denote that element with the
Ch e2r = g2r 6 H2r(s2r,~)
Proposition 4.4. For any element ~ 6 K~ m) 8
having the property that the Ch ~ is a homogeneous cohomology
class, the element
s em+ E 6 K~ mx S m+~) ~
(where ~ = O,1 such that m+~ = even) is aspherical.
Proof. Ch(~+ ) = Ch ~ x Chem+ E = (Ch~)xgm+ C ,
which is a homogeneous cohomology class whose degree exceeds the
middle dimension of X m x S m+e . The proposition 4.3. completes
the argument.
w The Index Theorem for Signature Operators.
5.1.
Index D + = (Chq) ( Z 2s
i=O 2~-4i(X2~))
We are going to show that the formula (5.1) follows

Teleman 707
from
i) Theorem 4.2. (Hirzebruch-Thom formula)
2) Bott isomorphism : K~ = ~ ,
+
3) Theorem 3.2. (Index D = Sig X, Hodge theory)
4) Theorem 3.4. (ii) (Excision)
5) Theorem 5.2. (Multiplicativity of the Index).
6) Theorem 5.3. (Index formula for spheres).
2s
Theorem 5.2. Let X i 6 HT , i = 1,2 ; let
be complex vector bundles and
~1 [] ~2 § X I x X 2
their external product.
Then
5.2. Index D~I [] ~2 = Index D~I " Index D~2
~i § Xi
Theorem 5.3. (Index formula for spheres). The index
formula (5.1) holds for even dimensional spheres, i.e. for ~ a
bundle over a sphere s 2r :
(5.3) Index D + =~Ch ~ [S 2r]
The proof of theorem 5.2. will be postponed to w In
order to prove theorem 5.3., it is enough to check it for the
generator of the reduced K-theory group, K~ = ~ , for
r = 1,2, .... Because the index of signature operators is inde-
pendent on the measure on the base space, we can take the stan-
dard Lebesque measure on S 2r and therefore the problem can be
reduced to the smooth case. For this computation see [14] ch. XV,
w (The author thanks A. Connes for the 2 r factor in(5.3.)).
Proof of Theorem 5.1.
5.2. The aspherical case.
We will prove (5.1) first for n an asphericalbundle:
n = f*~ ,

708 Teleman
where
X2f.~ S 2-~ -4i $2~-4i
(5.4) f : , $ +
By virtue of (A.10), we may suppose
the point P 6 IB 2s c S 2~-4i
o
Let
(5.5) V 4i f-l{po}
and if we thought of the embedding F
tion, we may consider that
5.6) V 4i x IB 2g-4i c X 2~
5.7) fl ({v} 163 : {v} 2s § I]32s
is the identity mapping for any v 6 V 4i , and
(5.8) f(X2s v4ix IB 2s _c S 2s ~{po}
f transverse regular to
(see 4.2.) as identifica-
The mapping f can be deformed continuously on the subset of
X2Z~ v4i x IB 2Z-4i consisting of those points x which are sent
by f into the lower hemisphere IB 2s (and, therefore, in
IB 2 s {po } ; this will be done by moving f(x) along the me-
ridian passing through f(x) northward, until the point reaches
the equator (think of P as the South pole); the function f
o
will be kept unchanged everywhere else). As a consequence, we may
strengthen (5.8.) and suppose that :
2~-4i S2s
(5.8.') f(X 2~ ~V 4i IB 2~-4i)_ _c IB+ c
This last homotopy does not modify the isomorphism class of f ~.
At this point, it is very easy to describe the bundle f~
For, we think of ~ as obtained by clutching together two product
bundles
2s ~n pr~ 21-4i
IB IB
2s
along the equator ~IB_ by a clutching function :
2~-4i
: ~IB § GL (n,~)
As f carries V 4i IB 2s onto IB 2Z-4i , and the complemen-
tary onto B+ Z-4i- , and as ~ has a product bundle structure on

Teleman 709
IB 2s f~ is a product bundle both over V 4i IB 2s
x and
9
i~complementary. If we take into account (5.7.), we can see that
f*~ is obtained by clutching together two product bundles with
fiber ~n
(V 4i x IB 2~-4i) x ~N pr~ V4ix IB 2~-4i
(X2s V 4i x IB2s x ~N
with a clutching function f~ :
v4i -4i
f*~ : x~IB 2 § GL(n,~)
(f~q0) (v,x) = ~(x)
pr~ X2s ~V 4i IB 2s
+
We are preparing to use excision for showing that Index Df~
depends only on V 4i and ~ For, let X ~2s be
~2s = v4i $2s ;
~2z may be thought as glueing together V 4i x IB 2s and
V 4i IB~ s along V 4i a IB 2Z-4i_ by using the identity map-
ping. Let ~ § ~2~ be the bundle obtained by clutching together
the product bundles with fibre ~r over the two pieces
2~-4i v4i 2~-4i
V 4i x IB , along x a IB , and with the clutching
isomorphism
therefore
V 4i a IB 2s
: x ~ GL(n,~)
~(v,x) = ~(x) ;
(5.9) ~ = I , ~ ,
V 4i
where ~4i + V4i is the product bundle of rank 1 o
We are almost ready to invoke the Corollary 3.6. for :
M1 = X2s , M2 = ~2s
~i = f~ ' 62 = ~
el= If* l , o2 =

710 Teleman
U = V 4i IB 2s , and
W = Int U
of course, U and W so chosen are not exactly what we need;
notice, however, that these two sets are necessary to show the
existence of bundle isomorphisms A21,A21,EI,E 2 which make a com-
mutative diagram (3.2); this is a homotopic problem in nature and,
if we can solve it for this choice of U and W we can solve it
for correct U and W obtained by thickening. This saves us
some more notation.
It will be convenient to say that the product bundles
used to describe f*$ and ~ are coo~dinas charts (in the sense
of Steenrod [16]); those two product bundles with bases
V 4i IB 2s will be called first coordinate charts, and they
will be denoted (f ~)i ' rasp. ~i ; the other two charts will
be called second coordinate charts, and will be denoted (f ~)2'
rasp.
~2 " The same conventions and notations will apply to the
product bundles If*s1, I~I- thought as obtained by coordinate
charts, respectively with the same bases, but with clutching func-
tions equal to the identity.
Now we are in a position to define the isomorphisms
A21,A21, EI,E2 involved in the Corollary 3.6.
We set :
A21 = identity : (f*~)l + ~i
A~I = identity : If*~.ll §
l I = identity : (f*~)N§
12 = identity : ~2 +I~12
Notice that on U ~W = V 4i ~IB 2~-4i these isomorphisms are
compatible with the clutching functions, and hence the diagram
(3.2) commutes for these bundles and isomorphisms.
From Corollary 3.6, (5.9) and Theorem 5.2, we get
+ +
Index = Index D =
Df*E-If*~ I 7- I~' I
+ +
(5.10) Index DIV[]~_IIv~ ] = Index Dlv [](~-]~I) =

Teleman 711
+ .Index D +
Index D~v ~-[~I ;
the bundle ~-I~I is over a sphere ; therefore, Theorem 3.2
and Theorem 5.3. will give :
ther :
Index D + = Sig V 4i 2 i-2i Ch(~- I~I)[S 2~-4i] =
= Sig V 4i 2 ~-2i Ch $ ([$2~-4i])
From the Hirzebruch-Thom formula (Theorem 4.2) we get fur-
Index D + = 2 s
f*~-I f*E I
Ch(f*~)(L2s i (X 2s ))
or, using again the Theorem 3.2 and Theorem 4.2. :
+
Index D
f
= Rank ~'Sig X 2s + Ch(f*~) (2s =
= Ch(f*~) (Lo(X2s + Ch(f*~) --s (2 L2s (X2~)) =
Ch(f*~) ( Z 2 ~-2i (X 2~
= L2s i ) ) ;
i=O
here we have used the Hirzebruch signature formula (Proposition
4.2 (ii)), and the fact that f*~ is an aspherical bundle. This
proves the index theorem for aspherical bundles.
5...3 Reduction to the aspherica ! case. Let ~ 6 K~
be such that Ch ~ is homogeneous. From Proposition 4.4. we know
that ~ ~ g2~ s K~163 xS2~) ~ is aspherical; Theorem 5.1.
for aspherical elements applies, and we have :
Index D+m82s (5_1) Ch(~82s ( E
i=O
22s163 (X2s S 2s )
IITheorem 5.2) ll(ch and L, are multiplicative
Index D~-Index D +
e2s
II (Theorem 5.3)
Index D~- + 24 .
00
(Ch~) E (2%-2iL2s -
i=O
Ch~( ~ 2s163 j(S 2s
j~
II (Ch82s ( ~ $2 s ~] )=l'The~ 5.3,Prop.4.2)
2Z-2i L 2 s
Ch ~ (i=EO 2s (X2s "

712 Teleman
which proves the Index Theorem for signature operators in the whole
generality.
6. Proof of Theorem 5.2.
6.1. Review of Baaj-Julg [4] and Hilsum [9]results.
S. Baaj and P. Julg [4] defined the elements of Kasparov's [II]
KK(A,B)-groups in terms of unbounded operators. Kasparov's cons-
truction applies to oth-order operators, while the Baaj-Julg cons-
truction is very natural for ISt-order operators ; the important
examples of operators arising from geometry are operators of order
> O For this reason at least, the Baaj-Julg construction is
very important. In a consecutive important development along these
lines, M. Hilsum [9] shows that the signature operators D +
constructed by the author in [19] fit naturally into the Baaj-
Julg construction, and gives functional-analysis new proofs to two
basic theorems from [19] and [20] .
For what we are concerned, the group KK(C~(X),~) suffi-
ces our needs (C~ = C -algebra of continuous complex-valued func-
tions on X) ; it is isomorphic to Ko(X ) , the K-homology group
of X in dimension 0 .
Let A be a C~-algebra. A cycle into the Baaj-Julg
KK(A,~)-group consists of a triple (H,~,D) , where :
i) H is a ~2-graded, i.e. H = H + ~ H- , infinite dimensional
separable Hilbert space,
2) 9 : A § L(H) is an unital C~-algebra homomorphism, compatible
with the graduation of H , i.e. ~(a) (H c H , Va 6 A ,
m
3) D is an unbounded linear operator on H such that
a) D(D), D(D ~) are dense in H (D = domain)
b) D is closed
c) 1 + D~D has dense range in H
d) D is an operator of degree I, i.e. D(D(D) n H c H ~
4) D = D ~
5) ~(a) (I+D2[ 1 is a compact operator, for any a s A ,
6) Do~(a) - ~(a) oD is an operator which can be extended to a
bounded operator on H for any a s A D , where A D is a dense

Teleman 713
subalgebra in A
perty.
The requirement 5) is equivalent to a Rellich-type pro-
To any Baaj-Julg cycle (H,~,D) there corresponds a
Kasparov cycle [ii] by the canonical mapping :
B : (H,~,D) + (H,~,D(I+D2) -I)
notice that D(I+D2) -I is a bounded, self-adjoint, Fredholm ope-
rator of degree i. The index of this cycle is by definition :
Index{ (D(I+D2)-IIH+) : H + § H-}
Kasparov defines a pairing (external product) :
KK(AI,~ ) x KK(A2,~ ) § KK(A 1 ~ A2,~ )
The Kasparov external product is difficult to perform, but the
corresponding Baaj-Julg product is very simple, because it is al-
gebraic :
THEOREM 6.1. (Baaj-Julg [5]) Let (Hi,~i,Di) , i = 1,2
be Baaj-Julg cycles over the C*-algebras A 1 , resp. A 2 .
(i) Then the external product given by the formula :
(HI,~I,DI)) (H2,~2,D 2) = (H 1 ~ H2,~I ~2,DI @i + 1 QD2)
is~ s~till a ~aaj-Julg cycle, and via B , it corresponds to
Kasparov's external product; in addition,
(ii) Index{(Hl,~l,Dl) x (H2,~2,D2) } =
= Index(Hl,~l,Dl).Index(H2,D2,~2 )
The statement (ii) follows from the fact that it is true
for the Kasparov multiplication, and from (i).
THEOREM 6.2 (M. Hilsum [9]) For any compact oriented
Lipschitz manifold of even dimension,
{W+o ~ wo , multiplication by Lipschitz functions, D + ~ D-}
is a Baaj-Julg cycle over the algebra of continuous functions.
The same result holds for signature operators with values
in a vector bundle.
6.2. Let Xi,~i , i = 1,2, be as in the Theorem 5.2. We

714 Teleman
consider the corresponding signature operators D + i = 1,2. We
~i '
can keep track of these operators by considering tlie graduated
Hilbert spaces
W O(xi,~i ) = W~(Xi,~i ) @ Wo(Xi,~i ) ,
in which W+(Xi'gi ) o , resp. Wo(X i,~i ) , has degree O, resp. i, and
D~i = D+~i ~ D~i- is thought as an operator of degree 1.
A direct calculation still valid for differential forms
with values in vector bundles, see [14], Ch. XV, w 6, Lemma 2,
involving the information given by the axiom A.II, shows that:
(6 .i)
where
X 1 with values in ~i :
@ ~r(x I) @A(XI ) A($ I) ;
r
on the differential forms of degree r , ~i
by (-i) r , see [14], p. 230.
we get
= 81 + ~i @ ,
D~im~ 2 D~ 1 D~ 2
~l is an involution on the space of differential forms on
From (6.1) and from the fact that ~i
(6.2) D~Im~2 = ~I~176 | 1 @ I | D~2]
is the multiplication
is an involution,
~i is a hermitian isomorphism which preserves the graduation, so
~lOD~l is still a self-adjoint, degree 1 operator.
We know from Theorem 6.2. and Theorem 6.1 (i) that the
bracket from (6.2) is the operator-component of a Baaj-Julg cycle;
by invoking hhat ~i is an isomorphism, and the Theorem 6.1 (ii),
we get
Index D~I~2 = Index[(~l~ | i + I | D~2J=
which proves Theorem 5.2.
= Index(~lOD l).Index D~2 =
Index "Index
= D~I D~ 2

Teleman 715
w Admissible Pseudomanifolds vs.HT. In this sec-
tion, we intend to check, briefly, (A.I.) - (A.12.) for admissible
pseudomanifolds. The fulfillment of these properties for these
spaces constitutes an important series of results due to
J. Cheeger [5] , [5'], [6] , [7] and M. Goreski-MacPherson [8].
(A.I.) Recall [6], that a closed pseudomanifold of
dimension n is a finite simplicial complex of dimension n in
which any (n-1)-simplex is a face of precisely two n-simplice~
(A.2.) If M n is a pseudomanifold of dimension n ,
let M (k) denote its K-skeleton . M n may always be thought
as a smooth Riemannian manifold away from the codimension two
skeleton of it. It is possible to introduce natural metrics on
M n , e.g. piecewise flat metrics, which are intimately related
to the geometry of M n . Such metrics define the required Borel
measure.
(A.3.) It makes sense, then, to speak about smooth
differential forms on M with support in M n\ M (n-2) . Here,
the classical operators d , , , 6 , A are defined.
~i(M) is, by definition, the space of L2-forms of
degree i on M , obtained by the completion of the space of
smooth differential forms with support away from the codimension
two skeletons.
(A.4) 9 The operator d i is densely defined in h i
let d i denote also its closure, and ~d i its domain.
i and homomorphisms d i
The spaces ~d ' , i = 0,1,2,...,
form the L2-cohomology complex ~(M)
If the pseudomanifold M satisfies an extra local
property regarding the metric geometry of the link of any simplex
in M , the pseudomanifold M is called admissible, see [6] ,
p. 127.
9 For example, the pseudomanifolds having only
even codimension strata, see [8], and cone-like metrics times
flat metrics, are admissible.

716 Teleman
The homology Hi(~(M)) , for M an admissible pseudo-
manifold, is the dual of the middle homology group IH~(M n)
defined in [8], see theorem 6.1. [6].
tiable functions on M
(A.5.) A(M) is, e.g. the algebra of PL-differen-
(A.6.) The operator * , densely defined on smooth
forms with compact support in M n \ M (n-2) , extends isometrical-
ly to the L2-spaces.
(A.7.) This is the property B) w [6] , p. 95 ; it
is verified in the proof of Theorem 5.1. [6]
(A.8.) The L2-cohomology is represented by harmonic
forms. The *-operator induces isomorphisms between the spaces
of harmonic forms in complementary dimensions. The pairing (A.8.)
is given by integration
M
this pairing is clearly nondegenerate because the ,-operator is
an isomorphism.
(A.9.) This follows from Theorem 3.1. [5] or Theorem
3.3. [5'3, which states that there exists an orthonormal system
of eigenforms for the Laplacian, in any dimension, and the corres-
ponding eigenvalues converge to +~ . The Rellich lemma is one of
the basic ingredients of the Baaj-Julg cycle.
(A. IO.), (A.IO'.). The middle intersection (co)homo-
logy of manifolds, is isomorphic to the singular (co)homology,
ef. 4.3. [8].
For the transversality arguments, at least for pseudo-
manifolds with even codiemnsion strata, see w [8]
(A.11.) , (A.12.). This is Theorem 7.1. [7]
bundle
X 2s
To summarize, we have the :
Theorem 7.1. (J. Cheeger [7]). For any complex vector
over the admissible closed, oriented pseudomanifold

Teleman 717
+
Index D = (Ch ~) (
n i= O
0o
2s163 ))
The author is in debt to Jeff Cheeger for useful
discussion regarding pseudomanifolds.
w Appendix. The multiplicativity property of the
L-classes follows from two basic facts
r.
(i) , (ii) explained here.
(i) If f. : M 9 § S 1 , i = 1,2, are two mappings,
1 1
consider the mapping
r1+r 2
fl ^ f2 : MI M2 § S
where
with
fl ^ f2 = p o (flxf2) ,
r I
p : S x S r2 § S rl A S r2 = S r1+r2
is the projection mapping; here A denotes the 6mash product
rl = r2 r 2
S ^ S r2 S rl x S /{a I} x S U S rl x {a2} ,
r.
1
a. 6. S
1
If f.
1
is transverse regular to
r.
1
x 9 6 S , and
1
x i r a i , i = 1,2, then fl ^ f2 is transverse regular to the
point (Sl,X 2) E S rl ^ S r2 , and
I .
3.
(fl A f2)-1(X1'X2 ) = f11(Xl ) x f21(X2 )
(ii) The KHnneth formula (A.12.)
REFERENCES
M.F. Atiyah : Global Theory of Elliptic Operators,
Proc. Int. Conf. Functional Analysis and Related
Topics, Tokyo, (1969), 21-30.
M.F. Atiyah, R. Bott, V. K. Patodi : On the Heat
Equation and the Index Theorem, Inventiones Math. 19,
(1973), 279-330
M.F. Atiyah, I. M. Singer : The Index of Elliptic
operators, Part I, Annals of Math. 87 (1968), 484-530.

718 Teleman
3'.
4.
5.
5f o
6.
7.
8.
9.
10.
11. 84
12
13.
14.
15.
16.
17.
M. F. Atiyah, I. M. Singer : The Index of Elliptic
Operators, Part III, Annals of Math. 87 (1968),
546-604.
S. Baaj. P. Julg : Th4orie bivariante de Kasparov et
Op4rateurs non born4s, Compt. Rend. Acad. Scien.,
Paris, 296 (1983), 875-878.
J. Cheeger : On the spectral geometry of spaces with
cone-like singularities, Proc. Nat. Acad. Sci- 76 (1979)
2103-6.
Idem, more extended version, preprint, 1979.
J. Cheeger : On the Hodge Theory of Riemannian Pseudo-
manifolds, Proc. Symp. Pure Mathem. Vol. 36, (1980),
91-146.
J. Cheeger : Spectral Geometry of Singular Riemannian
Spaces, J- Diff. Geom. 18 (1983) 575-657.
M. Goresky, R. MacPherson : Intersection Homology
Theory, Part I, Topology 19,(1980), 135-162.
M. Hilsum : Op4rateurs de Signature sur une Vari4t~
Lipschitzienne et Modules de Kasparov non born4s,
Compt. Rend. Acad. Scien., Paris,297 (1983) 49-52.
F. Hirzebruch : Topological Methods in Algebraic
Geometry, 3rd ed. Springer-Verlag, (1966).
G. Kasparov : The Operator K-functor and Extensions
of C*-algebras, Math. USSR Izvestija, 16 (1981), 3,
513-572.
R. Kirby, L. Siebenmann : Foundational Essays on
Topological Manifolds, Smoothings and Triangulations,
Annals of Math. Stud. 88 (1977), Princeton.
S.P. Novikov : Topological Invariance of Rational
Pontrjagin Classes, Doklady, 163, (1965), 2, 921-923
(English translation).
R. Palais : Seminar on the Atiyah-Singer Index Theorem.
Annals Math. Stud. 57, (1965), Princeton.
J. P. Serre : Groupes d'homotopie et Classes des Grou-
pes Ab~liens, Ann. of Math., 58, (1953), 198-231.
N. Steenrod : Topology of Fibre Bundles, Princeton
Univ. Press, Princeton, (1951).
D. Sullivan : Geometric Topology, Part I, M.I.T.,
(197~).

Teleman 719
18.
19.
20.
21.
D. Sullivan, N. Teleman : An Analytical Proof of
Novikov's Theorem on Rational Pontrjagin Classes,
Publ. Math. IHES, 58, (1983).
N. Teleman : The Index of Signature Operators on
Lipschitz Manifolds, Publ. Math. IHES, 58, (1983).
N. Teleman : The Index Theorem for Topological Mani-
folds, Acta Mathem., Vol. 153 (1984), 117-152.
R. Thom : Les Classes Caract~ristiques de Pontrjagin
des Vari4t4s Triangul4es, Symp. Int. Top. Alg. Mexico,
(1956), 54-67.
Department of Mathematics
State University of New York
Stony Brook, New York 11794
U.S.A.
Submitted: January i, 1985