I Integral Equations and Operator Theory

Integr. equ. oper. theory 39 (2001) 377-386
0378-620JU01/040377-10 $1.50+0.20/0
9 Birkh/iuser Verlag, Basel, 2001
I Integral Equations
and Operator Theory
ON COMMUTING COMPACT SELF-ADJOINT OPERATORS ON A
PONTRYAGIN SPACE
Tirthankar Bhattaeharyya and Toma~ Ko~ir
Suppose that Ai, A2, .. 9 An are compact commuting self-adjoint linear maps on
a Pontryagin space K of index k and that their joint root subspace M0 at the zero
eigenvalue in C n is a nondegenerate subspace. Then there exist joint invariant
subspaces H and F in K such that K = F @ H, H is a Hilbert space and F
is finite-dimensional space with k _< dimF _< (n + 2)k. We also consider the
structure of restrictions AjlF in the case k = 1.
1 Introduction
Let K be a Pontryagin space whose index of negativity (henceforward called index) is k
and A be a compact self-adjoint operator on K with non-degenarate root subspace at the
eigenvalue 0. Then K can be decomposed into an orthogonal direct sum of a Hilbert subspace
and a Pontryagin subspace both of which are invariant under A and this Pontryagin subspace
has dimension at most 3k. This has many applications among which we mention the study
of elliptic multiparameter problems [2]. Binding and Seddighi gave a complete proof of this
decomposition in [3] and in fact proved that non-degenaracy of the root subspace at 0 is
necessary and sufficient for such a decomposition. They show that the bound 3k can be
attained. We refer to the books [1, 4] for general theory of operators on a Pontryagin space.
We present a generalization of the decomposition to encompass a tuple of commuting
compact operators on a Pontryagin space. Such tuples occur naturally in applications to
boundary value problems for partial differential equation, say of Sturm-Liouville type, that
are coupled by several parameters. When such multiparameter boundary value problems
are of so-called elliptic type, their analysis leads to an n-tuple of commuting compact self-
adjoint operators on a Pontryagin space that is not a Hilbert space. We shall not elaborate
on the multiparameter aspects here. They can be found, for example, in [2] and [5]. In
this paper, our aim is to obtain a decomposition of K and also classify tuples of commuting
compact operators when k = 1. A compact normal operator can be thought of as a pair
of commuting compact self-adjoint operators. In the case of finite-dimensional Pontryagin
space of index 1, normal operators are completely classified in [8]. Thus the classification of
n-tuples of commuting compact self-adjoint operators on a Pontryagin space of index 1 is a

378 Bhattacharyya, KSsir
natural question.
There are two main results in this paper. The first one of them, Theorem 2.7 gives
a decomposition of the entire space into joint invariant subspaces one of which is a Hilbert
space H and the other, say F, is a Pontryagin space of index k and its dimension is at most
(n + 2)k. We give an example to show that this bound is indeed sharp. The structure of
F in the decomposition of Theorem 2.7 is described in further detail. The subspace F is
equal to a direct sum FI @ F2, where F1 is spanned by all joint root subspaces at nonreal
eigenvalues and the spectra of restrictions to F2 are real. Furthermore, the dimension of F1
is exactly twice the index of F1, while the dimension of F2 is bounded below by the index of
F2 and above by n + 2 times the index of F2. In particular, the bound (n + 2)k above can
be achieved only if all the eigenvalues are real. In Theorem 3.1 we classify the n-tuples of
commuting compact self-adjoint operators on a Pontryagin space of index 1.
2 Splitting of an invariant finite-dimensional subspace
with an invariant complement that is a Hilbert space
Let A = {At, A2,.. 9 A~} be a set of commuting compact self-adjoint linear maps on K. If L
is a subspace of K then L [ = {u C K; [u, v] = 0 for all v E L} is its orthogonal complement.
Here [., .] is the inner product on K. The subspace L is called nondegenerate if L N L [ = 0.
Any nondegenerate subspace of K is a Pontryagin space in its own right and we denote
by a(L) its index. A subspace L is called ortho-complemented if L + L [ is equal to K. If
A : K --+ K is a linear operator then we denote by A[*] its adjoint. One has [Au, v] = [u, A[*]v]
for all u, v C K.
If X C K is a nonempty set we denote by E(X) the closed linear span of X in K.
For a = (al, a2,..., s~) E C ~ we write
and
Ls(A, j) = N X ((A~ - s~ ... (~ - s~) 'o)
Ls(A) = [.J Ls(A,j).
j=l
Here Af(A) is the nullspace of a linear map A. Note that La(A) is the joint root subspaee
of A at s. We use the notation La(A,j) and La(A) if A = {A}. The tuple ~ denotes the
n-tuple obtained from s C C ~ by conjugating all its components. We define
{L~(A) if s ~
Me (A) = L5 (A) + L~(A) if a ~ IR.
For a E C ~ we define Ms(A) = n~=lM, j (Aj). We remark that for a E IR ~ we have Ls(A) =
Ma(A). If s, fl ~ ]R ~ then it follows from Lemma 2.1 below that Ms(A) = Mfl(A) if and
only if sj C {f~j, ~j} for all j. To avoid duplication when considering the subspaces Ms(A),
we assume that the imaginary parts of components aj, j = 1,..., n, of s are nonnegative.

Bhattacharyya, K6sir 379
In this paper we assume that the joint root subspaee Mo = Mo(A) at 0 = (0, 0,..., 0)
E C ~ is a nondegenerate subspace.
We say that an eigenvalue a of A is normal if La(A) is finite-dimensional and it has
a closed complement that is invariant for A. If a E C is a nonzero eigenvalue of a compact
linear map then it is a normal eigenvalue (see e.g. [10, p.190]). It follows that a nonzero
eigenvalue a C C ~ of A is a normal eigenvalue.
A subspace L C K is invariant for A ifAiL C L for all i. A closed invariant subspace
L for A is called decomposable if there exist nonzero closed subspaces L1 and L2, invariant
for A, such that L = L1 @ L2. If such L1 and L 2 do not exist we call L an indecomposable
subspace for A. Observe that subspaees La(A) for a r 0, and the closure Mo of Mo are
closed and invariant for A. Then it follows that if L is an indecomposable subspaee for A
the restrictions Ai]L have only one eigenvalue. Moreover, each subspace La(A) and Mo are
direct sums of indecomposable subspaces for A. If L is an indecomposable subspace for A
and L c La(A) then we say that L is an indecomposable subspace for A at a, and a is the
eigenvalue corresponding to L. If n -- 1 then an invariant subspace is indecomposable if and
only if it is a linear span of a single Jordan chain.
Now we prove a few auxiliary results that lead to the proof of our main results.
LEMMA 2.1. Ira ~ 0 is an eigenvalue of A then Ma(A) is nondegenerate.
PROOF. Since a ~ 0 there is an index I such that al ~ 0. We may assume without
loss of generality that l = 1. Then M~I (A1) = ~fle~, Mfl(A), where E1 is the set of all the
eigenvalues fl = (/31, f12,..., fin) of A such that imaginary parts of flj are nonnegative and
/31 = al. We know by [3, Lemma 1] that Mal (A1) is nondegenerate. For each eigenvalue
fl E E1 and fl ~ a there exists an index j such that aj ~/3j. Then MZj (Aj) C M~j (Aj) [ By
the fact that L [ C L~ j-] if L1 C L2, we have Mfl(A) C MZj (Aj) C M~j (Aj)[ C Ma(A) [1].
Therefore Ma(A) is an ortho-complemented subspace of M~ (A1). An ortho-complemented
subspace is nondegenerate (see [4, Cor. 1.9.5]). []
The following result is a consequence of the assumption that Mo is nondegenarate.
COROLLARY 2.2. For each j the subspace Mo(Aj) is nondegenarate and the clo-
sure of the linear span of Jordan chains of Aj is equal to I4.
PROOF. Observe that Mo(Aj) is a direct sum of Mo and the subspaces Ma(A),
a --- (ax, a2,..., an), such that aj = 0. By [1, Cor. 3.14] it follows that Ma(A) C Mo [
By our assumptions and Lemma 2.1 all the subspaces Ma(A) are nondegenerate. Therefore
Mo(Aj) is nondegenerate. Theorem 1 of [3] implies the second part of the statement. []
THEOREM 2.3. Ira ~ IR n is an eigenvalue of A then n(Ma(A)) = 89 dim Ma(A).
PROOF. Since a is normal it follows that Ma(A) is finite-dimensional. By Lemma
2.1 it is also nondegenerate. Suppose aj is a nonreal component of a. Then the restriction
of Aj to Ma(A) has a conjugate pair aj, ~j for its spectrum. The lemma then follows by [6,
Thin. 1.3.3]. []
Let a be an eigenvalue for a compact self-adjoint operator A on a Pontryagin space
and let J = {v0, vl,..., vz} be a Jordan chain at a. Then we follow [3] and call J negative if

380 Bhattacharyya, K6sir
[vo, v0] _~ 0 and positive if [Vo, Vo] > 0. Note that if c~ @ ]R or if 1 > 1 then [vo, vo] = 0 and J
is negative.
If L is an indecomposable subspace for a set A = {A1,A2,... ,A~} of compact
commuting self-adjoint operators then L is called positive if it contains only positive Jordan
chains for each Ai, otherwise it is called negative. Observe that if L is a positive indecom-
posable subspace for A then it is one-dimensional, spanned by a joint eigenvector of Ai, and
the corresponding eigenvalue is real.
LEMMA 2.4. Let K be a Pontryagin space of index k. Given a compact self-adjoint
operator A on K with nondegenerate root subspace Mo(A), the whole space K can be written
as an orthogonal direct sum K = H @ F where H is a HiIbert space and F is a finite-
dimensional Pontryagin space of index k such that both F and H are invariant for A and
k < dim F ~ 3k. Moreover, the Jordan canonical form of the restriction of A to F has at
most k blocks. In particular, there are at most k negative Jordan chains in a Jordan basis
for A.
PROOF. A maximal nonpositive subspace of K has dimension equal to k (see [1, 4]).
Since we assume that Mo(A) is nondegenerate it follows by [3, Thin. 1] that root vectors of
A are complete. By [1, Thins. 2.26 and 3.4] or [4, Thins. 4.6 and 4.9] it follows that a Jordan
chain J of A at a real eigenvalue has length l < 2k + 1 and the dimension of a maximal
nonpositive subspace in s is equal to if l is even and to [~] or [4] + 1 if l is odd. A
Jordan chain J of A at a nonreal eigenvalue a has length I _< k. For J there is a chain J for
A at ~ such that J and J are of equal length and s U-I) is a nondegenerate subspace (see
[6, Thin. 1.3.3]). Moreover, a maximal nonpositive subspace of s U J) has the dimension
equal to I. It follows now that the subspace F spanned by the union of all negative Jordan
chains is a Pontryagin space. The subspace H spanned by the remaining Jordan chains is a
Hilbert space. Since Jordan chains are complete it follows that K = F @ H and therefore
F has index k. Since the linear span of each chain in F always contains a one-dimensional
nonpositive subspace, it follows that the restriction A to F has at most k Jordan blocks. It
also follows by the above discussion that k _< dim F

Bhattacharyya, K6sir 381
PROOF. By [6, Thin. 1.3.3] it follows that in an appropriate basis for F1 we have
AI[F~ =
Ji 0 ... 0 0
o 4... o o
: '.. : :
o o ... 3~ o
0 0 .-. 0 0
where Jj are nilpotent Jordan blocks, and the inner product is given by the matrix
where
Pj=
P1 0 .-- 0 0
o .G... o o
: -.. : : ,
0 0 ... P~ 0
0 0 ... 0 I
0...01
0...10
:
1 ... 0 0
Note that AIIH1 ---- 0. By analogy, if we replace A1 by A2, there are a finite-dimensional
Pontryagin space F2 of index k and a Hilbert space //2 such that both /'2 and //2 are
invariant for A2 and K = F2 | Next we denote by E the subspace of K spanned by
/;1 and F2. It follows that E is a finite-dimensional subspace for A. It is invariant for Aj,
j = 1,2, since for v e E there exist fj e Fj and hje Hj such that v = fj + hj and
Ajv --= Ajfj C Fj. Observe that there is a
and G C //1 is a Hilbert space. Since A2
form
Bll
B21
A21E = :
Btl
u1
complement G of E in K such that K = G | E
commutes with A1 the restriction A21E is of the
U12
B22
Bl2
U2
9 " Bll W1
9 " B21 1472
: :
9 -- Bu W,
9 .. U~ C
where each block Bij is an upper-triangular Toeplitz matrix, Wj = , Uj = [ 0 ul ]
and wj, uj are column vectors (see [7, Thm. 9.1.1]). Blocks Bij, i,j = 1, 2,..., l, correspond
to the negative Jordan chains of A1. Since A2 is self-adjoint in K it follows that B*jPi =
PjBji, uj = iwj, j = 1, 2,...,1 and C = C [ Note that C is an operator on a Hilbert
space, therefore C = C*. Then we can assume that C is a diagonal matrix without changing
the structure of other blocks of A2. Next let E1 be the linear span of the set
{[0] [0] [0]}
Ul ~ U2 ' " " " ' ~l

382 Bhattacharyya, K6sir
and F the linear span of FI and U. It is clear that dim U _< l and thus dim F _< dim F1 + I.
Since C is diagonal there is a complement E2 of F in E that is spanned by eigenvectors of
A2. With respect to the decomposition E = Fi (9 U @ E2 the matrix for A2[E is of the form
A2[E= U C1 0 .
0 0 C2
But A2 is nilpotent and C2 diagonal, hence it follows that 6"2 = 0. Furthermore, F is
invariant for A and since FI C F it follows that F is a Pontryagin space of index k. Observe
that we can now choose a complement E~ of F in E so that E~ C HI. Finaly, we conclude
that H = G @ E~ is a Hilbert space such that H is invariant for A and H C H1. []
THEOREM 2.6. Ira C ]R '~ is an eigenvalue of A then there exist a finite-dimensi-
onal subspace Fa and a Hilbert space Ha such that both Fa and Ha are invariant for A,
Ma(A) = Fa | Ha and ~(Ma(A)) ~ dimFa _< (n + 2) ~ (Ma(A)).
PROOF. We prove the theorem by induction on n. For brevity we write Ma =
Ma(A) and ka = ~ (Ma). Assume first that n = 1. Applying Lemma 2.4 to AlMa, we
get a finite-dimensional Pontryagin space F1 of index ka and a Hilbert space H1 satisfying
Ma = F1 | H1, both F1 and Hi are invariant for A1 and ka

Bhattacharyya, K6sir 3 8 3
PROOF. Since Mo(A) is nondegenerate it follows that K = @aMa(A), where the
direct sum is over all eigenvalues of A with nonnegative imaginary parts. There are at
most k eigenvalues with a negative Jordan chain since a maximal nonpositive subspace has
dimension equal to k. The theorem then follows by Theorems 2.3 and 2.6, and remarks on
indecomposable subspaces for A preceeding Lemmas 2.1 and 2.4. []
The bounds in Theorem 2.7 coincide for n = 1 with those in [3]. For n = 2 observe
that A = A1 + iA2 is a normal operator on K. The bounds in Theorem 2.7 then coincide
with those given for a normal operator in [9, Thin. 1]. A normal operator A on a Pontryagin
I(A -t- n[*]) and A2 1 (A- A[*]) of commuting
space can be considered as a pair A1 = 7 =
self-adjoint operators. It is obvious that pairs A, A[*] and A1, A2 have the same joint invariant
subspaces. Moreover, a subspace is invariant for the pair A, A[*] if and only if it is a sum
of indecomposable invariant subspaces for A (see [8, 9]). Compare also the case n = 2 in
Theorem 3.1 below and [8, Thm. 1].
EXAMPLE 2.8. The bounds (n+2)k~ and (n+2)k in Theorems 2.6 and 2. 7 cannot,
in general, be improved. The proof of Lemma 2.5 suggests how to find ezampIes where the
bound is achieved. It is clear that the bound n + 2 for the dimension of F is attained if and
only if there is no non-real eigenvalues. For example, if k = 1 and n = 3 then the matrices
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0
0
0
' 0
0
0
00100
0 0 0 0 0
00000
01000
0 0 0 0 0
00000
0
0
0
' 0
0
0
00010
00000
00000
00000
01000
00000
commute and are self-adjoint with respect to the inner product [u, v] = (Pu, v), where (u, v)
is the standard scalar product in C 6 and
P =
0 0 1 0 0 0
0 1 0 0 0 0
1 0 0 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Note that the linear span of the first 5 coordinate vectors is the minimal invariant subspace
of the matrices that contains all their negative Jordan chains. Observe that 5 = (n + 2)k. []
3 Reduced form for commuting compact self-adjoint
operators on a Pontryagin space of index 1
If K is a Potryagin space of index 1 and A is an n-tuple of commuting compact self-adjoint
operators on K such that Mo(A) is nondegenerate, then Theorem 2.7 gives the existence

384 Bhattacharyya, KSsir
of a finite-dimensional Pontryagin subspace F of index 1 and a Hilbert space H satisfying
K = F | H, both F and H are invariant for A and 1 _< dim F < n + 2. Assume that F is
a minimal subspace with the required properties.
The restrictions AylH are compact commuting self-adjoint operators on a Hilbert
space and thus by the spectral theorem, they can be simultaneously diagonalised. We are
interested in structure of restrictions Aj IF. In the following theorem, (-, .) denotes a definite
inner product.
THEOREM 3.1. Suppose that A and F are as above. Then the spectrum of the
restrictions of A to F contains a single real eigenvalue or a pair of complex conjugate eigenvalues.
Assume that a = (al,as,...,a~) is an eigenvalue. Then one and only one of the
following is true:
1. If a q~ IR ~ then
AjIF=[ a~O -~jO 1
product on F ~- C 2 is given by [u, v] = (Pu, v}, where P = [ 0
and the inner
k 1
2. If c a ~ and dim. = 1 then AjlF = [ ]
[u,u] = -lul 2 foru e r-~ C.
3. If a e ]R ~ and f = dim F > 2 then
I a;
AjlF = 0 ajl
0 0
1 0]
(1)
and the inner product is given by
xj]
where I is the identity matrix of order
[00
f - 2, aj
]
C C f-2, {xj,a*aj} C ]P~ for i,j =
1,2,... ,n and al,as,... ,a,~ are linearly independent. The inner product on F -~ C f
is given by [u,v]= (Pu, v), where P= 0 I
100
0 .
PROOF. Note that the linear span of each negative Jordan chain contains at least
one nonpositive subspace of dimension one. It follows by minimality of F that the spectrum
of the restrictions of A to F contains a single real eigenvalue a or a pair of complex conjugate
eigenvalues a, ~. If a ~ ]R n then one of its components is nonreal, without loss we may
assume that al ~ ]1%. Since F has index 1 it follows that
andtheinnerpr~176 [ O1 01]' SinceAj
commute it follows that they are all diagonal, and thus of the form (1) (where some of c~j
may be real).
c~j

Bhattacharyya, KSsir 385
Next assume that a E ]R ~. If dim F = 1 then we obtain case 2. So suppose that
f = dimF > 2. By the minimality of F it follows that at least one of the operators Aj]F is
not diagonalizable. Without loss we may assume that A1 is such. Then there are nonzero
vectors vo, vl such that Ajvo = ajv0, j = 1, 2,..., n and Alvl = alVl + Vo. Then it follows
that [v0, v0] = 0. Since F is nondegenerate there exists a vector u C F such that [u, v0] ~ 0.
Then w -- ~-~-o4u- 1 ~ ru,u] V o is such that [w, v0] -- 1 and [w, w, ] = 0. Now let V = s w) [
Here and later in the proof the orthogonal complement is taken in F. We write W = s
and V~ = s for i = 0, 1. It is easy to verify that V0 [ = V0 @ V and F = V0 @ V | W. We
want to show that V0 @ V is an invariant subspace for all Aj. To do so choose z C V0 @ V.
Then [Ajz, vo] = [z, djvo] = aj[Z, Vo] = 0 and thus Ajz e V [ = Vo 9 Y. Then it follows
that with respect to the decomposition F = Vo @ V @ W we have
Ajlr = 0 B~ aj , j= 1,2,...,n,
0 0 aj
and that the inner product on F ~ C f is given
[001]
by [y, z] = {Py, z), where
P= 0 Q 0
1 0 0
and Q is a positive definite matrix. Since Aj commute and are self-adjoint it follows that
By are commuting linear maps on a Hilbert space. Thus Bj = ajI and we can assume
that Q = I. The conditions bj = aj and xj, a~ai E ]R hold because Aj are commuting and
self-adjoint. []
In the paragraph preceding Example 2.8 we explained how the case n = 2 is related
to a single normal operator on a Pontryagin space. Then an improvement of Theorem 3.1
for n = 2 can be deduced from the canonical form for a normal operator on a Pontryagin
space of index 1 given by Gohberg and Reichstein [8, Thm. I].
ACKNOWLEDGEMENTS
The authors wish to thank Prof. Paul Binding for suggesting the topic studied in
the paper. Research was supported in part by the Ministry of Science and Technology of
Slovenia and the National Board of Higher Mathematics, India.
References
[1] T. Ya. Azizov and I.S. Iokhvidov. Linear Operators in @aces with an Indefinite Metric,
John Wiley & Sons, 1989.
[2] P.A. Binding and K. Seddighi. Elliptic Multiparameter Eigenvalue Problems. Proc.
Edinburgh Math. Soc. 30 1987, 215-228.

386 Bhattacharyya, KSsir
[3] P.A. Binding and K. Seddighi. On Root Vectors of Self-Adjoint Pencils, J. Funct. Anal.
70 (1987), 117-125.
[4] J. Bogns Indefinite Inner Product Spaces, Springer-Verlag, 1974.
[5] M. Faierman. Two-parameter Eigenvalue Problems in Ordinary Differential Equations,
volume 205 of Pitman Research Notes in Mathematics, Longman Scientific and Techni-
cal, 1991.
[6] I. Gohberg, P. Lancaster, and L. Rodman. Matrices and Indefinite Scalar Products,
Birkh~user, 1983.
[7] I. Gohberg, P. Lancaster, and L. Rodman. Invariant Subspaces of Matrices with Appli-
cations. Wiley-Interscience, 1986.
[8] I. Gohberg and B. Reichstein. On Classification of Normal Matrices in an Indefinite
Scalar Product, Int. Equat. Oper. Theory 13 (1990), 364-394.
[9] O.V. Holtz. On Indecomposable Normal Matrices in Spaces with Indefinite Scalar Prod-
uct. Lin. Alg. Appl. 259 (1997), 155-168.
[10] F. Riesz and B. Sz-Nagy. Functional Analysis. Dover Publ., 1990, (a reprint of the 1955
original published by Frederick Ungar).
T. Bhattacharyya 1
Poorna Prajna Institute for Scientific Research
4 Sadshivnagar
Bangalore 560080
India.
e-marl: tirtha@math.iisc.ernet.in
and
T. Ko~ir
Department of Mathematics
University of Ljubljana
Jadranska 19
1000 Ljubljana
Slovenia.
e-maih tomaz.kosir@fms uni-lj.si
AMS Classification: 47B50
Submitted: November 30, 1999
Revised: May 3, 2000
1 Present address: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India.

Integr. equ. oper. theory 39 (200I) 387-395
0378-620X/01/040387-9 $1.50+0.20/0
9 Birkl~user Verlag, Basel, 2001
Integral Equations
and Operator Theory
APPROXIMATION OF APPROXIMATION NUMBERS
BY TRUNCATION
A. BSttcher, A.V. Chithra, M.N.N. Namboodiri
Let A be a bounded linear operator on some infinite-dimensional separable Hilbert space H
and let An be the orthogonal compression of A to the span of the first n elements of an
orthonormal basis of H. We show that, for each k _> 1, the approximation numbers sk(A~)
converge to the corresponding approximation number sk(A) as n --+ oo. This observation
implies almost at once some well known results on the spectral approximation of bounded
selfadjoint operators. For example, it allows us to identify the limits of all upper and lower
eigenvalues of Am in the case where A is selfadjoint. These limits give us all points of the
spectrum of a selfadjoint operator which lie outside the convex hull of the essential spectrum.
Moreover, it follows that the spectrum of a selfadjoint operator A with a connected essential
spectrum can be completely recovered from the eigenvalues of An as n goes to infinity.
1. Approximation numbers
Let H be a separable complex Hilbert space and let B(H) stand for the C*-algebra of all
bounded linear operators on H. For A C B(H), the kth approximation number sk(A) is
defined by
s~(A) := inf{llA-FI] : F c 13(H),rankF _ s2(A) > .... One can show (see [10, p. 212]) that
where the essential norm ]lAIless is given by
s~(A) --+ ltAIle~s as k --~ e~, (1)
IIAI]es~ := inf{I]d - KI]: K e ]C(H)},
~(H) denoting the ideal of the compact operators in B(H).
Let {ej}~= 1 be any othonormal basis of H, denote by Pn the orthogonal projection of
H onto span {el,..., e~}, and put A~ := P~AP~. We think of A~ as a truncation of A and
identify An C B(H) with an n x n matrix, that is, with an operator on C ~. It is easily seen
that
sk(A~) := inf{I]P~AP~ - F[[ : F e B(H),rankF < k- 1}
= inf{ltA~ - F~It : F, e/3(C~),rankF~

388 B6ttcher, Chithra, Namboodiri
Here is the key result of this note.
Theorem 1.1 If A E B(H) then, for each k > 1,
lim s~(A~) = sk(A).
n --~ O0
Theorem 1.1 is trivial if k -- 1 or if A E/C(H). It was established by one of the authors
and S. Roch in the case where ]lAIless = ]IAII, which is in particular true if A is a Toeplitz
operator (see [2], [3, Theorem 5.6], [5, Theorem 4.13]); notice that in this case s~(A) = Ilmll
for all k, so that Theorem 1.1 amounts to saying that
limsk(A~) = [IAI[ for each k > 1.
The following proof is a modification of the argument of [2]. We first need an auxiliary
result. The inner product in H will be denoted by (., .).
_ E ~o uniformly bounded sequence of
Lemma 1.2. Fix an integer k > 1 and let { ~}n=l be a
operators F~ E B(H) such that rank Fn < k - 1 for all n. Then there exists an operator
F C B(H) with rankF < k - 1 which enjoys the following property: for each x, y C H, the
number (y, Fx) is a partial limit of the sequence {(y, Fnx) }n~=p
For a proof see [3, Lemma 5.7] or [5, Lemma 4.12].
Proof of Theorem 1.1. Put d = sk(A) and dn = sk(An). Given e > 0, there is an
F C B(H) such that rankF _< k - 1 and IIA - Fll < d+e. Since
[[A. - P~FP~II = IIP~AP~ - P~FP~]I ~ I[A - Fll < d + c and rankP.FP~ 0 can be chosen as small as desired, we obtain
that
lim sup dn< d. (2)
From (2) we get the assertion in the case d = 0. So suppose d > 0. Contrary to what we
want, we assume that there is a e E (0, d) such that dnj _ c for infinitely many nj. Then,
given any e E (0, d - e), we can find Fn~ E B(H) such that rank F,~ < k - 1 and
IIP,~jAPnj - Fnjll < c + z < d. (3)
Lemma 1.2 implies the existence of an operator F C/3(H) of rank at most k - 1 such that,
for each x, y E H, the number (y, Fx) is a partial limit of the sequence {(y, F~jx)}~= 1. Now
let Ilxll = IlYll = 1. Then
I(y,P~jAP~jx) - (y, Fnjx)l < IIPnjAP w -F~jl ] < c + e,
and the afore-mentioned property of F yields the estimate
I(Y, Ax) - (y, Fx) l

B6ttcher, Chithra, Namboodiri 389
Hence NA - F N < c + s and thus d < c + e, which is impossible by virtue of (3).
contradiction proves that
liminfd~ > d. 9
n--~oo
2. Singular values
For A c B(H), the operator IAI c B(H) is defined as IAI := (A'A) 1/2. Suppose first that A
is a matrix, A E B(C~). The eigenvalues of IAI are then referred to as the singular values
of A. It is well known that if
~I(IAI) > A2(IAI) >... > ~(IAI)
are the eigenvalues of IAI in nonincreasing order, then
This
Ak(IAI) = s~(A) for k -- 1,...,n. (4)
In the infinite-dimensional case, things are as follows. Let sp A and spess A stand for the
spectrum and the essential spectrum of A E B(H), respectively:
spA :-- {A c C : A - )~I is not invertible in B(H)},
spes~ A := {A C C : A- M + IC(H) is not invertible in B(H)/]C(H)}.
The following theorem relates approximation numbers to singular values and thus to pure
spectral characteristics.
Theorem 2.1 (Gohberg, Goldberg, Kaashoek). The set sp IAI \ [0, IIA]less] is at most
countable, IIAlless is the only possible accumulation point, and all points of the set are
eigenvalues of finite (algebraic) multiplicity of IA]. Let
)h(]A[) _> A2(IAI) _>... (5)
be these eigenvalues in nonincreasin9 order (multiplicities taken into account) and let N E
{0, 1, 2,...} U {oo} be the number of terms in (5). Then
sk(A) = { ~k(]AI) if g = co or 1 < k < N,
IIAIless /fNN+l.
For a proof see [10, pp. 2o4 and 212-214] and note that II IAI I[e,~ = Ildll~- We also
remark that an amusing consequence of Theorem 2.1 is that the approximation numbers
enjoy the C*-algebra property: sk(A*A) = (sk(A)) 2 for every k >_ 1.
Combining (4) with Theorems 1.1 and 2.1 we arrive at the following result.
Corollary 2.2. Let A E B(H) and let Ak(]AI) be given by (5). Then
{ A~(IA]) /fg=c~ or l

390 BSttcher, Chitin'a, Namboodiri
3. Spectral approximation for selfadjoint operators
The approximation of isolated eigenvalues of finite (algebraic) multiplicity has been studied
for a long time. The first general results on this topic we are aware of are due to Gohberg and
Krein [11], [12], who considered compact (but not necessarily selfadjoint) operators. Osborn
[13] also established several pioneering results in this direction, including error estimates.
Things are more complicated for non-compact operators, and Chatelin's monograph [6] is
a comprehensive and authoritative source of this field.
In the case of selfadjoint operators, it is well known that the part of the spectrum outside
the convex hull of the essential spectrum can be found by spectral approximation; see the
books by Chatelin [6], Davies [7], and Greenlee [8]. The problem of locating eigenvalues
in gaps of the essential spectrum is much more delicate; sample papers devoted to this
question are [9] (abstract context), [14], [15] (differential operators), [16] (banded block
Toeplitz operators). The main results of this and the next sections (Theorems 3.1 and 4.1)
are folklore. We cite these results here because we have not found them explicitly in the
literature and because they can astonishingly easily be deduced from Corollorary 2.2.
Thus, let now A c B(H) be selfadjoint, A = A*. We put
m:= inf x), M:= sup(Ax, x),
Ilxll=l (Ax' ilxll=l
and we define tJ and # as the minimum and the maximum of SPess A, respectively. Thus,
conv sPe~s A = [zJ, p], where cony stands for convex hull. Clearly, m < z~ < p _< M, and we
have the inclusions
{~,#}Cspe~ sAC[~,#], {m,M}CspAc[m,M].
Since (Ax, x) > m(x, x) for all x E H, we see that A - mI = [A - mI]. Applying Theorem
2.1 to the operator A - m[, we arrive at the conclusion that the set spa M (#,M] is at
most countable, that it consists of isolated points only, and that each point of this set is an
eigenvalue of A with finite (algebraic) multiplicity. We denote these eigenvalues by
A+(A) _... _> A~(An)

BSttcher, Chithra, Namboodiri 391
Theorem 3.1. Let A C B(H) be selfadjoint. Then
In particular,
and
lim Ak(An)={ A+(A) if R=co or l u> limA~+l_k(An) for all k > l
7~--~ oo -- n--+oo
lim lim Ak(A~) = #, lim lim An+l-k(An) = P.
k--+OO n--+oo k-+oo n-+oo
Proof. We apply Corollary 2.2 to A - mI = ]A - rnI[. The operator P~(A - mI)Pn
can be identified with the matrix An - mIn, where In is the n x n identity matrix. As
(A=x,z) >_ m(x,x) for all x c C ~, it follows that A~ - mI~ = [A~ - rain[. Thus, by
Corollary 2.2,
- - lim Ak(A n mIn) = ~ A+(A - mI) if R = co or 1 < k < _R,
~-+oo [ IlA-mIHes s if RR+l,
Because Ak(A~ - mIn) = Ak(An) - m, A+(A - mI) = A+(A) - m, I[d- tailless = # - m,
we get the desired result for Ak(A~). Similarly, considering MI~ - An and using that
Ak(MIn -An) = M- A~+I-k(A~), we arrive at the result for An+l-k(An). The assertion
on the iterated limits follows from (1) and Corollary 2.2. 9
This theorem in conjunction with (6) and (7) shows that the part of sp A which lies
outside the convex hull of SPess A can be recovered from the upper and lower eigenvalues of
A n as rt --+ 0

392 BSttcher, Chithra, Namboodiri
Example 3.2. Let an be the nth Fourier coefficient of the characteristic function of the
upper half of the complex unit circle T,
and put
1 . 1-(-1) n f 1/2 if n=0,
aN = -~ fo e-*ne dO - - ~ 1/(~rin) if n is odd,
2~rin 0 if n is even and nonzero,
A=
al
a-1
a2
a-2
a3
a_ 3
a-1 al a-2 a2 a-3
ao a2 a-1 a3 a-2
a-2 ao a-3 al a-4
ax a3 ao a4 a-1
a-3 a-1 a-4 ao a-5
a2 a4 al a5 ao
a-4 a-2 a-5 a-1 a-6
a3
a4 ...
a2 ...
a5 .,,
al ...
a6 ...
ao ...
One can show that A induces a bounded selfadjoint operator on l 2 for which
sp A = spes S A = {0, 1},
liminf spAn = timsup spAn = [0, 1]. 9
(oh) are + b 2.
Example 3.3. The two eigenvalues of the matrix B(a,b) := b -a
2 2
Choose any sequence {an}n~=l of numbers an G (0, 1) and define bn 9 (0, 1) by a n + b n = 1.
Put
d = diag (B(al, bl), B(a2, b2), B(aa, ba),...).
Then sp A = SPes s A = {-1, 1}, and since sp A2,~ = {-1, 1} and sp Aem-1 = {-1, am, 1},
we see that liminf spAn = {-1, 1}, whereas lira sup spAn is the union of {-1, 1} and the
set of all partial limits of the sequence {an}. 9
The following corollary is an immediate consequence of Theorem 3.1 and (6), (7). It shows
that outside the convex hull of the essential spectrum the limiting sets coincide with the
spectrum.
Corollary 3.4. If A E B(H) is selfadjoint, then
lim inf sp An \ cony spess A = lira sup sp An \ cony spess A -- sp A \conv SPess A. 9
4. Selfadjoint operators with connected essential spectrum
As the following theorem reveals, spectral approximation works perfectly for operators
whose essential spectrum is connected.
Theorem 4.1. Let A E 13(H) be selfadjoint and suppose SPess A is connected. Then
lira inf sp An = lim sup sp An = sp A. (9)

BSttcher, Chithra, Namboodiri 393
Moreover, each point A E spe~ s A is essential, that is, for each e > 0 the number of points in
sp As N (A-s, A +e) goes to infinity as n -+ ec, and each point A E sp A \ SPe~ A is transient:
there exists an r > 0 and a natural number p such that the set sp An n (A- e, A + c) contains
at most p points for all sufficiently large n.
Proof. To prove (9), we are by virtue of (8) left with the inclusion lira sup spA~ C spA.
Pick A ~spA. Then A C [m,u) or A E (#,M]. For the sake of definiteness, suppose that
A E (#, M] and that, with the notation as in (6), A++I(A) < A < A+(A) for some r. Put
e = (1/3) min(A - Ar+I(A),A + r + (A) - A). as At(As) --+ A+(A) and A~+I(An) -+ A++I(A) due
to Theorem 3.1, there is an no such that At(As) > A + e and A~+I(As) < A - r for all
n >_ no. Consequently, sp An M (A - r A + e) -- 0 whenever n >_ no, which implies that A
does not belong to lim sup sp A~.
Arveson [1] proved that each point in SPes ~ A is essential, independently of whether
spe~A is connected or not. Finally, Theorem 3.1 shows that if A = A+(A) or A = A/(A),
then the number of points in sp AsN (A-e, A+e) does not exceed the (algebraic) multiplicity
of A for all sufficiently large n whenever e > 0 is small enough. 9
If A E B(H) is selfadjoint and invertible, then the norms IlA~Xl] need not stay uniformly
bounded as n -+ oo. However, this never happens for operators with connected essential
spectrum.
Corollary 4.2. ff A E B(H) is selfadjoint and spess A is connected, then
for every A E C \ sp A.
liin I](As - AI~)-III = II(A - AI)-IN
Proof. Since I](B - AI)-I][ = 1/dist (A, sp B) for selfadjoint operators B, this follows
easily from Theorems 3.1 and 4.1. 9
Here are a few concrete classes of operators with connected essential spectrum.
Example 4.3. For a real-valued function a E L~176 define the Fourier coefficients as as
in Example 3.2. The Toeplitz matrix
T(a) :=
a0 a-I a-2 ... "~
al
a2
ao
al
a-1
a0
...
J
9 9 9
induces a bounded and selfadjoint operator on 12. It is well known that
sp T(a) = SPess T(a) : [essinf a, esssup a]. 9
Example 4.4. Let a and b be piecewise continuous functions in L~(T) and let (as}~=l
b oo
and { s)n=l be their sequences of Fourier coefficients. If as = a_n for all n and bk = bk for

394 B6ttcher, Chithra, Namboodiri
all k _> 1, the the Toeplitz plus Hankel operator
a0 a-1 a-2 ...
~ j'rta~+H(b):= al ao a-1 ..-
a2 al a0 9 9 9
+
bl
b2
b~
b2 b3 ...)
b 3 .,.
is bounded and selfadjoint on the space 12. Fredholm criteria for T(a) + H(b) were established
by S. Power and B. Silbermann (see, e.g., [4, p. 209]). They imply that the essential
spectrum of T(a) + H(b) is connected. 9
Example 4.5. Let a : T 2 -+ R be a real-valued continuous function and let {a,~,n}m ~ .....
be its sequence of Fourier coefficients. The quarter-plane Toeplitz operator generated by a
is the bounded operator T++(a) on 12(Z+ Z+) defined by
(T++(a)x)i,j := ~ ai-k,j-zXk,l (i,j > 0),
k,l>_O
Simonenko as well as Douglas and Howe proved that
sp T++ (a) = 8pess T++ (a) : [min a, max a]
(see, e.g., [4, p. 353] or [5, p. 210]). 9
If A is any of the operators of Examples 4.3 to 4.5 and ifK is any compact and selfadjoint
operator, then Theorem 4.1 shows that sp (A+ K) can be found by spectral approximation.
References
[1] W. Arveson: C*-algebras in numerical linear algebra. J. Funct. Analysis 122 (1994),
333-360.
[2] A. BSttcher: On the approximation numbers of large Toeplitz matrices. Documenta
Mathematica 2 (1997), 1-29.
[3] A. BSttcher and S.M. Grudsky: Toeplitz Matrices, Asymptotic Linear Algebra, and
Functional Analysis. Hindustan Book Agency, New Delhi and Birkhs Verlag, Basel
2000.
[4] A. BSttcher and B. Silbermann: Analysis of Toeplitz Operators. Springer-Verlag, Berlin
1990.
[5] A. BSttcher and B. Silbermann: Introduction to Large Truncated Toeplitz Matrices.
Springer-Verlag, New York 1999.
[6] F. Chatelin: Spectral Approximation of Linear Operators. Academic Press, New York
and London 1983.
[7] E.B. Davies: Spectral Theory and Differential Operators. Cambridge University Press,
Cambridge 1995.

B6ttcher, Chithra, Namboodiri 395
IS]
[9]
[10]
[11]
[12]
W.M. Greenlee: Approzimation of Eigenvalues by Variational Methods. Rijksuniver-
siteit Utrecht, Mathematical Institute, Utrecht 1979.
W.M. Greenlee: A convergent variational method of eigenvalue approximation. Arch.
Rational Mach. Anal. 81 (1983), 279-287.
I. Gohberg, S. Goldberg, and M.A. Kaashoek: Classes of Linear Operators. Vol. I.
Birkhguser Verlag, Basel 1990.
I. Gohberg and M.G. Krein: The fundamentals on defect numbers, root numbers, and
indices of linear operators. Uspehi Matem. Nauk 12 (1957), 43-118 [Russian]; Engl.
transl.: Amer. Math. Soc. Transl. (2) 13 (1960), 185-264.
I. Gohberg and M.G. Krein: Introduction to the Theory of Nonselfadjoint Operators in
Hilbert Space. Nauka, Moscow 1965 [Russian]; Engl. transl.: Amer. Math. Soc. Transl.
of Math. Monographs, Vol. 18, Providence, RI, 1969.
[13] J.E. Osborn: Spectral approximation for compact operators. Math. Comput. 29 (1975),
712-725.
[14]
[15]
[16]
G. Stolz and J. Weidmann: Approximation of isolated eigenvalues of ordinary differ-
ential operators. J. Reine Angew. Math. 445 (1993), 31-44.
G. Stolz and J. Weidmann: Approximation of isolated eigenvalues of general singular
ordinary differential operators. Results Math. 28 (1995), 345-358.
P. Zizler, K.F. Taylor, and S. Arimoto: The Courant-Fisher theorem and the spectrum
of selfadjoint block band Toeplitz operators. Integral Equations and Operator Theory
28 (1997), 245-250.
A. BSttcher
Fakult~t fiir Mathematik
TU Chemnitz
09107 Chemnitz
Germany
aboettchQmathematik.tu-chemnitz.de
MSC 2000: Primary 47 A 58
Secondary 15 A 18, 47 A 75, 47 B 06, 47 B 35
Submitted: April 30, 2000
M.N.N. Namboodiri and A.V. Chithra
Department of Mathematics
Cochin University of Science and Technology
Cochin 682 022, Kerala
India
nambu@cusat.ac.in

Integr. equ. oper. theory 39 (2001) 396-412
0378-620X/01/040396-17 $1.50+0.20/0
9 Birkh~user Verlag, Basel, 2001
J Integral Equations
and Operator Theory
ACCRETIVE PERTURBATIONS AND ERROR ESTIMATES FOR THE
TROTTER PRODUCT FORMULA*
VINCENT CACHIA, HAGEN NEIDHARDT, and VALENTIN A. ZAGREBNOV
We study the operator-norm error bound estimate for the exponential Trotter
product formula in the case of accretive perturbations. Let A be a semibounded
from below self-adjoint operator in a separable Hilbert space. Let B be a closed
maximal accretive operator such that, together with B*, they are Kato-small
with respect to A with relative bounds less than one. We show that in this
case the operator-norm error bound estimate for the exponential Trotter product
formula is the same as for the self-adjoint B [12]:
(e-tA/ne-tB/n)n e-t(A+B) < L lnn
- , n = 2, 3, ....
n
We verify that the operator -(A + B) generates a holomorphic contraction semi-
group. One gets similar results when B is substituted by B*.
1 Introduction and Setup of the Problem
Let -A and -B be generators of strongly continuous semigroups on a separable Hilbert space
Y). For many years the convergence of the Trotter product formula for those semigroups:
T - lira (e-tA/ne-tB/n) n = e -tH, (1.1)
n---~+oo
has been of interest only in the strong topology T -- s, see [3], [4], [18], where H is the sum of
the operators A and B defined in a suitable sense. The most general result in this direction
is due to Kato [9], [10], who has shown that the convergence in the strong topology always
takes place if the operators A and B are self-adjoint and non-negative, or more generally,
if they are m-sectorial (here and below we follow the terminology of the Kato's book [8]).
Moreover, he identified H as a form-sum of A and B and generalized the product formula
(1.1) to the case when the exponential functions are replaced by functions of a certain class,
which is often called the Kato-class. Then the corresponding formula is called the Trotter-
Kato product formula. However, certain applications of the Trotter or the more genera/
Trotter-Kato product formul require that
*To the memory of Toslo Kato

Cachia, Neidhardt, Zagrebnov 397
(a) the convergence in (I. I) would be in the operator-norm or in stronger norms topologies,
for instance in the trace- or in the Hilbert-Schmidt-norm,
(b) error estimates in these norms, in particular, in operator-norm, would be available,
(c) the convergence and the error estimates are valid not only for self-adjoint generators
N and B.
Until now the problems (a) and (b) have been solved in a satisfactory manner only
for non-negative self-adjoint operators A and B. In papers [11]-[15] the conditions (partially
necessary and sufficient) are found for convergence of the Trotter-Kato product formula in
the operator-norm or in norms of symmetrically-normed ideals of compact operators, see
also [5]. The operator-norm error bound estimates are given in [6], [12], [13] and [17], and
for the trace-norm in [7] and [15]. On the other hand, there are only few results concerning
the problem (c). The first result in this direction was due to Sophus Lie who has shown
the convergence of the exponential product formula (I.I) for arbitrary matrices A and B.
In fact, his proof provides the operator-norm error bound estimate for a couple of~ounded
operators A and B on a complete normed space:
~ - e-"ll-< --, C n= 1,2,..., (1.2)
n
where H = A+B, see [16]. So, the Lie's result solves problems (a) - (c) for bounded operators
on a Banach space. In [1] the problem (a) is solved in the operator-norm topology 7 = H' II
for a class of unbounded m-sectorial operators A and B with semi-angles OA, OB E (0, ~/2)
in a Hilbert space. A solution of the problem (b) in a Banach space is presented in [2] for
particular domain conditions. Namely, let -A be generator of a holomorphic contraction
semigroup and let -B be generator of a contraction semigroup satisfying the conditions:
dora(B) _~ dom(A ~ for some ~ e [0, 1) and dora(B*) 2 dora(A*). Then there is a constant
Ca such that
- < c nl_~ , n=2,3,..., (1.3)
unifor y in t [0,T], 0 < T < where H = A + B on dora(H) = dora(A), and
s(a = 0) = 2, s(a ~ 0) = 1. The condition dora(B) _~ dom(A ~) implies that B is
relatively bounded with respect to A with the relative bound a = 0. Thus, compared to the
corresponding results for the self-adjoint case in a gilbert space, see [12] and [13], the result
(1.3) is not optimal. For instance in [12] it is proven that if A _~ I, B ~__ 0 and B is relatively
bounded with respect to A with a < 1, then there is a constant C such that
(e-tA/"e-tB/")" -- e -tH < cln(nn) , n = 2,3,..., (1.4)
uniformly in t _> 0, where H = A + B on dora(H) = dom(A).
In what follows we suppose that A = A* is a semibounded from below densely de-
fined self-adjoint operator in a Hilbert space -9. It generates a semigroup {UA(t) = e-tA}t>O.
Without loss of generality one can suppose that A _> I. Then it is known (see, e.g. [8]) that

398 Cachia, Neidhardt, Zagrebnov
UA(t) is a contraction holomorphic semigroup of angle w = ~r/2, i.e. II~A(t ~ 0)ll ~ 1
and UA(t) has a holomorphic extension from t >__ 0 into the open complex half-plane:
s~=~/~ = {z e c: z r 0 a~d larg(z)l < ~/2}.
About the generator B we suppose that :
(i) domain dom(B) _D dom(A) and there exists a > 0 such that
IIB~II _< ,~ItA~II, ~ c dom(A);
(ii) domain dom(B*) _D dom(A) and there exists a, > 0 such that
[lB*ull 0
one has
tI(B + ~)-111 _ X -1. (1.7)
The latter implies that {UB(t) = e-tB}t>o is a (nonself-adjoint) contraction semigroup, but
B is not necessarily sectoriah 0B may be ~r/2.
Notice that under conditions (i)-(iii) the convergence of the Trotter product formula
(1.1) is known ([3], [4], [18]) only in the strong topology T = s. In our main Theorem (see
Section 3) we prove that under conditions (i)-(iiQ and a, a, < 1 there is a constant L > 0
such that uniformly in t > 0 one gets (cf (1.4))
(1.5)
(1.~)
(e-tA/'~e-tB/~)~ -- e-tH

Cachia, Neidhardt, Zagrebnov 399
It turns out that this is the case if and only if the spectrum or(H) of H is contained
in the closed sector S,~/2-~o, i.e. a(H) c_ S,q=_~,, and for each 0 E [0, cJ) there is a constant
Mo > 0 such that
for z E S~/2+o, see e.g. [8, Ch.IX].
II(H + z)_lll _< Mo (2.1)
Theorem 2.1 Let A > I and let B be a closed accretive operator satisfying condition (i)
with a < 1. Then the operator -H = -(A + B) is boundedly invertible generator of a
contraction holomorphic semigroup of angle w = arccosa, in the sense that lie-trill < 1 for
t>O.
Pro@ Our goal is to verify (2.1) for z E S~/2+0, 0 C (0,w), where w = arceosa < ~r/2. To
that end we use the representation
Since
(H + z)f = {I + BA-1A(A + z)-~}(A + z)f, f E dora(H) = dora(A). (2.2)
a straightforward computation shows that
Let z C S~/2+o and Re z < 0, then
Hence (2.4) leads to the estimate
I]A(A + z)-ll[ _< sup>,_>1 ~-7+z -< supA_>_o ~+z
1 forRez_>0
HA(A + z)-l[] -< lzl/l-,~mzt for ~ez < 0, ~mz # 0
(2.3)
(2.4)
[zl 1
I~mzl -< cosO" (2.5)
IIA(A + z)-Zll < ~ (2.6)
- cos 0
for z E S~12+o and ~ez < 0. If Nez > 0, then (2.4) yields the estimate
Therefore the estimate (2.6) holds for any z E S,~/2+o.
Using the condition (i) we find the estimate
1
IIA(A + z)-lll _< 1 _< cos-----0' (2.7)
[[BA_IA(A + z)_,H < ItBA_XIIItA(A + z)_Xl I < a < 1 (2.8)
- - cos 0
for z E S~/2+0. Hence, the operator I + B(A + z) -1 is invertible for any z E S~/2+o and we
have the estimate
cos 0
I1(~ + B(A + z)-1)-111 _< -cos0 - - ~
(2.9)

400 Cachia, Neidhardt, Zagrebnov
Then from (2.2) and (2.9) we obtain that z e S~/2+o C p(-H) (the resolvent set of the
operator -H), the representation
and the estimate
Since A is self-adjoint, we have
(H -}- z) -1 = (A + z)-l(z + B(A + z)-l) -1, (2.1o)
cos
I[(H + z)--ll[ < --II(A + z)-Xll 9
- cos 0 - a
1 1
H(z + A)-xlI < cos---O Iz---/
for any z r S~/2+e. Thus by (2.11), (2.12) we find
for all z E S~/2+e with
(2.11)
(2.12)
II(H + ~)-lIf < M~ (2.1a)
-Izl
1
- - (2.14)
Me := cos 0 - a"
This shows that -H is the generator of a holomorphic semigroup with angle ~ = arccos a.
Since B is accretive and A is self-adjoint positive, the operator H = A + B is
accretive. Since A+ H is boundedly invertible for A > 0, H is m-accretive, i.e., -H generates
a contraction semigroup. Finally, by the representation H = (I + BA-I)A with IIBA-~[I <
a < 1, the operator H is boundedly invertible.
[]
Remark: Since ~(H) c S~/2-~ implies ~(H*) C S~/2-~, the characterisation (2.1) entails
that -H* is also the generator of a holomorphic semigroup.
Below we denote by D = {z C C: Iz] < 1} and D1/2 = {z C C: ]z- 1/2] < 1/2},
two discs in the complex plane C.
Lemma 2.2 Let A > I. Then for any r > 0
{ zED\D;/ (2.15)
11 z-e-~] - ~>0
Letz=x+iyand
1 - e -x 1 - e -~
m(~) '- Lz - e-~--------~ - ~/(x - e-~)2 + y2' ~ > 0. (2.17)

Cachia, Neidhardt, Zagrebnov 401
If z E II3 \ D1/2, then m'(A) _> 0 for ~ >_ 0. Therefore re(A) is a non-decreasing function such
that
1
supra(A) = lira re(A)=- (2.18)
~_>o ~+~ Izl'
which proves the first estimate (2.15). If z E D~/2 \ [0, 1], then re(A) attains its maximum at
A0 such that
Inserting (2.19)into (2.17)we obtain
which yields
e -~~ ~- (2.19)
1-x
~(Ao) _ I1 - zl (2.20)
I~mzl'
Ii-zl
0 _< re(A) < i~mz----~, (2.21)
The inequality (2.21) verifies the second estimate in (2.15). []
Since for A > 7 and an m-accretive B the operator
T(T) = e-~-Ae -rB, T > O, (2.22)
is a contraction, its spectrum ~(T(~-)) _ ]D, ~- > 0. However, for t3 small relative to A we
can show more. For that we introduce the family of closed convex sets in C, for r E [0, 1],
defined by
E~ := E ~ U E~, (2.23)
where
and
E ~ := {z E Dl/2: I-~mzl < rll - zl} (2.24)
E) := {z E ]D \ D1/2: Izl _< r}, (2.25)
see Figure 1. We remark that the family ET increases with r: for r < r' one gets Er C ET,.
On the other hand, we denote by R~ the decreasing family of domains ID \ Er.
Theorem 2.3 Let A >_ I and let B be an m-accretive operator satisfying the condition (ii)
with a, < 1. Then cr(T(T) ) C E~. for each ~- > O. Moreover, for any r E (a., 1), one has the
estimate
I](z - r(',-))-lll
< r-r. II(z - e-rA)-lll, z E ~ = D \ ~,.. (2.26)
Pro@ One has to prove that each point z E D \ Ea. = Ra. belongs to the resolvent set
p(T(~-)) of T(T) for any w > 0. To that end we use the representation
z - T(T) = z -- e -rA + e-'A(I -- e-rB). (2.27)

402 Cachia, Neidhardt, Zagrebnov
If r > O, then the operator (I - e -~-A) is invertible. Let
and
x(~,,) := (r _ ~-~)(z - e-~) -~
Y(r) :-- (I - e-rA)-le-*A(I -- e-*S).
Then by (2.27)-(2.29) one obviously gets the representation
For each f 6 dom(B) one has
Hence, we get
for z C/~ fq D1/2 and r > 0.
z - T(~) = (~ - e-'~){Z + X(% z)Y(~)}.
(I- e-~'B)f = fO T dsBe-*B f.
(2.28)
(2.29)
(2.30)
(2.31)
A-I(I - e-*B)f = ~0 ~i- dsA-1Be-*Bf. (2.32)
By condition (ii) the representation (2.32) leads to the estimate
Using for (2.29) the representation
we obtain
IIA-l(I - ~-%II -< ~a,. (2.33)
Y(r) = (Z - e-~A)-IAe-~AA-I(I -- e-~B), (2.34)
I]Y(~)II < ~a, ll( ~-~A)-IA~-~I]. (2.35)
By the functional calculus for self-adjoint operators one verifies that
rlt(I - e-~A)-IAe-~AI[ < 1 (2.36)
for each r > 0, and by consequence the estimates (2.35), (2.36) give
IIYO-)II-< a,. (2.3r)
Let r > a,. To estimate X(r, z) (2.28) we have to consider two cases:
(1) Let z E D1/: n R~. Then I~mz] > r]l - z I. Since z E D1/2, by Lemma 2.2 we
find that
I1 - z[
IIX(r,z)ll_ ~7.71"1' (2.38)
Hence one gets 1
IIX(r, z) ll < - (2.39)
T

Cachia, Neidhardt, Zagrebnov 403
(2) Let z E -Rr \ D1/2. Since ]z[ > r, by Lemma 2.2 we find
1 1
IlX( ,z)ll < < -
-- IX[ r"
(2.40)
Therefore we see that the estimate
1
IIX( ~, z)ll < -,
T
(2.41)
cf (2.39), (2.40), is valid for any z 9 P~ and ~- > 0. Moreover, by (2.37) and (2.41) we have
a.
Hx(% < IIx( , z)NflY( )EI < - < 1 (2.42)
71
for z e R~, f > 0. This yields that the operator {I + Y(T)X(T, z)} is invertible for z e
R~. = [.Jr>~./~ and ~- > 0, with the following estimate for z E R~, r > a.:
1
I[(I + X(%z)Y(T'z))-lll < 1 -a./r' T > 0. (2.43)
Since the operator (z - e -~A) is invertible for z E R~. and ~- > 0, we obtain from
the representation (2.30) that the operator z - T(7) is invertible for z 9 R~., T > 0, and the
announced estimate (2.26) is valid for z 9 R~, r 9 (a,, 1). []
Let us define the closed contour Fr = 0E~, which is the boundary of the set E~.
Outside the disc D1/2, it coincides with the arc of radius r with the centre at 0, whereas
inside D1/2, this curve consists of two segments of tangents to the preceding arc passing the
point 1 (see Figure 1).
Figure i: Illustration of the set Ea. (shaded domain) with boundary Pc. = cgE~., where
a, = since, as well as of our choice of the contour Pr in the resolvent set p(T(~-)), where
r = sinfl > a,. The contour Fr consists of the two segments of tangent straight lines (i, A)
and (1, B) and the are (A, B) of radius r. The dotted circle 0D1/2 corresponds to the set of
tangent points for the different values of r E [0, i].

404 Cachia, Neidhardt, Zagrebnov
Theorem 2.4 Let A >_ I and let B be an m-accretive operator obeying condition (ii) with
a. < 1. Then there is a constant C such that
holds for any ~- > O.
I]T(~-)k(I -
T(~-))II
C
_< k +---G' k = 0, 1, 2,..., (2.44)
Proof. By Theorem 2.3, one gets that for any r > a, and ~- > 0, the contour Fr lies in p(T(T)),
the resolvent set of T(~-). Using the Dunford-Taylor calculus we get the representation
1 ]~ zk(1 _ z)(z - T(T)k(I-T(T)) = ~ ,
We estimate the norm of the integral (2.45) with help of inequality (2.26):
- T(T))-ldz.
(2.45)
frzk(l_z)(z - T(~_))_ldz < l fr, Izk(1 -z)lrr--~* I'(z --e-~-A)-lHldzl" (2.46)
Since the integral in the right hand side of (2.46) is invariant with respect to con-
jugation z --+ ~, it is sufficient to estimate the integral on the path F+: the branch of the
contour F, with @m z > 0. This branch consists of the segment (1, A) and the arc @/2-/3, ~)
of radius r, see Figure 1.
Let us parametrize (1, A) by
z = 1 - se -i~, 0 < s < cos/3, with r = sin/3. (2.47)
Since for (2.47) and the self-adjoint e -~A one has II(z - e-~A)-lll

Cachia, Neidhardt, Zagrebnov 405
In the last inequality we use that: supk>0(k + 1)e -kln~ = x/(elnx) for x > 1.
Finally, combining (2.46), (2.50) and (2.51) we obtain:
2@~ fr zk(1-z)(z-T(7))-ldz < _
( eosfl 1/r "~ r 1
2 \27rsinfl(1- sinfl) + ecosflln(1/r)] r -a, k + 1
which gives the announced estimate (2.44) with
( rcosZ 1 h 2
C = \27r sinfl(1 - sinfl) + ecosflln(1/r) J r - a,"
(2.52)
(2.53)
Remark: Let T(~-) = e-~Ae -~B be replaced by F(T) = e-~Be-~A. Then Theorems 2.3 and
2.4 remain valid if B satisfies, instead of (ii), the condition (i) with a < 1. For that one
has to consider the adjoint family F(~-)* = e-~Ae -~-B*. Then the proofs of the estimates
corresponding to (2.26), (2.44) carry through verbatim by a simple replacement of a, by a.
3 Proof of the main Theorem
First we recall the following property of holomorphic semigroups (see e.g. [8, Ch.IX]):
Proposition 3.1 Let -H be the generator of a bounded holomorphic semigroup. Then
ran(e -~) C dom(H), and we have the estimate:
ilg _,HI] < __CH t > 0 (3.1)
-- t '
Lemma 3.2 Let A be a self-adjoint operator, A > I. Let operator B be m-accretive, sat-
isfying (ii) with a, < 1. Then operator I - T(~-) is boundedly invertible for each ~- > 0
and
[[( T(~-))-I~II-< 1 -- a,
Proof. We use the representation
I - T('r) = I - e -r + e-r -- e-r (3.3)
Since II~-~AII < C" < 1, the operator Z - e -~A is boundedly invertible provided ~- > 0.
Hence, one gets the representation
I - T(T) = (I -- e -~A) (I + Y(T)), (3.4)
where we have used the notation (2.29). By (2.37) and a, < 1 we obtain that I - T(T) is
also invertible. Moreover, we have the estimate :
[]

406 Cachia, Neidhardt, Zagrebnov
1 (I - e -~A)-lu u E -9 (3.5)
[[(I - T(T))-lu[t 0 that
(f - e-~A) -1 u _ 0, we find
m:I
T(T) ~ - U(T) ~ : T(T)~-I(T(T) -- U(T)) (3.12)
n--i
+ ~ T(~) .... ~(I - T(T))(I - T(T))-~ (T(T) -- U(T))U(T) ~.
m~l

Cachia, Neidhardt, Zagrebnov 407
This leads to the estimate
_ -
IIT(~-F -
n--1
u(,-)'~ll ~ IIT(~-)~-~(T(z ) - u(~-))ll
+ X: IIT(~-) .... 1(i T(~-))IIII(/- T(T))-I(T(T)
m=l
Then by (3.2) we obtain
Since
one gets
Hence
Similarly we get
[l(I - T(~-))-I(T(7) - U(,-))U(~-)mll
- g(~-))u(~-)'~ll
9
{ 1 IJ}
1 H(T(T) g(~-))g(~-)"ll+ IIA-I(T(~ -) U(~-))U(~-)" .
l-a, "r
(T(T) -- U(T) )U(T)" : (T(T) -- U(T) )A-1AH-1HU (T) m,
[I(T(~-) - u(~-))u(~-)mll
(3.13)
(3.14)
(3.15)
II(T(~-) - U(,-))U(~-)'~II ~ I[(T(~) - U(~-))A-11111AH-11111HU(~-)'~II. (3.16)
Applying the estimate (3.1) of Proposition 3.1 to the last factor in (3.16) we obtain
II(T(~_) _ U(~_))U(~_)mll _< II(T(T ) _ U(~_))A_I[ I IIAH_~IIG,~ 1. (3.17)
T m
By AH -1 : (I+BA-1) -1 and IIBA-1[I _< a < 1 one gets
_< I[(T(~-) - U('r))A-1]] (1 c~, ----a)T m" 1 (3.19)
IIA-I(T(~-) - U(~-))U(~-)'~II -< IIA-I(T(~-) -
Inserting (3.19) and (3.20) into (3.14) we find
U(~-))A-~II
CH 1
(1--a)~-7~' (3.20)
II(Z - T(T))-I(T(T) -- u(~-))u(~-)'~ll (3.21)
{1 1 } CH 1
a,)(1 - a) ,~
< II(T(~-) - U(~-))A-111 + ~ I[A-I(T(T) - u(~-))A-1[I (1 -
Applying to the estimate (3.21) Lemma 3.3 and Lemma 3.4 we find
1
IIAH-111-< 1- a'
(3.18)
L1 + L2 1 L3 (3.22)
Jl(Z - T(T))-I(T(T) -- U(~-))U(~-)mll

408 Cachia, Neidhardt, Zagrebnov
where we put L3 := CH(L1 + L2)/(1 - a,)(1 - a). Further, by Theorem 2.4 we have
C
iiT(,r ) .... 1(/_ T(,r))ll < --, n = 2,3,...,
n--'m,
Inserting (3.22) and (3.23) into (3.13) we get
Since
m = 0, 1,2,...,n- 1. (3.23)
1
IIT(,r)" - U(,r)'~]l _< IIT(,r)n-l(T@) - U(,r))]l + CL3 ~, (n _ re)m"
m=l
n--1
1
(~- ~)~
~1%:i
_ _2~_1

Cachia, Neidhardt, Zagrebnov 409
Inserting (3.31) into (3.26) we find the estimate (3.10):
- u( -)nll _< L ln(n)
--, n = 3,4,.... (3.32)
n
Here L := C(2 + L1)(1 - a,) -1 + 4CL3, and we use that 1In < ln(n)ln to estimate (3.31),
for n = 3, 4,... []
Corollary 3.6 Let A and B satisfy the conditions of Theorem 3.5. Then there are constants
L ~, L" such that the estimates
hold uniformly in t >_ O.
(e-tBlne-tA/~) ~ e -~H < L' in(n)
- --, n = 3, 4,..., (3.33)
n
(e_tA/2ne_tB/ne_tA/2n)n e_tH ~-- L, ! ln(n)
- --, n = 8,4,.... (8.84)
n
Proof. Since B* satisfies conditions of Theorem 3.5, we have
(e-tAl~e-tB'l") n -- e -tH* ~_ L' ln(n). (3.35)
n
Notice that H* = A + B* is well-defined and m-accretive. By taking the adjoint sequence
of operators, we find (3.33). In order to obtain (3.34), it is sufficient to note that the line of
reasoning in the proof of Theorem 3.5 follows thought verbatim for the symmetrized formula.
[]
4 Conclusion
The aim of the present paper was to show that the operator-norm error bound estimates for
the exponential Trotter product formula persist if for a couple A = A*, B we pass from a self-
adjoint A-small perturbation B to an m-accretive A-small operator B. This generalizes the
previous result [12] to the case of a nonself-adjoint B under conditions that IIBuH I, i.e. dora(A) C_ dora(V) and
Jlvfll ~ aJlAflI, f e dora(A), for some a e (0, 1). (4.1)
Then the operators B(z) = zV and B(z)* = -2V are obviously m-accretive, and they satisfy
the conditions (i)-(ii) for ~ez > 0. If, in addition, ]z I _< 1, then the constants a, a. for B(z)
remain smaller than 1. Applying Theorem 3.5 we find, for instance, that the error estimate
H(e-tA/~e-t~v/") ~ - e-tH(z) N

410 Cachia, Neidhardt, Zagrebnov
holds uniformly in t > 0 and in z such that Rez _> 0, Izl _< 1, where the operator H(z) =
A + zV is defined on dom(H(z)) = dom(A). In the particular case z = i, we are faced with
the perturbation of a self-adjoint contraction semigroup by- a unitary group.
Since in the main Theorem a, a* may be strictly positive, the present paper is also
a generalization of [2], where the estimate (1.3) was found for B that is infinitesimally
Kato-small with respect to A. Notice that in [2] it is not supposed that the operator A is
self-adjoint, but only that it generates a contraction holomorphic semigroup. In particular,
this is the case when A is an m-seetorial operator with vertex in 0, see [8, Ch.IX].
Therefore, it is relevant to ask: why do we need A = A* in the main Theorem of
the present paper ? First, this condition is important for the estimate of II (z- e-~A) -1 II, see
(2.48) and (2.51) in the proof of Theorem 2.4. Second, it makes possible to use the spectral
theorem to obtain the estimates (2.15), see Lemma 2.2, and (3.2), see Lemma 3.2.
Suppose, for example, that instead to be self-adjoint the operator A is m-sectorial.
Then operator-norm continuity of the holomorphic semigroup UA(t) = e -tA for t > 0, al-
lows to localize the spectrum of this semigroup a(e -tA) but not the numerical range @(e -tA) !
However, to estimate the norm of the resolvent (z-e-~A) ~1, the localization of the spectrum
is not sufficient, except for a normal (in particular self-adjoint) operator A, when O(e -tA)
is the convex hull of a(e-tA). Thus, in the present paper we have solved the problem only
for the couple of self-adjoint A and m-accretive B, i.e. for the limiting case of angles OA = 0
and OB = ~r/2.
Although the above mentioned result of [2] allows to pass to an m-sectorial operator
A with 0 < OA < ~r/2, the price is that B and B* should be infinitesimally A-small, if one
insists on the error bound estimate. If not, then [1] for two m-sectorial operators A and B (
OA, OB e (0, ~r/2) ) such that:
- - either (I + A) -1 or (I + A)-I(I + B) -1 is compact,
- - or A and B satisfy conditions
(i) NeA,~eB>~/>0;
(ii) dom(~e A) C_ dom(,~m A) ;
(iii) dom(~e B) C_ dom(~m B) ;
(iv) dom((~e A) ~) c_ dom((~m B) ~) for some a E (1/2, 1] ;
(v) II(~e B)%II _< cll(~e A)~ull, u e dom((~e A)~), 0 < e < 1 and some a C (1/2, 1],
one gets
t1.11 - 1]~oo(e-tAl~e-tSl~)~ = e-tH" (4.3)
The operator-norm convergence in (4.3) is uniform in t for any compact set of the sector
S~ = {z E C: z 7 ~ 0 and larg(z)l < w}, where w = ~r/2--max{OA, OB} and H is the form-sum

Cachia, Neidhardt, Zagrebnov 411
of A and B.
We conclude by remark that our main Theorem 3.5 is valid if, instead of e -tA/n, one
considers operator f(tA/n). Here f : [0, +oc)--+[0, 1] is a Borel function from the Kato-class
such that, cf.[12] :
as well as
Then we get
Co := sup/5~
~>0 t i - f(x) J
C1 :~- sup{1-f(x)}
< + ~ x
< +oo
: -- < +oo
(4.4)
(4.5)
(4.6)
(4.7)
0 < Coa < 1. (4.8)
(f(tA/n)e-tB/n) n -- e -t(A+B) 3 (4.9)
n
uniformly in t _> 0. The proof needs only minor modifications of Lemmata 3.2-3.4 as it is
outlined in [12].
Acknowledgements
H.N. would like to thank Deutsche Forschungsgemeinschaft for financial support and Centre
de Physique Th~orique-CNRS-Luminy for hospitality.
References
[1] Cachia, V.; Zagrebnov, V.A.: Operator-norm convergence of the Trotter product for-
mula for sectorial generators, to appear in Lett. Math. Phys. (2000).
[2] Cachia, V.; Zagrebnov, V. A.: Operator-norm convergence of the Trotter product for-
mula for holomorphic semigroups, to appear in J. Operator Theory (2000).
[3] Chernoff, P.R.: Note on product formulas for operator semigroups. J. Funet. Anal. 2
(196s), 238-24z
[4] Chernoff, P.R.: Product formulas, nonlinear semigroups and addition of unbounded
operators. Mere. Am. Math. Soc. 140 (1974), 1-121.
[5] HiM, F.: Trace norm convergence of exponential product formula Lett. Math. Phys. 33
(1995), 147-158.

412 Cachia, Neidhardt, Zagrebnov
[6] Ichinose, T.; Tamura, H.: Error estimate in operator norm for Trotter-Kato product
formula. Integr. Equ. Oper. Theory 27 (1997), 195-207.
[7] Ichinose, T.; Tamura, H.: Error bound in Trace norm for Trotter-Kato product formula
of Gibbs semigroups. Asymptotic Analysis 17 (1998), 239-266.
[8] Kato T.: Perturbation theory for linear operators, Springer Verlag, Berlin 1966.
[9] Kato, T: On the Trotter-Lie product formula. Proc. Japan Acad. 50 (1974), 694-698.
[10] Kato, T.: Trotter's product formula for an arbitrary pair of self-adjoint contraction
semigroups. Topics in Funct. Anal., Ad. Math. Suppl. Studies Vol. 3, 185-195 (I.Gohberg
and M.Kac eds.). Acad. Press, New York 1978.
[11] Neidhardt, H.; Zagrebnov V.A.: The Trotter product formula for Gibbs semigroups.
Comm. Math. Phys. 131 (1990), 333-346.
[12] Neidhardt, H.; Zagrebnov V.A.: On error estimates for the Trotter-Kato product for-
mula. Lett. Math. Phys. 44 (1998), 169-186.
[13] Neidhardt, H.; Zagrebnov V.A.: Fractional powers of self-adjoint operators and Trotter-
Kato product formula. Integr. Equ. Oper. Theory, 35 (1999), 209-231.
[14] Neidhardt, H.; Zagrebnov V.A.: Trotter-Kato product formula and operator-norm con-
vergence. Comm. Math. Phys. 205 (1999), 129-159.
[15] Neidhardt, H.; Zagrebnov V.A.: Trotter-Kato product formula and symmetrically
normed ideals. J. Funct. Anal. 167 (1999), 113-167.
[16] Reed, M.; Simon, B.: Methods of Modern Mathematical Physics, Vol.I: Functional
Analysis, Acad. Press, New York 1972.
[17] Rogava, D.L.: Error bounds for Trotter-type formulas for self-adjoint operators.Funct.
Anal. Application 27, No. 3 (1993), 217-219.
[18] Trotter, H.F.: On the products of semigroups of operators. Proc. Am. Math. Soe. 10
(1959), 545-551.
Vincent Cachia
Centre de Physique Th~orique, CNRS-Luminy-Case 907,
F-13288 Marseille Cedex 9, France
Email: cachiaQcpt.univ-mrs, fr
Hagen Neidhardt
Weierstrafi-Institut fiir Angewandte Analysis und Stochastik,
Mohrenstr. 39, D-10117 Berlin, Germany
Email: neidhardQwias-berlin.de
Valentin A. Zagrebnov
D@artment de Physique, Universi% de la M~diterran~e (Aix-Marseille II) and
Centre de Physique Th~orique, CNRS-Luminy-Case 907,
F-13288 Marseille Cedex 9, France
Email: zagrebnov@cpt.univ-mrs.fr
AMS Classification: 47D03, 47B25, 35K22, 41A80
Submitted: March 1, 2000

Integr. equ. oper. theory 39 (2001) 413-420
0378-620X/01/040413-8 $1.50+0.20/0
9 Birkh~user Verlag, Basel, 2001
SQUARE OF w-HYPONORMAL OPERATORS
M. Ch5 and T. Huruya
i Integral Equations
and Operator Theory
In this paper, we show that ff T is a w-hyponormal operator, then T 2 is also
w-hyponormal.
1. Introduction
Let 7-/be a complex separable Hilbert space and B(7-t) be the set of all bounded
linear operators on 7-/. For an operator T E B(7-t), let T = UtT I be the polar decomposition
of T. The operator T = ITi~UITI~ is said to be the Althuge transformation of T. An
operator T is said to be w-hyponormal if liPI >_ ITI _ IT* I. In [5], Aluthge and Wang
proved that if T is an invertible w-hyponormal operator, then T2 is also w-hyponormal. In
[8] Kim gave a simple proof of it. And in [6] ChS, Huruya and Kim proved that if T is
a w-hyponormal operator such that ker(T) = {0}, then T ~ is also w-hyponormal. About
powers of a w-hyponormal operator T, they showed Example 3 of [6]:
Let A and B be positive operators on 7-/such that A > B. Let a Hilbert space ]C
be t: = ~ ?-/~, where ?-/~ = 7-/for every n E Z. Let U be the bilateral shift on If and
operators {D~} be
B (,, 1).
For x = (.--, x-l, x0, xl,--.) e E, let an operator D on E be (Dx)~ = D~x~ and T = UD.
Then T and T k are w-hyponormal for every k E N.
In this paper, we show the following:
THEOREM 1. If T is a w-hyponormal operator on ?-l, then T 2 is al~o w-
hyponormal.

414 Ch6, Huluya
2. Proof
For the proof of Theorem 1, we prepare some results.
THEOREM A (Theorem 4 of [2]). IfT is w-hyponormal, then IT 2 ] > ITI 2 and
IT*F > IT*~l.
THEOREM B (Corollary 1.2 of [5 0. An operator T is w-hyponormal if and
9 1 1 1 I ~ I I
only iflTI _> (ITI~IT*IITI~)~ and (IT*I~I7 IIT*I~)~ > IT*I-
First we show the following
THEOREM 2. Let A and B be positive operators o~ 7-(. Let a Hilbert space ]C
o|
be ~ = (~ 75~, where Tl~ = 7-[ for every n E Z. Let U be the bilateral shift on ]C and
operators (Do} be
B (n 1).
For x = (... ,z_l,xo, x~,...) E ]C, let an operator D on ]C be (Dx)~ = D~x~ and T = UD.
If T is w-hyponormal, then T 2 is also w-hyponormal.
Remark. In Theorem 2, we do not assume the order A _ B.
For the proof of Theorem 2, we need following lemma about the Aluthge transformation.
LEMMA 3. Let U and V be partial isometries. Let T = U]T I = V[T I be polar
1 1
decompositions of T. Then ITI89189 = ITI~VITI~.
{z~} of
we have
PROOF. Let T~ be the closure of ITI~(?-/). If x E 7~, then there exists a sequence
1
ITI89 such that lim x~ = x. Let y~ E 7-( be x~= ITl~y~ (n = 1,2,...). Then
n--~oo
= Ty~ : V}TI~x. Since 7-/= n @ 7~ and UITt89 = 0 : VITt89 for
1 1
x ET& L, we have ITI89 : ITI~VITI~.
PROOF OF THEOREM 2. Using the canonical orthogonal basis {e~ : n E Z}
of ~2(Z), we have ?/o = 7-/| e| and ]C = 7-I | I~2(Z). Let x E 7/. Since ITI = D, then
ITl(x | e~) = Dox | for every n E Z, so that
{ Bx| (n>O)
ITl(x | eo) = A~ | e. (~ < 1).
Hence we have ITI2(x| = D2~x| Since U(x| = x| and U*(x| 0 = x|
we have
TT*(x | e,~) = U1Tt2U*(x | e~) = UITt2(x @ eo-1)

Ch5, Huruya 415
Hence
Next we have
Hence we have
and
= U(D~_lx | e,~-l) = D~_lx | e,.
IT*l(x | e~) = D,~_ix | e,~ and lT*12(x | an) = D~_~x | en.
T2(x | e, 0 : T(D,~x @ en+l) = U[T](D,x | e,~+~) = D,~+ID,~x | e,~+2.
2 1
lT2l(x | e.) = (D,~D,~+ID,~)~x @ e,~
2 1_
IT*Zl(z | en) = (Dn_IDn_2Dn_I)2X | en.
Since T is w-hyponormal, by Theorem A we have IT21 > IT[ 2 and IT*I 2 > IT*21, that is,
and
2 ! 2
(D,~D,~+ID,~)~ > D,~
2 2 1
D,~_ 1 > (D~_lD~_2D~_l)2
for every n E Z. Letting n = 0 of (2) and n = 2 of (3), we have
and
(BARB) 89 Z B ~
A 2 > (ABeA)89
Let AB = WIAB ] be the polar decompositions of AB. Define the operator V on K: by
and
V(x | en )
f z | e,+2 (n # 0)
Wx| (n=O)
I
v*(~| z| (~#2)
W*x| (n= 2).
By (I) it is easy to see that T 2 = VIT21. Since V is partial isometry, we have
T "-2 = IT21~V[T2]] by Lemma 3.
To show that T 2 is w-hyponormal, we have only to prove that
for each n.
For n ~ 0 we have
N N ,
IT%~e. > [T211n| _> IT 2 ll~|
T2*~(x | e,~) = IT~I89189 (x | e~)
(I)
(2)
(3)
(4)
(5)
(6)

416 Chs, Huruya
]T2I-~ 9 ~- 2 1- ]T2]~V.]T2I((D,~D~+ID,d88174
---- 2V IT [V((D~D,~+ID,~)4x| ) =
2 I * 2 x 2 !
= IT [2 V ((Dn+2D,~+aD,~+2)~ (D,~D~,+ID,~) 4x | e~+2)
= IT2[ 89
Also for n = 0 it holds that
Similarly for n # 2 we have
and for n = 2
Therefore, we have
and
2 1 2 1
((D.~+2D,~+3D=+u)~(D,~D,~+ID,~) ~x | e~)
2 i 2 1 2 I
= (D,~D,~+ID,~) ~ (D,~+2Dn+3Dn+2) ~ (D,~D~+ 1D~) Zx | e,~.
T~*T~(x | eo) = ID D2D ~88 n2D "89 ~2,~ ,88
~, 0 1 O] vv ~ 2zJ 3 2) I/Vl, lJO/JlL20) x~e0.
9 2 ! 2 ! 2 1
~*(x | e=) = (D~D~+~DJ.(D~_~D=_~D~_2)2 (D~D~+~D~)~ | e~,
~ * ~ * 2 -
T2T 2 (x | e2) = (D2D~D2)88 ~ W (D2D3D2) 4 x | e2.
2 ! 2 1 2 !
_ {(D,~D,~+I D,~), (D,~+2Dn+aD~+2)2 (D~D,,+I D,~) ~ }~ x | e~
IT21(x| = 2 ' 9 ~ ' ~. ~ '
{(DoD~Do)~W (D2D3D~)~W(DoD~Do)~} ~x | eo
2 1
IT~l(x | e~) = (D=D,~+ID.~)~x | e,~
2 ! 2 ! 2 ! !
IT2*I(x | en) = { (DnDn+lnn)4 (D,~+2D.+aD,~+2)~ (D~Dn+ 1D~)4 }2 x | e~
2 1 2 :t . 2 1 1
{ (D2DaD2)~W(DoD1Do)~W ( D2D3D2)~ } ~x | e2
We compare for every n E Z. First, for n < -3, it is easy to see
Hence (6) holds for n < -3.
For n = -2 it holds that
and
]T"-2i(x | e,~) = IT2l(x | e=) = I~*1(~ | e~) = B2x | ~.
2 i z
IT21(x | e_2) = (B(BA B)~ B)~x | e-2
[T2l(x | e-21 = [~*l(x | e-s) = B2x | e-2.
Since by (4) we have (BA2B)89 > B 2, by L6wner's inequality we have
(B(BA2B)89189 > B 2.
(~ # 0)
(~ = 0),
(~ # 2)
(~ = 2).
(7)

Chh, Huruya 417
Hence (6) holds for n = -2.
For n = -1 we have
N 2 1
[TUl(x | :- (BA B)~x | e_ 1 and IT=l(x | ~_~) : I~*l(x | ~-~) : m=x | e_,.
Hence (6) it holds for n = -1 by (4).
For n = 0 we have
and
IT-~l(x | e0) = ((BA2B)88188 | eo
2 1
IT21(x | e0) = ( BA B)~x | eo,
]T-~*[(x @ e0) = ((BA2B)88188 89
By (4), it is easy to see that (BA2B)89 > ((BA2B)88188 Since AB = W]AB[, we
have W]ABIW* = I(AB)*] = ]BAI, so that
and similarly
W(BA2B) 89 * = W((AB)*AB)89 = WlABIW*
= IBAI : ((BA)*(BA))89 -- (AB2A)89
W*(AB2A)89 = (BA2B)89
Since we have A ~ > (AB2A)89 by (5), it follows that
Hence, (6) holds for n = 0.
For n -- 1, we have
W*A2W > W*(AB2A)89 = (BA2B)89
N 2 1
IT2](x | ei) = ]T2I(x | el) ---- A2x | el and IT'2*](x | el) =- (AB A)~x | el.
Hence (6) holds for n -- 1 by (5).
For n -- 2, we have
and
[T2I(x | e~) = tT2[(x | e2) = A2x | e2
I~*](x | e2) = (AW(BA2B)89189 | e2.
Since by (8) it holds W(BA2B) 89 * = 2 ~- 9
(AB A)2, it follows from (5) that (6) holds for n = 2.
Finally, for n > 3 we have
Hence (6) holds for n > 3.
This completes the proof.
[T~21(x | ~) -= ]T~l(x | e,~) = ]T'~*l(x | e,,) = A2x | e~.
(s)

418 Ch6, Huruya
PROPOSITION 4. Let T be a w-hyponormal operator on 7-l. Let a I-Iilbert space
1C be 1C = G 7-l~, where 7~ = "If for every n E Z. Let U be the bilateral shift on ]C and
n:--oo
operators {D~} be
D,~ = f [T*I (n < O)
IXJ (n_> 1).
[
For x= (... ,x-l,xo, xl,.. .) E 1C, let an operator D on 1C be (Dx)~ = D=x~ and R = UD.
Then R is w-hyponormal.
PROOF. By the canonical orthogonal basis {e= : n E Z} of g2(Z), we have ]C =
7-t | g2(Z). Put A = IT] and B = ]T*]. Then the operator R on ~ is
Since we have
it holds that
and
Therefore
and
n(x| Bx| (n_ 1).
[
Bx| (n~ 1)
n*(x | e~) = A~ | e~-i (n _> 2),
{ Bx| (n 1)
IR*i(x| Bx| (n_ 2).
B= | e~ (n _< O)
! $ 1 ! 1 1 1
Ax | e,~ (n >_ 2)
Bx | e~ (n _< O)
. 1 1 1 I 1
(In*l~lR[ln I~)~(~| = (B~AB~)~x| (n= 1)
Ax | e,~ (n >_ 2).
1 1 1
Since T is w-hyponormal, by Theorem B we have A >_ (A~BA~)~ and (B89189189 >_ B. In
comparison with every n E Z, we have
,1 ,l !
IRI >_ (]RI89189189 and (IR 12JR]JR [2)2 _~ JR* I.
Therefore, by Theorem B, R is w-hyponormal.
PROOF OF THEOREM 1. Let R be the operator weighted shift defined by
Proposition 4. Since R is w-hyponormal by Proposition 4, (5) of Theorem 2 implies that
ITI 2 >_ (ITIIT*I21TO 89

ChS, Humya 419
Hence by Douglas' result there exists a positive operator A such that (]TIIT*I2[T])89 =
ITIA]TI. For ~ > 0, let S~ = (ITI 2 +c)-89189 2 -I-E)-89 By the above inequality
and the same argument of [?, Theorem 2], we have
and there exist strongly limits
& IT2*89 (10)
holds without the assumption kerT = {0}. Therefore, by (9), (10) and Theorem B, T 2 is
w-hyponormal. This completes the proof.
Hence, by Theorem 1 we have following corollary.
COROLLARY 5. If T be a w-hyponormal operator, then T 2~ is also w-hyponormal
for every natural number n.

420 ChS, Huruya
References
[1] A. Aluthge, On p-hyponoT"maI operators for 0 < p < 1, Integr. Equat. Oper. Th.
13(1990), 307-315.
[2] A. Aluthge and D. Wang, An operator inequatity which implies paranorTnality, Math.
Inequal. Appl. 2(1999), 113-119.
[3] A. Aluthge and D. Wang, Powers ofp-hyponormal operators, J. Inequal. Appl. 3(1999),
279-284.
[4] A. Aluthge and D. Wang, w-hyponormaI operators, Integr. Equat. Opel Th. to appear.
[5] A. Aluthge and D. Wang, w-hyponormal operators II, preprint.
[6] M. ChS, T. Huruya and Y. O. Kim, A note on w-hyponormal opera~ors, J. Ineqnal.
Appl. to appear.
[7] ~1" Furuta, A > B > 0 assures (BrAPBr) 1/q ~ B (p+2r)/q for r > O,p >_ O,q >_ 1 with
+ 2r)q >_ p ~-2r, Proc. hmer. Math. Soc. 101(1987), 85-88.
[8] Y. O. Kim, An application of Furuta inequality, Nihonkai Math. J. 10(1999), 195-198.
Muneo Ch5
Department of Mathematics
Kanagawa University
Yokohama 221-8686
JAPAN
e-mail: chiyom01 @kanagawa-u. ac.jp
Tadasi Huruya
Faculty of Education and Human Sciences
Niigata University
Niigata 950-2181
JAPAN
e-mail: huruya@ed.niigat a-u. ac.jp
2000 Mathematics Subject Classification: 47B20, 47A30 and 47A63.
Submitted: February 1, 2000

Integr. equ. oper. theory 39 (2001) 421-440
0378-620X/01/040421-20 $1.50+0.20/0
9 Birkh~iuser Verlag, Basel, 2001
RANK-ONE PERTURBATIONS OF DIAGONAL OPERATORS
Eugen J. Ionascu
I Integral Equations
and Operator Theory
We study rank-one perturbations of diagonal Hilbert space operators mainly from
the standpoint of invariant subspace problem. In addition to proving some general properties
of these operators, we identify the normal operators and contractions in this class. We
show that two well known results about the eigenvalues of rank-one perturbations and
one-codimension compressions of selfadjoint compact operators are equivalent. Sufficient
conditions are given for existence of nontrivial invariant subspaces for this class of operators.
I. PRELIMINARIES
We let 7-( be a separable, infinite dimensional, complex Hilbert space, and let
s denote the algebra of all bounded linear operators on 7/. If u, v C 7/, we shall write
u | v for the operator of rank one defined by
(uOv)x=u, x~7/,
where denotes the inner product of the Hilbert space 7/. The class Af of operators
T in s which can be written in the form T = N + (u | v), where N is a normal
operator and (u | v) ~ 0 is still not very well understood. Indeed, even the smaller class
of operators of the above form, where N is a diagonalizable normal operator, is not in a
much better situation, despite the structural simplicity of diagonalizable operators. In this
paper we are interested in this second class of operators which will be denoted simply by 79.
Some spectral properties, examples, applications and the equivalence between two known
results about the eigenvalues of rank-one perturbations and one-codimensional compressions
of selfadjoint compact operators are discussed in Section 2. We characterize those operators
in 7:) which are normal and prove that under mild assumptions they have the single value
extension property in Section 3. In Section 4 we give a characterization for an operator in a
relatively natural subclass of 7) to be a contraction. Finally in the last section, we combine
our previous results with known reductions that one would naturally make in dealing with

422 Ionascu
the invariant subspaee problem for this class of operators and give a sufficient condition for
the existence of a non-trivial invariant subspace based on the Lomonosov's theorem [21].
Similar problems concerning operators in the class Af, or rank-one perturbations
of different classes of operators such as isometries, selfadjoint compact operators, selfadjoint
Toeplitz operators, shift restriction operators, cyclic operators, differential operators, (or
Volterra operator) have been studied in a series of papers of which we cite only a few of
them: [3], [4], [16]-[19], [22]-[25], [28]-[32]. It is worth mentioning that the class of rank-
one perturbations of bounded (or unbounded) selfadjoint operators has been extensively
studied and many interesting spectral properties have been established in various works
(see for instance [8]-[Ii], [14], [15], [29], [30]).
We let {en}n~176 denote an orthonormal basis for 7~ which will remain fixed through-
A oo
out the paper. We also let { -}n=l be an arbitrary bounded sequence of complex numbers
and throughout the remainder of the paper we shall write Diag({A~}) for the unique op-
erator D satisfying Den = A~e~, n E IN. We shall denote henceforth by 7:)o the subset of
L(~) consisting of all operators T which can be written in the form
(1) iF = Diag({)~n}) + u | v, u # O, v # O.
Throughout the paper we shall suppose that u and v are nonzero vectors in ?t and their
expansions with respect to the (ordered, orthonormal) basis {en} are
(2) ~= ~e~, ~=E~e~.
n=l n=l
Note that up to unitary equivalence, 7)o consists exactly of a11 sums N + 7~, where N is a
normal operator whose eigenvectors span ~ and R is an operator of rank one. Note also
that the inclusion T) CAf is a strict one. One way to see this is to make use of Kato and
Rosenblum's result (cf. [20]) stating that the absolutely continuous parts of a selfadjoint
operator and its selfadjoint trace class perturbation are unitarfiy equivalent.
Observe that the expression for T in (I) is not necessarily unique. If we restrict
our study, though, to the class 7)1 of those operators in :D o which admit a representation as
in (i) with u and v having nonzero components an and ~n for all n E IN, we have uniqueness
in the following sense.
PROPOSITION 1.1. IfT E 2)1 then the representation (1) for T is unique in the
sense that if T = Diag({A~}) § (u | v) = Diag({A$}) + (u' | v'), then Diag({An}) =
Diag({AS} ) and (u | v) = (u' | v').
PROOF. We may assume T = Di~g({~})+(~| = Di~g({~'})+ (~'| where
all the Fourier coefficients of u and v in (2) are not zero. This means that Diag({A~}) -
Di~g({~'}) = Di~g({~ - ~'}) = (~' | r - (~ | ~) h~ rank at most two. Thus, there
exist different positive integers hi, n2 such that Ak = A~ for all k C IN \ {n~, n2}. Moreover
the range of S = Diag({A~ - AS} ) is contained in V{e,u, e~2} , and so we may have three

Ionascu 423
essentially different situations. If the range of S is (0) we are done. If the range of S is one-
dimensional--say, spanned by e~1, then since (u'| - (uev) would have a two-dimensional
range if {u, u'} and {v, v ~} are linearly independent sets of vectors, we get that either u
and u ~ are linearly dependent or v and v ~ are. Let us suppose that u and u I are linearly
dependent. Then u = ~nlen~ and u' -- fl=le=1. But this cannot happen since we have
assumed that < u, ek >~ 0 for all k E ]IN-. Similarly the case in which v and v ~ are linearly
dependent is ruled out. If the range of S were two-dimensional, then V{u, u ~} = V{en~, e~2},
and again we would have a contradiction. []
2. SPECTRAL PROPERTIES
The next two propositions show that when looking for nontrivial invariant subspaces for
operators in D0, one can then restrict his attention to the subset 7) 2 of :DI consisting of
those operators T = D + (u @ v) in T) I such that D has uniform multiplicity one (i.e., if
D = Diag({A~}), then all of the numbers An, n E IN, are pairwise distinct).
PROPOSITION 2.1. Suppose T = Diag((An}) + (u @ v) E Do is not a normal
operator, and for some no E ]N, a~o = 0 or fl~o = O. Then T* [resp. T] has point spectrum
and T and T* have nontrivial hyperinvariant subspaces (n.h.s).
PROOF. In case ~no =< u, | >= 0, we have
T* eno = -A~oeno + (v | u)e~ o = -A~oeno + < e~o, u > v = -~noeno,
which shows that ap(T*), the point spectrum ofT*, is nonempty, and since T* is non-normal,
the eigenspace associated with ~o is a n.h.s, for T*. Its orthogonal complement is thus
hyperinvariant for T. The case j3~ o = 0 is handled similarly. []
eigenvahies A=.
For a diagonal operator D -- Diag(~n}) we denote by A(D) the set of all its
PROPOSITION 2.2. If T = D + (u | v) E D1 and at least one A E A(D) has
multiplicity larger than 1, then T has A in its point spectrum.
PROOF. Suppose A -- A,,o -- A,u, no ~ nl. Then (T- A)e),~o --< e;~o ,v >
u = ~o u, and (T - A)e~ =< e~,v > u = ~lu. Hence, if fl~o ~ 0 and/?n~ ~ 0 then
(T - #)(&~e~ ~ - &oe~) -- 0. In any case T - A is not injective, and then A E ~rp(T). []
For an operator T C ~1 given by (1), an interesting phenomenon happens with the
isolated eigenvalues of Diag(A~): they are not in the spectrum of T. The following theorem
gives necessary and sufficient conditions for a point # in or(D) (T -- D + (u | v) E Do) to
be in ~(T) (resolvent set).
THEOREM 2.3. Suppose we have T ---- D + (u | v) E :Do and # E or(D). Then
# E ~(T) if and only if the following two conditions are satisfied:
(i) # is an isolated eigenvalue of D, A~o, of multiplicity one,
(ii) ~o --< v, e~ o >~ 0 and ~o --< u, e~o >~ O.

424 Ionascu
PROOF. For the necessity part of this theorem, let us assume first that (i) is not
satisfied. We have three cases: (a) # is not an eigenvalue; (b) # is an eigenvaiue but is not
isolated, and (c) # is an isolated eigenvalue but has multiplicity larger than I. In the cases
(a) and (b), there exists a sequence of distinct eigenvalues {An~}k>1 such that Ank --~ #.
Then, since (T - #)e~ = (~nk - #)e;~+ < e;~k , v > u we have
II(T-,)e;,~ _< I~,~, -~1 +l < e~.~,v > IIl~ll ~ 0,
as k goes to infinity. This says in particular that T - # is not bounded below (if it is
injeetive), and then it cannot be invertible. In other words, # E G(T). In the case (c), if we
have ~ = .~.0 = .X~l, then (T - ~)e~0 =< e~.o, v > u = ~oU, and (T - ~)e~ ---< e~l, v >
u = ~--~u. Hence, if ~0 ~ 0 and ~ ~ 0 then (T - #)(~e~.0 - ~-~e~) = 0. In any case
T - # is not injective, and then again # E G(T).
Suppose now that (i) holds but (ii) doesn't. First, if ~no = 0, we get as above
(T - p.)e~o = 0, and so # E G(T). If o~n0 = 0, then (T* - ~)e;~o = 0, and then ~ E G(T*),
or equivalently, # E G(T).
For the sufficiency, we assume now that (i) and (ii) hold. We want to show that
T - # is invertible. Since # is an isolated point in o-(D) and D is normal, D - # and hence
T - #, is Fredholm with index zero. Thus it suffices to show that # is not an eigenvalue for
T. If (T - #)x = (D - A~o)X+ < x, v > u = 0, then by our hypothesis, a~o # 0, it follows
that < x, v >= 0. So, x = ?e~ 0 with ff # 0, and this contradicts the hypothesis/3~ o # 0. 9
and only if
We characterize now the point spectrum of an operator T in 191[resp. 192].
PROPOSITION 2.4. For )~ E (~, A is an eigenvalue for T = D + (u | v) E 191 if
(0 ~ e R~we(D - ~), ~nd
(ii) < x, v > +1 = 0 for at least one vector x E ~ satisfyin 9 u = ( D - A )x.
Equivalently, )~ is an eigenvaIue for T = Dia9({)~n}) + (u | v) E 192 if and only if
(iii) ~k r A(D),
(iv) z~~ < oo, a,~
(v) E;,.eA(D) :,_~,.. =
1.
PROOF. For the necessity part, let A E 9 be an eigenvalue for T and x ~ ~ \ {0},
such that Tx = )~x. Then < x, v > u = (A - D)x. We cannot have < x, v >= 0 because
- - 1
we obtain then )~ = A~~ x = ~%, ~ E~ \ {0}, and then ~io =< %,v >= ~ < x,v >= 0
which is not possible since T E 19~. Hence, if we write ~ = -< ~,~>x, then u = (D - A)~
and +l=0.
For the sufficiency part, we can assume that there exists x ~ ?f such that u =
(D-zk)xand+l=0. Thenx#0andTx=Dx+u=u+)~x-u= )~x.

Ionascu 425
Finally, suppose (i) is valid and A E A(D). Then u = (D - )~I)x = (D - lno)X
for some x E 7-/, and so a~0 = 0 which contradicts that T C :D1. It follows that A ~ A(D)
and the rest of the equivalence between (i) together with (ii) and (iii)-(v) is now obvious. []
For T : D + u | v E :D1, the diagonal operator D and the rank-one operator
are uniquely determined by T (Theorem 1.1) and so we can define the function fT(z) =<
(zI -- D)-lu, v >, for z E C\A(D). This function is clearly an analytic function and it can
be written as a Borel series ([2]):
(3) s (z) = z
n=l
COROLLARY 2.5. Assume T = D + u | v E 7)1 and A C ~\A(D). Then A is an
eigenvalue for T if and only/f fT(A) = 1.
PROOF. Since A E ~\A(D), part (i) in Proposition 2.4 is satisfied. Taking
x = (D - AI)-lu in part (ii) of Proposition 2.4 we obtain the corollary. []
The next corollary describes the spectrum of an operator T E :D2.
COROLLARY 2.6. If T = D + (u | v) E 7)2 then
(4) a(T) = A(D)' U {z E~\A(D); fT(z) = 1},
where A(D)' denotes the derived set of A(D).
PROOF. In general for an operator A E ~:(7-/), or(A) = cre(A ) U crp(A) U crp(A*)*,
where ifA C~, A* = {~:z E A} (cf. [6], p. 51). Since T E :D2, we have ~re(T ) = ~(D) =
A(D)', and so by Corollary 2.5, one inclusion necessary to establish (4) follows. For the
other inclusion, let us assume A E a(T) = a~(T) U ap(T) U ~rp(T*)*. Since ~r~(T) = A(D)',
we can assume that ;~ r ~r~(T). Suppose then that ~ E crp(T). If 3~ E Crp(T) M A(D), by
Proposition 2.4, A r A(D) and so ~ E A(D)' = Re(T) which contradicts our assumption. It
follows that ~ E ap(T)\A(D) and so by Corollary 2.5, fT()~) = 1. Since fT(z) = fT*(-5), for
all z E r \ A(D), one takes care likewise of the case )~ E ~p(T*)*. []
Example ([33]) Let T = Diag(An) + u | u where D = Diag()~) and u are
constructed in the following way. First we consider a family of open disjoint (and non
tangent) disks {D~}ne ~ (D~ is centered at ~ and has radius r~) contained in the unit disk
]13 --- [z E C : Izl < I} and such that the set 113\ U~e~ ~ has Lebesgue measure zero. Such
a family can be constructed using an induction argument, covering at each step a closed set
of whose measure is a fixed nonzero fraction of the measure of the open set uncovered by
the disks constructed at previous steps. Moreover, one can refine the argument in order to
satisfy the condition ~ne~ r~ < cx~.
The diagonal operator D is defined by the sequence {An} constructed above and
u is given as in (2) where c~n = rn, n E IN. We want to compute the point spectrum of T.
In order to do this let us observe that the essential spectrum of T is A(D) ~ = ~\ U~E ~ D,~.

426 Ionascu
Also we need the following formula which can be proved easily by a change of variables to
polar coordinates:
{ ~---2-2 if Iz-al

Ionascu 427
In particular, ifT : D + (u | v) E t)1 and fT(A) # 1 for some A C C\A(D), we have
(A - T) -~ =
(7) (A -- D) -1 - (IT(A) -- 1)-I((A - D)-lu (9 ((A - D)*)-lv).
PROOF. If < A-lu, v > +1 = 0, then u # 0 and since S(A-lu) = O, it is clear
that S is not invertible. On the other hand, if < A-lu, v > +1 # 0, then it is enough to
cheek that (6) gives the inverse of S:
[d + (u(gv)][A -1 - 1 (A-lu| (A*)-lv)] = I+ (u(9 (A*)-lv) -
< A-1% v > +1
t < A-lu, v > (u | (A*)-lv) = I.
< A-lu, v > +1 (u (9 (A*)-lv) - < A_lu ' v > +1
The second part of the lemma clearly follows from the first part. []
For T = D + u (9 v e :Do we define the function FT(Z) :< (zI T)-~u, v > for
z E C\a(T). We have the following relation between the functions FT and fT.
we have
PROPOSITION 2.8. Assume T = D+(u| E ~91. Then for all z e T~\ ( ~(T) U or(D))
(8) FT(Z)= IT(z)
1 - fT(z)
Moreover, ifr E A(D)\A(D)' (r . Ano), . then FT(r . . --1, dFr(Z]dz ,w (C~n0/gno)-- -1, and if
T E Z)2 we have
(9)
1
(T - r = D - 1__~ | e~o - ~--eno | ~*v+
(]{'nO ,"nO
OZ -- "
where D : ~kr -- r k | ek.
--I
O~k-~k
k#no
PROOF. Formula (8) can be easily derived from (7). Each r e A(D)\A(D)' is an
isolated eigenvalue of multiplicity one for D, and hence by Theorem 2.3, T - r is invertible.
We have r - D = r - T + (u | v) and then by Lemma 2.7, < (r - T)-~u, v > +1 : 0, which
proves that FT(r = --1. To compute dFr dz {,~ ~w we differentiate (8) at a point z different of
and take the limit as z --+ r
(z-OVa(z) =_
-~z = (1 - fT(z)) 2 [z -- r -- (z -- r 2
The equality (9) follows from (7) by a similar argument of passing to the limit as z ---* r []
As an application to formula (9) we will show the equivalence of two interesting
facts from the theory of selfadjoint compact operators. The first result appears in [32] (see
also [16]) and the second result was proved independently by several authors (cf. [7], [13]
and [26]).

428 Ionascu
THEOREM 2.9. (i) Let {~k}ke~ and {Pk}ke~ be two distinct monotone in-
creasing sequences of real numbers, each having zero as the limit point. Further assume
that (pk} belongs to (Pk, ~k+l) for each k E IN. Then if A is a selfadjoint compact op-
erator on a separable Hilbert space H having the sequence ~ (k ~ IN) as its eigenvalues
(with multiplicity one), there exists a vector x ~ H such that A + x | x has precisely
the eigenvaIues {#~}ke~.
(ii) Let {~}~ and {#~}k~ be two distinct monotone decreasing sequences of real num-
bers, each having zero as the limit point and such that {p~} belongs to (~+1, ~'~) for
each k ~ IN. Then if A is a selfadjoint compact operator on a Hilbert space H having
the eigenvalues ~k (k ~ IN) (with multiplicity one), there exists a vector y ~ H such
that if P denotes the orthogonal projection on the one-dimensional space spanned by the
vector y, the compact operator (I - P)A(I - P)I(Z-P)(H) has exactly as its eigenvalues
the. sequence {#~}~e~.
PROOP. For the implication (i)=>(ii) we assume that {~'k}k~, {#~}k~ and A
are as in (ii) and let us take the diagonal operator D on H whose eigenvalues are {,~k}k~
where %; =-i, Ak+l = (l+#k) -1-1for k E ]hi. Then by (i) we can findx suchthat
T = D + x | x has exactly the eigenvalues {(1 + ~k) -1 - 1}k~. We take ~ = 51 and
apply formula (9) for D, u = v = z and no = 1. Let Q be the orthogonal projection
on el. We see that (I - Q)(T - (i)-i(i _ Q)I( is a diagonal whose eigenvalues are
precisely {~-;~_;}k>2 i = {1+ #k}~e~. Hence, by spectral mapping theorem the operator
S = (T - r _ I is compact and has the eigenvalues {L'k}ke~. Thus, we can find an
unitary operator U such that U*SU = A. To finish the proof we take y = U*ei and observe
that (I - P)A(I - P) = U*(I - Q)S(I - Q)U, where P is the orthogonal projection on the
one-dimensional space spanned by y.
For the implication (ii)~(i), let {~k}k~m, {#k}ke~ and d be as in (i). Without loss
of generality, we can assume that A is a diagonal operator with respect to the basis {ek}ke~
and ~i = -I. Let B be an arbitrary compact operator on H which has {(#k + 1) -i - 1}ke~
has its only eigenvalues (multiplicity one). Using (ii) we can find y :-- Yi E H such that
(I-P)A(I-P)I(f_p)(H) has precisely {(uk+i+l)-i- 1}ke~ as its eigenvalues. Let {Yk+i}k~
be an orthonormal basis in (I - P)(H) with respect to which (I - P)A(I - P)[(Z-P)(H)
diagonalizes. Then the matrix of B + I with respect to the basis {Yk}kc~ looks exactly as
the right hand side of (9) (for D = A, Ak = ~k (k E IN), 4 = ~'1, u = v and e,~ 0 = 1).
We shall show that we can determine the coefficients of u such that these two matrices
coincide (which will give a unitarily equivalence between the operators which admit this
same representation matrix in different orthonormal basis). Let us write the representation

Ionascu 429
of B as follows
B + I = b1~1 | yl + ~ bkyl | yk + ~ ~Y~ | Yl + ~(~+1 + 1)%~ | y~.
k>2 k>2 k_>2
If we compare this with (9) we obtain that ak = -al(vk + 1)b~, (k _> 2) and then
(10) 1 = bl - ~(~k + 1)lbkl =.
15~12 k>2
This will allow us to solve for 51 if the right hand side of (10) is not zero. Suppose by way
of contradiction that this is not true. Then a simple computation shows that (B + I)z = 0
where z = Yl - ~k>2(uk + 1)b-~k and so B + I admits the value 0 as one of its eigenvalues
but by our assumption the only eigenvalues of B + I are the elements of the sequence
{(#k + 1)-l}ke~. This proves that we have a solution for u e H and so by spectral theorem
A + u | u has precisely the eigenvalues {#k}k~. 9
3. NORMALITY, DECOMPOSABILITY, AND THE SVEP
We begin this section with a characterization of normal operators in No which,
in particular, applies to the normal operators in :D o .
PROPOSITION 3.1. Let T = N + (u | v) E s where N is a normal operator
and u, v are nonzero vectors in 7-[. Then T is a normal operator if and only if either
(i) u and v are linearly dependent and u is an eigenvector for .~(aN*), where
Ot ~ ~ j or
(ii) u, v are linearly independent vectors and there exist 5, fl E 9 such that
01)
(12)
where ~(~) = -1/2.
PROOF. We observe that the equation T*T = TT* is equivalent to
N*u | v + v | N*u + Ilull2v | v =
N~|174 I1~11~ | ~.
It is a simple computation to check that (12) is satisfied if (i) or (ii) is true.
Let us assume that T is a normal operator. We distinguish two distinct cases.
CASE I. We assume that u, v are linearly dependent. Thus, there exists 5 E ~ such that
u = 5v (5 =< u,v >/llvll2). Since Ilvll2u | u = lal2llvll2v | v = Hul]2v | V, if we write
w = (aN* - NY)v (= 2.~(hN*)v), (12) becomes w | v = -v | w. This last equality holds
if and only if w = itv for some t E ]R and (i) is proved.
CASE II. We assume that u, v are linearly independent vectors. From (12) we get that
v=u, xE(V{u,v})

430 Ionascu
Hence < iV*u, x >=< Nv, x >= 0 for every x E (V{u, v}) which means that
(13) N*u = allU -t- a12v, Nv = a21u q- a22v,
for some a/j E C. Substituting in (12) we obtain that the aij satisfy the following relations:
all = a2--~, a12 + ~-i~ + Ilull 2 = a21 + ~-57 + Ilvll ~ = 0.
So, if we write an = ~ and a12 = -]]u]]2/2 + is1, am = -IIvI]2/2 + is2, where sl, s2 e JR,
(13) implies that (N*-~I)u = (-Ilull~/2 +isl)v and (N-~I)v = (-IIvll2/2 + is2)~. Thus
(N - c~I)*(N - c~I)u = (-]1u]]2/2 + isl)(-I[ull2/2 + is2)u which implies that sl/HuH 2 =
-s~/Ilvll ~. zf we write t = 1/4 + (1/llull4Ds~ and # = -1/2 + sign(sl)i~/t - 1/4 then clearly
u and v satisfy (11). (Here, we used the notation sign for the real valued function defined
by sign(x) = 1 if x > 0, sign(x) = -1 if x < 0 and sign(O) = 0.) []
COROLLARY 3.2. T = D + u | V E :D1 is normal if and only if either
(a) there exist c~ E (~ and ~ E IR such that A(D) lies on the line {z E (~ : .~(c~-2) = t}, and
U ~-- O~V~ or
(b) there exist a E 9 and t E IR such that A(D) lies on the circle {z e (~ : I z - a] = t},
t E ]R, and
tulllu[I = e~(D - oI)(v/llvll),
where ~ E [0, 7r) is determined by the equation ~(te~O/llull Ilvll) = -1/2.
PROOF. Suppose that (a) or (b) holds. Then either .~(aD*) = tI or ID-~Zl = tz.
If (a) holds then (i) in Proposition 3.1 holds and hence T is a normal operator. If (b) holds
then an easy computation shows that (11) holds for/3 = teiO/l[ul[ Ilvll. The two relations in
(11) alone imply that (12) holds and so T is normal.
On the other hand if T is normal then, by Proposition 3.1, (i) or (ii) holds. In case
(i) is true then ~(aD*)u = tu for some t E IR. Thus ~'(o~-~oe~) = tan for all n E IN and since
a~ r 0 for every n in IN we obtain that A(D) is a subset of the line {z E (l:J : .~(~g) = t} and
(a) follows. If (ii) holds, we get from (11) that (D-c~I)*(D-aI)v = IMl~llvll21~l%, and by
a similar argument as above, we get that A(D) is a subset of the circle {z E e: Iz- a I = t},
where t = Ilull Ilvlll~l. Then, the other part of (b) follows easily from (11). []
It is worth mentioning that actually if A(D) is a subset of a line or of a circle
then T = D + u | v is a decomposable operator (el. Theorem 5.2, [5]). Moreover, T has the
property (Triang0) (of. Theorem 6.16, [5]), i.e., for any pair S~ C & of invariant subspaces
for T such that dim($1/S2) > 1 there exists another invariant subspace $3 of T verifying
Another interesting question about the class :Do is whether we have the decompos-
ability property for operators in :Do whose spectrum is not necessarily an arc of an analytic
curve. It is known ([5]) that every decomposable operator has the following property.

Ionascu 431
DEFINITION 3.3. We say that an operator T E s has the single valued ex-
tension property (notation: SVEP) if the only vector-valued analytic function f : O --+ ~,
where G is an arbitrary open connected subset of (B, which satisfies the equality
is the function identically equal to zero.
(T- zI)f(z) : O, z E G,
PROPOSITION 3.4. Every operator T : D+(u| E lPi for which the set(B\A(D)
is connected has the SVEP.
PROOF. Let f : G ~ 7-I be an analytic function such that (T - zI)f(z) : 0
for every z E G. If G F] ((I] \ A(D)) # ~ then by Corollary 2.5, T - zI is invertible for
all z E (G\A(D))\{z E (B\A(D);fr(z) = 1} and so f(z) = 0 for all z E (G\A(D))\{z E
~]\A(D); fT(z) = 1}. The function fT cannot be identically equal to 1 on the connected set
9 C \ A(D) because limlzl_~co fT(z) = 0. Hence the set {z; fT(z) = 1} is discrete arid since G
is connected it follows that f is identically zero.
We may assume that actually G C A(D). If we expand f in the basis {en} as
f = ~,~~176 1 fne,~, where fn : G --+ C are scalar-valued analytic functions, we get
(14) (A~ - z)f~(z)+ < f(z), v > ~n = o, z e a, ~ 9 ~.
If we take z = ,~ 9 G fl A(D) in the above equation, we obtain that < f(A~), v >= 0 for
all A~ 9 G F1 A(D). Since the set A(D) is dense in A(D) and G C A(D), the set G F1 A(D)
is clearly dense in G. Hence < f(z), v >= 0 for all z 9 G. Thus (14) implies that for every
integer n 9 ]iN, f~(z) = 0 for all z 9 G\A(D). Since each fn is a continuous function and
G\A(D) is dense in G, it follows that f,, is identically equal to zero on G for every n 9 ]N
and so is f. []
4. CONTRACTIONS IN ~)0(]D)
In this section we consider the class 7)0(]D) of the operators T--- D+u| 9 2Po for
which A(D) C ~. In this section we will characterize the contraction operators in T)0(ID).
The following proposition provides one such characterization and leads us to Corrolary 4.3
which gives a simple sufficient condition for an operator T 9 :Do(ID)MT~2 to be a contraction.
g
PROPOSITION 4.1. T = D + u| 9 :Do(]]:)) is a contraction operator if and only
(15) I i - s < ~(s), D~(s) > I > vq, s 9 (0, 1),
Ile(s) llll~(s)lt
where ~(s) = (I - ~D*D)-II~ and 5(s) = (S - sD*D)-il~v, or equivalently, in case r 9
7Po(]D ) F1 2)2, if and only if
(i6)
1 ,~=~ ~ (1 - s ak ~) > s (k__~l (1 - slakl~)) (k~__~l (], _ slAkl=)), s 9 (0, ].).

432 Ionascu
PROOF. Clearly T is a contraction if and only if T*T is a contraction. Since
T*T is a positive selfadjoint operator, T*T is a contraction if and only if its spectrum is
contained in the interval [0, i]. A simple computation shows that
T*T = D*D + (D*u + IM] %) | ~, + v | D*u.
Hence, o's(T'T) = ~re(D*D ) C or(D'D) C [0, 1] and so T*T (o'(T*T) = os(T'T) U crp(T*T))
has its spectrum contained in the interval [0, 1] if and only if its point spectrum does not
intersect the interval (1, oo). We need the following lemrna.
LEMMA 4.2. Let A E s be invertible and S = A + (a | b) + (c | d) for some
vectors a, b, c, d E 7-l. Then the following are equivalent:
(i) S is not invertible,
(ii) ker(S) # O,
[1+ < A-la, b > < A-lc, b > ]
(iii) the determinant of the matrix L < A-la, d > 1+ < A-lc, d >J is zero.
PROOF OF LEMMA 2.7. Since S : A (I + (A-la | b) + (A-% | d)), S is not in-
vertible if and only if I + (A-la | b) + (A-lc | d) is not invertible. Using the Fredholrn
theory, this latter operator being Fredholrn of index zero, it is not invertible if and only
if its kernel is not the (0) subspace. Hence (i) and (ii) are equivalent. For the equiv-
alence of (ii) with (iii), let x e 7-I be a vector such that Sx = O. This implies that
x+ < x, b > A-la+ < x, d > A-le : 0. Taking the inner product of this equation with b
and d respectively, we get the following system of equations with the the unknowns < x, b >
and < x, d >:
(l+)+=0
< A-la, d >< x,b > +(1+ < A-lc, d >) < x,d >= 0.
Therefore if we assume that (ii) is true, then
x:- A-la- A-~c#O
and so at least one of the numbers < x, b > or < x, d > is not zero. This implies that
the above homogeneous system has a nontrivial solution. This fact is equivalent with
the statement (iii). Let us assume that (iii) is true. Then there is a nontrivial solution
of the above homogeneous system of equations--say < x, b >: a and < X, d >: /~.
Hence x = -aA-la - ~A-lc is not the zero vector and a simple calculation shows that
(I + (A-la | b) + (A-lc | d)) x = 0 or Sx = O. m
We apply Lemma 2.7 for the case A = D*D - tI, a = D*u + l]u]12v, b = c = v,
and d = D'u, where t E JR, t > 1. Hence, T*T is a contraction if and only if the determinant
of the matrix
i+ ti)-l(D. + > < (D'D- tI)- v, > ]
< (D D - tI)-l(D*u + II~IP~), D*u > i+ < (D*D - tI)-=v, D*u > J

Ionascu 433
equals zero for no t e (I, oo). If we multiply the second column of this matrix by INI 2 and
subtract it from the first column, the determinant is the same as the determinant of the
resulting matrix
[ l+ < (D*D-tf)-lD*u,v > < (D*D-tI)-iv, v > ]
< (D*D - tI)-lD*u, D*u > -IluI[ 2 1+ < (D*D - tI)-%, D*u > "
The (2, 1) entry can be written differently as follows:
< (D*D - tI)-lD*u, D*u > -Ilu]] 2 =< (D*D - tI)-lD*Du, u >-I[u[[ 2 =
< (D*D - tI)-l(D*D - tI)u, u > -Ilu]l 2 + t < (D*D - tI)-lu, u >
= t < (D*D - tI)-lu, u >.
If we observe that the (1, 1) entry is the complex conjugate of the (2, 2) entry, we obtain
that T*T is a contraction operator if and only if the equation (in t)
]1+ < (D*D - tf)-lD*u,v > 12 - t < (D*D - tI)-lu, u >< (D*D - tI)-lv, v >= 0
has no solution in the interval (1, oo). Finally, if we change variables by setting s = 1/t,
s E (0, 1), the above equation becomes
[1-s < (I-sD*D)-lD*u,v > [2
< (I- sD*D)-lu, u >< (I- sD*D)-lv, v >
which implies (15) since both members of the above equMity are continuous functions of s
and the sign of the inequMity is determined when s = 0. The inequality (16) follows form
(15) taking into account the explicit form of the operator D. []
COROLLARY 4.3, Assume that for T = D 4- (u | v) C 2Po (]D) rh'D2 the coordinates
of u and v satisfy the inequality
Then T is a contraction operator.
if
(1 -I~l~)/ (1 - ]-~k12)] -< 3 - 2v~0.171572876 ....
PROOF. Using Proposition 4.1 we get that T is a contraction operator if and only
slle(s)ll211v(s)ll ~ <
(18) 1 - 2she < e(s), Dr(s) > +s=l < e(s), D~(s) > I s,
for every s e (0, 1). We observe that (18) is satisfied if I1~(1)11 and IIV(1)ll are finite numbers
satisfying
11~(1)11211v(1)11 ~ + 211e(1)llll~(z)ll < 1.
This last inequality is clearly satisfied if we have (17). []
COROLLARY 4.4. Assume that T = D + (u | v) E "Do(]])) N "D2 is a contraction
operator. Then the following inequality holds for every s E (0, 1) :
(19) (1 - slM2)/ (1 - ~1~1 ~) < 41 - v~) 2

434 Ionascu
PROOF. If T is a contraction operator then we have (18), which implies that
sll~(s)II~II~(s)ll ~ < 1 + 211~(s)IIlI~(s)[] + s~lI~(s)ll~II~(s)ll ~, s e (0, 1).
This last inequality is equivalent to (19) by simple computations. []
5. INVARIANT SUBSPACES
If A E L(~-{) and z E 7-{ we write Ca(A) = V~=0{Anx}. A vector x E 7-{ is
called cyclic for A if G~(A) -- T/. The following proposition characterizes those operators
T = D + (u | v) e :Do for which Lat(T) M Lat(D) # (0).
PROPOSITION 5.1. If T : D+ (u | v) C :Do then D and u | v have a common
n.i.s if and only ifC~(D) # 7-[ or C~(D*) # ~.
PROOF. One can easily find
subspace S is invariant for u | v if and
acommonn.i.s, for D and u| Ifu
all the invariant subspaces of u | v. Namely, a
only ifu C 8 or v_1_S. Let us assume that 8is
E S we get that Cu(D) ~ ~, and if v _L S, ~q
is nontrivial invariant for D* containing v. Hence in this case dr(D*) ~ T{. This proves
the necessity. For the sufficiency, we just have to observe that du(D) and (Cv(D*)) are
common invariant subspaces for D and u | v. []
The following proposition is a particular case of Brain's result [6] and answers
the natural question whether an arbitrary diagonal operator admits a cyclic vector. For
completeness we include here a simple proof of this fact which is a simplified version of the
proof of Brain's result given in [6].
PROPOSITION 5.2. Let D = Diag({An}) E L(T[) such that every value in A(D)
has multiplicity one. Then there exits a cyclic vector for D.
PROOF. We consider the operator Mz, the multiplication with the variable on
L2(X,7), where X = A(D) and 77 = ~=1 oo ~6:~. 1 Define V : T{ ---+ L2(X, 7) by Vx = fz
where f~(z) = nx~ if z = A,~ and zero otherwise, x = xzei + x2e2 + ... E ~: We have for
each x e 7t, ][Vx[] 2 = ]]/~l] 2 = Ix ]f~(z)i2&l(z) = ~oo~=1 -~]X~(A~)I 2 = E~=l ]x~l 2 = [[xII 2.
Clearly, V is an unitary operator and VDV -1 = Mz, which implies that it suffices to show
that M~ has a cyclic vector. For each n E IN, denote K,~ = {A1, A2, ..., AN}. Since all the
eigenvalues An are assumed to be distinct, the following system of linear equations has a
unique solution in Co, Cl, ..., c~:
(20) A~ = Co + cl),j + ... + c~A~, j = 1, 2, ..., n.
Let p~(z) = Co + clz + ... + c~z '~, where the coefficients co, cl, ..., c~ are satisfying (20). Using
this notation, (20) can by written as ~ = p~(z) on Kn. We now construct a Borel measure
v on X with the following properties:
(a) u is a measure absolutely continuous with respect to %
dv
(b) ~ := r is essentially bounded @/]),
(c) the function l(z) = t is a cyclic vector for M~ acting on L2(X, u).

Ionascu 435
First we choose a~ = (maxl_ ... _> a~, > 0. It is easy to observe that (a) is
satisfied, and in order to check the second property we take r = a~-i if z = ),~, n > 2,
l if z = A1 and zero anywhere else. Hence, 0

436 Ionascu
When one searches for invariant subspaces for an operator T it is useful to have
a description of its commutant {T}' := {A E Z:(7-L) : AT = TA}.
PROPOSITION 5.4. Let T = D + (u | v) C 1)2, and A E s Then A E {Ty
if and only if there exist a sequence of complex numbers {t,~}ne~ and a positive constant C
such that
(i) for every square-summable sequence {~k}k_>l we have
(21) E f J- E _l,k~n n
where %,n := ~ for k r n, (k, n E IN),
(ii) for every k E IN,
(22) gek = skek + -ilk ~ a~Tk,ne~,
n>_l,nr
where the sequence defined bY
(23) = - E k e
n>l,nek
is a bounded sequence.
PROOF. The equality AT = TA can be written equivalently as
(24) AD - Dd = (u | A'v) - (du | v).
For the necessity part, let {tk} be defined by the equation Au = ~k~=l tk~kek. For every
integer k __ 1, we have < (AD - DA)ek, ek >= 0 and then from (24) we obtain
< ek, A*v >< u, ek :> -- < Ck~ V :>< Au, ek >= 0,
which in turn implies that < ek, A*v >= tk-flk. Hence, using (24) again, we get
(25) (Ak -- D)Aek = ~k(tku-- Au) =-ilk k a,~(t~ - t~)e~, k > 1,
n>_l,nek
which implies that we can express Aek as in (22). Taking the inner product of both sides of
(22) with v, we obtain that sk is given by (23). To obtain the inequality (21) we first need
to observe that sk =< Aek, ek > (by (22) and so {sk} is a bounded sequence. Thus, the
inequality (21) follows easily from the boundedness of the operator A - D, where D is the
diagonal operator defined by Dek = s~ek, k G IN.
For Sufficiency, we observe that the linear operator A defined by (22) is bounded
because of (21) and the hypothesis that {sk} is bounded. Then from (22) and (23) we get
that < ek, A*v >= tkfl--k and Au = ~k~176 tkakek. Using these two relations and (22), we
obtain (25) which is equivalent to (24). []
Next we would like to combine Proposition 5.4 with Lomonosov's theorem (cf.
[21]) to obtain sufficient conditions for existence of n.i.s, for operators in /)2. For this

Ionascu 437
purpose we introduce some more notation. Let 7-L(U) be the set of analytic functions on
the open set U(C ~). For a fixed w E U we define a linear transformations on ~(U),
r -. r(r by
(26) r(r
r162
= i l r
-r ~ ~ .
if z = ~,
z e u, eer(u).
For T E 7)2 given by (I), and U such that A(D) C U we define another linear transformation
on ~(U) by
(27) BT(r = fA(D) F(r z E U U A(D), r E 7-/(U),
where u is the atomic measure supported on A(D) given by u = E~>I ~/~--5~.
THEOREM 5.5. Let T E 7)2 given by (1) and BT defined by (27). Suppose there
exists a function r E ~(U), with U D A(D), such that BTr = r and r is not zero on
A(D). Then T has a nontriviaI invariant subspace.
PROOF. Let us consider t~ = r n E ]hi, and let Ar be the operator A
defined as in (22) and (23). We will show that Ar satisfies (21) and it is a nonzero compact
operator. By Proposition 5.4, T commutes with a nonzero compact operator and then using
Lomonosov's theorem T admits a n.i.s.
Suppose that Ar = 0. Then, from the proof of Proposition 5.4, we have A~u =
~ner~ a,,t,,e~, and so t~ = 0 for all n E IN. By Proposition 5.3 we can assume that or(T) =
A(D)' and A(D)' is connected. Thus we can consider 0" to be the connected component of
U containing A(D)'. Hence, r = 0 on U since A(D) must have an accumulation point in
A(D)' C U (U is connected). s cannot contain but finitely many points of h(D) where
r must be zero because ~ - 0, n E IN. This contradicts our assumption on r and so Ar
is not zero.
Since r E 7-/(U) and A(D) C U, there exists a constant C1 > 0 such that
]F(r _< C1 for all z,w E A(D) and so, with the notation from Proposition 5.4,
iTk,~[ _< C1 for every k, n E IN, k r n. Then, using Cauchy's inequality, we have
n [k>l,k#n n kkl,k#n kkl,k~:n tc
where C = c2]lu][2Hvil2. This proves that inequality (21) is satisfied. Also, the sequence
defined by (23) is bounded since {tn} is clearly bounded and for every k E IN
Z ~.N.~,. -< C~ll~llll,-,ll.
r~>_l,n#k
Then, by Proposition 5.4, A O commutes with T. From (23), for every k C IN we have
s~ = r - ~ ~Z~--~,. = r - B~(r + ~Zr
n>l,n~:

438 Ionascu
which simplifies to sk akflkr (k) because of our hypothesis on %b. Clearly, lim~-,oo s~ = 0
and so the diagonal operator D (De~ = Skek, k E IN) is a compact operator. Since Ar =
D + B where B is defined by
Bek=-fik ~ anTk,~en, k E IN,
n>_l,n~k
it suffices to show that B is a compact operator. In fact, B is a Hilbert-Schmit operator
since
X] IIa~kll ~ = X] [&l ~ ~'. I~kl21"rk,,~l 2 < c,
kcIN kc~'q n>_l,n~ak
which finishes our proof. []
COROLLARY 5.6. Let T 6 l?2 given by (1) such that A(D) C ID. Suppose that
fT (cf. (3)) is bounded on ~\]D and let Tr be the Toeplitz operator on H2(]D) of symbol
r = fT(~) for ~ E OID. In addition we assume that the equation Tr162 = r has a solution
r E H2(ID) which is analytic on an open set U DID) and not zero on A(D). Then there
exists a n.i.s for T.
PROOF. The assumption on fT insures that r is in L~(O]D) and so the Toeplitz
operator Tr is well defined. Indeed, for z E ]D we have fT(89 = z ~=1 oo ~ 1-~ = ~k%o mk zk+l,
where mk are the moments of the measure . ( i.e., mk= f~D Ckd'(~), k e IN U {0}). So,
z --+ fT(1/z) is a bounded analytic function on ]D, and thus r E L~176 In fact, Tr
is a co-analytic Toeplitz operator. We want to show that B T and Tr act the same way
on functions r E H2(ID) which are analytic on open neighborhoods of ~. Forsooth, if
r = ~a=0~176 akzk E H2(ID) is such a function, we have
(2s)
T~(r 0): P~2 (r176162176 : P.~ m~e

Ionascu 439
REFERENCES
[1] N. Benamara and N. Nikolski, Resolvent tests for similarity to normal operators, Proc. London Math.
Soc. 78(1999), 585-626.
[2] L. Brown, A. Shields and K. Zeller, On absolutely convergent exponential sums, Trans. Amer. Math.
Soc., 96(1960), 162-183.
[3] K. Clancey, Seminormal operators, Lecture Notes in Math., vol. 742, Springer-Verlag, New York, 1979.
[4] D. Clark, One-dimensional perturbations of restricted shifts, J. Analyse Math. 25(1972), 169-191.
[5] I. Colojoara and C. Foias, Theory of generalized spectral operators, Science Publishers, New York, 1968.
[6] J. B. Conway, A course in functional analysis, Springer-Verlag, New York, 1985.
[7] C. Davis, Eigenvalues of compressions, Bull. Math. de la Soc. Sci. Math. Phys. de la R.P.Roumaine
(N.S.), 3(51) (1959), 3-5.
[8] R. Del Rio, S. Jitomirskaya, Y. Lasta, and B. Simon, Operators with singular continuous spectrum. IV.
Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69(1996), 153-200.
[9] R. Del Rio, N. Makarov and B. Simon, Operators with singular continuous spectrum. II. Rank one
operators, Comm. Math. Phys. 165(1994), 59-67.
[10] R. Del Rio and B. Simon, Point spectrum and mixed spectral types for rank one perturbations, Proc.
Amer. Math. Soc., 125(1997), 3593-3599.
[11] W. F. Donoghue, On the perturbation of spectra, Comm: Pure, Appl. Math. 18(1965), 559-579.
[12] J. Eschmeier and M. Putinar, Bishop's condition (~) and rich eztensions of linear operators, Indiana
Univ. Math. J., 37 (1988), 325-348.
[13] K. Fan and G. Pall, Imbedding conditions for Hermitian and normal matrices, Canad. J. Math. 9(1957),
298-304.
[14] S. Hassi and H. de Snoo, On rank one perturbations of selfadjoint operators, Integral Equations
Operator Theory, 29(1997), 288-300.
[15] S. Hassi, H. de Shoo, and A. Willemsma, Smooth rank one perturbations of selfadjoint operators, Proc.
Amer. Math. Soc. 126(1998), 2663-2675.
[16] H. Hochstadt, One dimensionaIperturbations of compact operators, Proc. Amer. Math. Soc. 37(1973),
465-467.
[17] G. Islamov, Properties of one-rank perturbations, Izv. Vyssh. Uchebn. Zaved. Mat. 4(1989), 29-35.
[18] W. Johnston, A condition for absence of singular spectrum with an application to perturbations of
selfadjoint Toeplitz operators, Amer. J. Math. 113 (1991), 243-267.
[19] V. Kapustin, One-dimensional perturbations of singular unitary operators, Zap. Nanchn. Sem. S.-
Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 232 (1996), 118-122, 216.
[20] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1976.
[21] V. Lomonosov, On invariant subspaces o f families of operators commuting with a completely continuous
operator, Funk. Anal. i Prilozen 7(1973), 55-56 (Russian).
[22] N. G. IvIakarov, One-dimensional perturbations of singular unitary operators, Acta Sci. Matt~. 42(1988),
459-463.
[23] N. G. Makarov, Perturbations of normal operators and the stability of the continuous spectrum, Izv.
Akad. Nauk SSSR Ser. Mat. 50 (1986), 1178-1203, 1343.
[24] M. Malamud, Remarks on the spectrum of one-dimensional perturbations of Volterra, Mat. Fiz. No. 32
(1982), 99-105.
[25] Y. Mikityuk, The singular spectrum of selfadjoint operators, (Russian) Dokl. Akad. Nauk SSSt~ 303
(1988), 33-36; translation in Soviet Math. Dokl. 38 (t989), 472-475.
[26] L. Mirsky, Matrices with prescribed characteristic roots and diagonal elements, J. London Math. Soc.
33(1958), 14-21.
[27] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, American Elsevier, New
York, 1970.
[28] Nakamura, Yoshihiro, One-dimensional perturbations of the shift, Integral Equations Operator Theory
3(1993), 337-403.
[29] B. Simon, Spectral analysis of rank one perturbations and applications. Mathematical quantum theory.
IL Schrdinger operators, CRM Proc. Lecture Notes, 8 Amer. Math. Soc., Providence, RI, (1995),
109-149.

440 Ionascu
[30] B. Simon, Operators with singular continuous spectrum. VII. Examples with borderline time decay,
Comm. Math. Phys. 176(1996), 713-722.
[31] J. G~ Stampfli, One-dimensional perturbations of operators, Pacific J. Math., 115(1984), 481-491.
[32] H. Vasudeva, One dimensional perturbations of compact operators, Proc. Amer. Math. Soc. 57 (1976),
58-60.
[33] J. Wolff, Sur les sgries ~ Ak/(z- ak), C. R. Acad. Sci. Paris, 173 (1921), 1057-158, 1327-1328.
The University of Georgia
Department of Mathematics
Athens, GA 30602, USA
e-mail: ionaseu~alpha.math.uga.edu
Mathematical Subject classification: Primary 47Axx; Secondary 47A15, 47A55, 30B50.
Submitted: January 4, 1999

Integr. equ. oper. theory 39 (2001) 441-474
0378-620X/01/040441-34 $1.50+0.20/0
9 Birkhfiuser Verlag, Basel, 2001
I Integral Equations
and Operator Theory
A SHIFT-INVARIANT ALGEBRA OF SINGULAR INTEGRAL
OPERATORS WITH OSCILLATING COEFFICIENTS *
YURI I. KARLOVICH and ENRIQUE RAMiREZ DE ARELLANO
Let B be the Banach algebra of all bounded linear operators on the weighted
Lebesgue space LP(T, w) with an arbitrary Muckenhoupt weight w on the unit
circle T, and Pd the Banach subalgebra of B generated by the operators of mul-
tiplication by piecewise continuous coefficients and the operators eh,~STe'~,kI
(h E R, ), E T) where ST is the Cauchy singular integral operator and eh,x(t) =
exp(h(t+A)/(t-A)), t E T. The paper is devoted to a symbol calculus, Fredholm
criteria and an index formula for the operators in the algebra 9/and its matrix
analogue 9INxN. These shift-invariant algebras arise naturally in studying the
algebras of singular integral operators with coefficients admitting semi-almost
periodic discontinuities and shifts being diffeomorphisms of T onto itself with
second Taylor derivatives.
1 Introduction.
Let P be the real line R = (-co, +co) or the unit circle T in C with length measure.
A measurable function w : P -+ [0, co] is called a weight if the preimage w-l({0, co})
has measure zero. We will deal with the weighted Lebesgue spaces LP(F,w)(1 < p < co)
equipped with the norm
/ f \ 1/p
Fix p E (1, eo). A weight w : Y ~ [0, co] is called a Mnckenhoupt weight, that is,
w E Ap(I'), if
*Partially supported by CONACYT grant, C~tedra Patrimonial, No. 990017-EX and by CONACYT
project 32726-E, M4xico.

442 Karlovich, Ramirez de Arellano
where r(t,s) := {~- e r. I~ - tJ < ~} and p-1 + q-1 = 1. As is wen known (see, e.g., [14],
[10], [3]), the Cauchy singular integral operator Sr defined by
($r!o)(t) := lim 1 /r T(r)
o~0 ~ \r(t,,) ~---t dr, t E r, (1.9.)
is bounded on LP(P,w) if and only if i < p < co and w E Ap(r).
Let PC := PC(T) be the C*-algebra of functions in L~(T) which have one-sided
limits at each point of T, and let SAPT be the smallest C*-subalgebra of L~ containing
PC and all the functions eh,)~ (h E R, A E T) where
(ht+A (1.3)
eh,.~:T\{.X}--+T, t~+exp\ t---~]"
Clearly eh,x has a periodic discontinuity at the point )t if h 5~ 0.
Let B(X) be the Banach algebra of all bounded linear operators acting on a Banach
space X, and let G be the Banach algebra of singular integral operators with PC coefficients
on LP(T,w), 1 < p < cr w E Ap(T). Our aim is to study the minimal extension 92 of the
algebra g in B(LP(T, w)) which is invariant under the transformations
A ~ BzAB21, A ~ eh,aAeh, ~ (Z E T, h E R, A E T) (1.4)
where B, is the rotation operator on L'(T,~) given by (Bn,)(t) = ~,(~t), t e T.
The algebra 92 arises naturally in studying the Banach algebra of singular integral
operators with SAPT coefficients on the spaces LP(T, w) and its extensions containing the
operators Bg (g E 7)), where Bg : T ~-+ ~ogis a shift operator on LP(T,w) and 79 is the set of
orientation-preserving diffeomorphisms of T onto itself which have second Taylor derivatives
(cf. [16], [17]). By definition (see [19, Section 2.3.3]), a diffeomorphism g : T --+ T has a
second Taylor derivative on T if for every to E T there exists the finite limit
lim 2(t - to)-2[g(t) - g(to) - g'(to)(t - to)]. (1.5)
t-+t o
The algebra 92 contains singular integral operators with PC coefficients, the operators
eh,:,STeh,~I (h E R, A E T) and hence some operators W~,(b) similar to convolution operators
of the form .T-lb.T" where ~- is the Fourier transform,
(S~)(z) := (2~)-~'f~e-'~ z e It, (1.6)
and b is a piecewise continuous function on R being a Fourier multiplier in the space LP(R, @)
with a weight ~ E Ap(R) corresponding to the weight w E Ap(T). In this connection the
algebra 92 contains algebras parameterized by points A E T which are similar to the Banach
subalgebra ~3 C B(L'(R, 6)) generated by operators of the form
A = a~-lb~ (1.7)
with piecewise constant functions a, b E L~176 having finite sets of jumps. The Fredholm
theory for the algebra ~3 has been constructed in [7], [8] (see also its weighted L p analogues

Karlovich, Ramlrez de Arellano 443
in [21], [5], [6]). An L p theory for a line analogue of the algebra 92 was developed in [16],
[1~].
This paper is devoted to a symbol calculus and the Fredholm theory for the Banach
algebra 92 in the case of weighted Lebesgue spaces LP(T, w) with general Muckenhoupt
weights w. An operator A E 92 is said to be Fredholm if it has a closed image and finite
kernel and cokernel dimensions. In that case
Ind A := dim ker A - dim coker A.
In contrast to [7], [8], [21], [5], [6] the symbols of A e 92 are defined on the set
9Y~:= (tU{t}xRxT-/(0,1;L'~-,~'+))U(Tx{co}x{0, l} )
where 7-/(0, 1; v~', v +) are horns bounded by two circular arcs with endpoints 0 and 1, shape
of which depends on p and indices of powerlikeness of Muckenhoupt weight w (see (5.1)).
The paper is organized as follows. In Section 2 we give a definition of the algebra 92
via its generators. In Section 3, applying the Allan-Douglas local principle (see, e.g., [4]) we
get a Fredholm criterion for operators A E 92 in terms of the invertibility of corresponding
cosets being elements of some local algebras (see Corollary 3.4). Section 4 is devoted to
studying the invertibility in the above-mentioned local algebras. In Section 5 we construct a
symbol calculus for the algebra 93 and establish a Fredholm criterion for operators A E 92 in
terms of their symbols. These results also extend to the algebras 92NxN of N N operator
matrices with entries in 92. Sections 6-8 deal with an index formula for operators A C 92NxN.
In Section 6 we introduce an analogue os the Cauchy index for the symbols of Fredholm
operators A E 92NxN and state an index formula for mentioned operators in terms of the
defined indices of their symbols. Section 7 is devoted to the proof of the index formula for
operators from the two important subalgebras ~NxN and [fl3;~]NxN of the algebra 92NxN. In
the general setting the index formula is proved in Section 8.
In a forthcoming paper we will give some applications of results obtained in this
paper to the Fredholm theory of some convolution type operators with shifts and oscillations.
2 The algebra P.I.
Fix 1 < p < co and w E Ap(T). Take 92 to be the Banach subalgebra of B := B(LP(T,w))
which is generated by all the operators aI (a E PC) and eh,~STeh,~I (h e R, A C T).
Hence 93 contains the Banach algebra g and also the ideal K: of all compact operators in
B(LV(T,w)) (see, e.g., [12]). It is clear that the algebra 92 is the minimal extension of the
algebra r which is invariant under the transformations (1.4).
Below we shall give another useful definition of the algebra 92.
Let 1~ be the one-point and 1~ the two-point compactification of R = (-co, +co).
Consider the family of the homeomorphisms
#~ : T ~ lk, t~-it_- t+ A ~
(A e T). (2.1)

444 Karlovich, Ramlrez de Arellano
Obviously, Z;I(~) = ~ g-~' 9 e ft. Put
Clearly, the operators
~(~):=[~+i,1-2/,~0(~--s ~eR. (2.2)
(~-i)
V~:L'(T,w)-+LV(R, ex), (Vx~o)(~):=~--~o ~-~, xeR, (2.3)
are isomorphisms for all ~ E T. Then in view of the equality
v~s~vf ~ = sR, (2.4)
the operator Srt is bounded on the space LP(R, ~), and hence ~ E Ap(R).
Let ~ E Ap(R). A function a E L~ is called a Fourier multiplier on LP(R, ~) if
the operator
W(a) := 5~-laY, (2.5)
given on functions in L2(R)g/LP(R, e), extends to a bounded operator on the space LP(R, e)-
It is well known that the set MP(~) of all Fourier multipliers on LP(R, ~) is a Banach algebra
under the norm
IlallM, c~) := IIW(a)IlBc.~CR.o))-
One can show (see, e.g., [4]) that M~'(~o) contorts an functions ~ C L'~(R) with ~nite tot~
variation Var(a), and
liall~,'C~) -< e:,,,~(il,~li~-~c~)+ var(~)) (2.6)
where cv, 0 is a constant independent of a.
Let PC(R) be the C*-Mgebra of functions in L~(R) which have one-sided limits
at each point of R, let C~(l~) (respectively, C~(R)) be the closure in MP(p) of the set
of all functions a G C(l~) (respectively, a E C(l~)) with Var(a) < co, and let PC~(R)
stand for the closure in MP(0) of the set of all piecewise constant functions on R which
have at most finite sets of jumps. One can show (see, e.g., [5]) that Cf(l~) C C(l~),
C~(I~) C C(R), PC~CR) C PC(R), and the Banach algebras C~(I~), C~(I~) and PC~(R)
are inverse closed in L~
Since according to (2.5)
sR = W(sgn ~), dh'S~e-'~I = W(sgn (~ - h)), (2.7)
we get from (2.4) and the equality V~,eh,~,V~ 1 = eih'rI that
~,~S~e;,U = V,-~ ~ e-'~V.,~= V~-lW(sgn(~ - h))V~. (2.8)
Consequently, the Banach algebra ~ C B(LP(T, w)) is generated by the multiplication op-
erators aI(a ~ PC) and all the operators
W~(b) := V~-~W(b)Va ()~ C r b C PC~x(R)). (2.9)

Karlovich, RamJrez de Arellano 445
3 Localization.
In this section we apply the Allan-Douglas local principle (see, e.g., [4]) to study the Fred-
holmness of operators A E 9/. As is known, an operator A C 92 is Fredholm if and only if
the coset A '~ := A + K~ is invertible in the quotient (Calldn) algebra B "~ := B/1C.
Let IC(LP(R, ~)) stand for the ideal of all compact operators on the space LP(R, ~).
By analogy with Lemmas 7.1-7.4 in [8] one can prove
Lemma 3.1. Let 1 < p < oo and ~ E Ap(l=t.).
(i) Ira e PC(R), b e PC~(I=t) and a(-t-oo) = b( = O, then
aW@, W(b)~I e ~C(LP(R, ~)).
(ii) x/~ e c(i~), b c pc~(R) or ~ e PC(R), b e c~(i~), then
aW(b) - W(b)aI e 1C(LP(R., ~)).
(iii) Ira C C(ff~), b e C~(R), then aW(b) - W(b)aI 9 K:(LP(R,e)).
Let Z be the Banach subalgebra of 92 generated by all operators of the form
aW~(b) (a 9 C(T), b 9 C;~(R), ~ 9 T). (3.1)
It is clear that/C C Z. In what follows we write A --- B if A - B is a compact operator.
For b 9 PC(R), put
b ~ := b(+~)x+ + b(-~)x_, b ~ := b- b ~ (3.2)
where X are the characteristic functions of semi-axes R = {z 9 tt : x > 0}. Then for
each A 9 T, we infer from (2.7) and (2.4) that
W~(b ~) = Y~-lW(b(+oo)x+ + b(-oo)x_)V~
= Vrl(b(+c~)P + + b(-c~)P~)Vx = b(+oo)P+ + b(-oo)P_
where P~ := (Z SR)/2 and P := (I ST)~2.
Lemma 3.2. For every Z 9 Z and every A 9 2{,
Proof. From Lemma 3.1(ii) we infer that
(3.3)
ZA ~ AZ. (3.4)
aA~_AaI for all a 9 A 9 (3.5)
/.From Lemma 3.1(ii) we also obtain that

446 Karlovich, Ramirez de Arellano
for every a E PC, A G T, and b e C0v ~ (1~).
It follows from Lemma 3.1(i) that ifa e C(T), b e PC0Va(R), and a(A) = 1,
b(+oo) = O, then
~w~(b) ~_ w~(b). (3.7)
In addition, ffA, r e T, b~ e pCV~(R), b~ e PCV (R), and b~(+oo) = b~( = 0, then
W),(b;OW.~(b~. ) ,.o 0 for A # v. (3.8)
Indeed, choosing functions ax, a~ E C(T) such that ax(A) = a~(r) = 1 and a~a,- = 0, we get
in view of (3.7) and (3.5),
Wx(b:OW~.(b~. ) = a:~W),(bx)a,.W.~(b~.) ~ a~a,W;~(b),)W.~(b,-) = O.
Moreover, if only ba e PCV~(R) and b~ e PCoV (R), then
W;,(bx)W,-(b~-) ~ W~.(b.~)Wx(bx) for all A, ~- E T. (3.9)
Indeed, for every b C PC(R) and al ~,~ C T, by (3.3),
Then (3.2), (3.10) and (3.8) imply that
w~(b o~ = w~(b-). (3.10)
W~(b~)W~(b~) = W~(~)W~(bT) + W~(b~)W,-(b ~ + W~(b~ ~
~- W~(b~)W~(bT) + W~(b~)W~(6 o) = W~(bT)W~(b:,) + W~(b~
Finally, (3.5), (3.6) and (3.9)imply (3.4). 9
Let 9X ~ := Pg/K: and Z ~ = Z/K. Lemma 3.2 implies that Z '~ is a central subalgebra
of 9U. Hence Z '~ is a commutative Banach algebra, and its maximal ideal space M(Z ~) is
homeomorphie to 12 := T 1~. The Geffand topology on ~2 is the following: a neighborhood
base of (to, z0) E T 1~ is given by
{(to, z) ca: I~-~o1

Karlovich, Ramirez de Arellano 447
local type in B. By Lemma 3.2, 92 C s Obviously, an operator A E 92 is Fredholm if and
only if the coset A ~ = A + ]6 is invertible in the quotient algebra/2~ := s
With every point (t, a~) 6 T x 1% we associate a closed two-sided ideal J,,, of the
algebra 1: ~r by
J,.~, :: { [(a,I + Wt(b~))L]~: a, e C(T), b. 60[,(1{), (3.11)
at(t) : br : 0, L e/:}
if tET, xER,
ff tET, ~:oo.
In virtue of the Allan-Douglas local principle, the coset A ~ E P.I ~ is invertible in s
if and only if for every (t, ~) E T x 1%, the coset A~,~ := A '~ + J,,~ is invertible in the quotient
algebra s := E~/J,,~.
Let g, (t 6 T) be the Banach subalgebra of 9.1 generated by singular integral opera-
tors on LP(T, w) with coefficients in PC, := PC M C(T \ {t}), and let 92, be the Banach sub-
algebra of 92 generated by the operators of the form aW,(b), where a E PC, and b E C~, (1{).
Clearly, g,,92, C s 92, D Z D/6 and, by Lemma 3.1(iii), the quotient algebra 927 := 92,/K:
is commutative.
To every point (t, a~) E T 15~ we correspond the Banach algebras
{AT, o e Z[,o: A e z,} if
~"~ := {A[,~ E s162 A E 91,} if
(t, z) E T 1%,
(t, ~) e T {~}.
(3.12)
For a E PC and t E T, let 5' be a function in PC, such that 5'(t 4- 0) = a(t 4- 0).
Analogously, for b E PC(1%) put
bC~(x) :=b(-c~)(1-tanhx)/2+b(+c~)(l+tanhx)/2, x e 1%. (3.13)
Obviously, ~( = b(+~) and, in view of(2.6), ~ e C~(P~) for an ~ e &(1%), 1 < p < ~.
Lemma 3.3. For every (t, x) E T x I{ the mapping l%~ : A ~-+ #,.~(A) defined on the
generators A = aWe(b) (~ ~ PC, b e PC;~(1%), ~ e T) 492 by
{ [a'(b(~ + 0)p+ + ~(~ - 0)p_)] L, t = ~, 9 e R,
#,,~(aW~(b)) :: [5'(b~(x + 0)P+ + b~(x - 0)P_)],:o, t C T \ {A}, 9 E 1%, (3.14)
extends to a surjective Banach algebra homomorphism ;h,~ : 92 --+ ~3,,~, and for all A E 92,
sup Ilm,.(A)ll~,,. ~ ]IA~II := inf{llA+ KII: K e ]C}. (3.15)
(*,.)eTx/t
Proof. Let (t, z) 6 T 1%. Consider the natural Banach algebra homomorphisms
u,,~ : 92 --+ 92~ --+ 92,~,., A ~-~ A '~ ~-+ A,~

448 Karlovich, Ram/rez de Arellano
where Pit, . := { A ~ t,. E s ~ : A E 92}. Obviously,
It remains to prove that
sup . [IALII ___ IIA"]I _< IIAII. (3.16)
(t,~) ~,r x P,.
I%x(A) = ~[,J r%~,(A) ~t,,, for all A E 92 (3.17)
where r := [e=,tI] "~ if x E R, and ~t,~o := [I]'E Then, by virtue of (3.11),
whence in view of (3.14) and (3.12),
J,,0 = r J,,~ ~t,~ in case z e R, (3.18)
which will imply by (3.17) that #t,.(92) = ~3,,.. Finally, (3.15) will follow from (3.17) and
(3.16).
It is sufficient to prove (3.17) only for generators of the algebra 92. Thus, let
A = aW),(b) where a E PC, b E PC~(R), A E T.
Let t = A and z E R. Put b~(r):= b(r +~), 7" E R. It is easily seen, b~ E PC~(R)
and due to (3.11),
[w~(b~ - b(~ + o)x+ - b(~ - 0)x_)] ~ E J~,o. (3.10)
On the other hand, if b E C~a(I~), b(0) = i and {~(oo) = 0, we derive from Lemma 3.1(i)
that [(a - 5~)W~(b)] ~ = 0, whence in view of (3.11),
Taking into account (3.19)-(3.20) we get
[(a -- h~)I] " = [(a - aX)Wx(1 - b)]~ E J;~,0. (3.20)
--1 7r
~:~,~,[aW),(b)] ~;~,,~ - [a~(b(z + 0)P+ + b(x - 0)P_]"
= [~w~(b~)] ~ - [a~w~(b(~ + 0)x+ - b(~ - o)x_)] ~ (3.9.1)
= [(~ - a~)w~(b~)] ~ + [a~w~(b. - b(~ + 0)x+ - b(~ - 0)x_)] ~ c J~,o,
which proves (3.17) in case t = A, 9 E R.
Let t e T \ {),} and a E R. Then, by (3.10),
where b ~ and b ~ axe given in (3.2). Hence
w~(b) = w,(b ~) + w~(b~ (3.9.9.)
C~,,: [aWx(b)]'r = ~~,2 [aWt(boo)]'r~t,~. + ~,2 [aWx(b~162 (3.23)
Consider a function 5 E C(T) such that a(t) = 1 and a(;q = 0. Since [aW~(b~ " = 0 in
view of Lemma 3.1(i) and [(1 - 5)I]" E Jt,~ by analogy with (3.20), we infer from (3.11) that
[aWx(b~ " -- [a(1 - a)Wx(b~ " E ]t,~. (3.9.4)

Karlovich, Ramirez de Arellano 449
Similarly one can prove that
Relations (3.24) and (3.18) give that
On the other hand, (3.21) implies that
[W~(b~ '~ = [(1 - a)Wa(b~ ~ ~ J,.~ if t r A. (3.25)
~,7~ [aW,(b~)Fr - [a*(b~(, + 0)P+ + b~(, - 0)P_)F e J,,o. (3.27)
Combining (3.23), (3.26) and (3.27) we get (3.17) in case t e T \ {A}, 9 e R.
If t = .~ and x = 0% then we easily deduce from (3.11) and (3.13) that
whence
[(a - a~)w~(b)V, [aXW~(b - g~)F c J~,oo
[aW~(b)F-[a~w~(g~176 ~ e J~,~
If t e T \ {A} and ~ = oo, then, taking into account (3.2) and (3.25) , we again get
[(a - a')W~(b)] ~, [a~W~(b oo - g~176 , [a~w~(b~ c Y~,~.
Consequently, using (3.22) we obtain
= [(a - agW~(b)F + [a~W~(~ ~ - ~)1~ + [a~w~(b~ ~ ~ &~. 9
Lemma 3.3, the Allan-Douglas local principle and (3.17) imply immediately the
following.
Corollary 3.4. An operator A E 9.1 is Fredholra if and only if
(a) /~t.(A) E ~3t ~ is invertibte in s for every (t, ~) e T x R, and
t~O
(b) ttt,~(A) E ~t,oo is invertible in s /or every t E T.
4 Invertibility in local algebras.
First we shall study the invertibility of cosets tt~,~(A) 6 fl3t,~ for operators A E 9.1 in the case
(t, x) E T x R. In virtue of (3.14) the Banach algebras ~3t,~ -=/z,,~(9./) C s are generated
by the identity e := [I]t~0 and two idempotents
r := [P+]t,o and s := IX I],,o (4.1)
where 2 ~ ~ PC~ = PC n C(T \ {t}) and 2~(t + O) = 1, 2'(t - O) = O.
To study the invertibility of elements b E ~3t,~ in ~3t,~ and/:t~,0 we use the following
particular version (see [3, Theorem 8.7]) of the two projections theorem established in [9],
[13].

450 Karlovich, Ramirez de Arcllano
Theorem 4.1. Let 2` be a Banach algebra with identity e and let r and s be idempotents in
2, Let further ~ stand for the smallest closed subalgebra of 2, containing e, r and s Put
x := rs~ + (e - r)(e - s)(e - ~) (4.2)
and suppose the points 0 and i are cluster points of the spectrum sps X. Define the map
~r,: {e, r, ~} --> C 2x2 for z E C \ {0, i} by
(10)
~(e) : 0 1 ' a,(r)= 0 0 , r -z 1--z
and for z e {0, 1} by
az(e) = 0 1 '
o',(r)= 0 0 , o-~(s)= 0 1-z ) "
(a) For each z E spc X the map crz extends to a Banach algebra homomorphism ~r~ of 9~
into C ~x2.
(b) An element b E ~ is invertibIe in 2. if and only if crz(b) is invertible in C 2x2 for every
z 6 sp~ X.
(c) An element b E fB is invertible in fB if and only if cz(b) is invertible in C z for every
z E sp~ X.
(d) If spcX = spX then the algebra fB is inverse closed in s
We shall apply Theorem 4.1 with 2` = s ~ = ~t,, and the idempotents (4.1).
Thus it remains to determine the spectrum of the element (4.2) in s and ~t,,.
Let 1 < p < 0% F = T, and let [F(t,e)[ be the length of the portion F(t,r (t E F,
e > 0). With a weight w 6 Ap(F) and every point t 6 F one can associate (see, e.g., [3,
Theorem 3.4 and Section 3.6]) a regular submultiplicative function Y,~ : (0, oo) -+ (0, co)
given by
1 f log w(r)]&.[ 1 / log w(T)ldvO '
(Vt~ := limsupR_+o exp Ir(t,.R)l Ir(t,n)l c,,~)
r(t,~R)
(4.5)
and its indices
log(V?w)(x) log (V,~
at := at(w) := lira fit := #it(w) := lira (4.6)
x-,o log m ' x-,o~ log x
which are called the indices of powerlikeness of the Muckenhoupt weight w at the point
t E F. As is known [3],
-I/p < ~, < #t < I/q. (4.7)
Put
vt := v;-(p, w) := l/p + at, u, + :=ut+(p,w):=l/p+flt. (4.8)

Karlovich, Ramirez de Arellano 451
In virtue of (4.7) we have
0 < v~- _

452 Karlovich, Rarnirez de Arellano
where/%(~ot) = ~ot(T \ {t}) O {0, 1} is the essential range of ~ot, and the horn 7-/(0, 1; v~-, v +)
is given by (4.10). Hence, taking into account that {0,1} C 7-/(0, 1; v~-, ~+), we get
spr A = sp=.. T(cpt) U {1} = ~pt(T \ {t}) U n(O, 1; v~-g+).
On the other hand, by Corollary 3.4,
sp~A = ( U spz
(r,~)eTxtt
Since A = ~otWx(x+) + Wx(X-), it follows from Lemma 3.3 that
[~ot ]t,o, ~ > 0
A,~,o, 9 = 0
#,,~(A) = Z~
t,0'
~ < 0 '
[~,wt(~+) + W~ ,(x-)],,oo, - ~r ~ = co
{ [~,(r)Z]~,0, 9 > o
[Tt(r)P+ + P-]~,0, m = 0
~,o(A) = /:,o, ~ < o '
[w~(~,(r)2+ x-)].,0, 9 co
where r e W \ {t} and ~(~) := (1 tanh ~)/2. It is eas~y seen that for each (., ~)
(T x 1~) \ {(t, 0)}, the cosets #.#(A) belong to commutative algebras, and therefore
{1,o}, ~>o I' {~,(r)}, 9 >o
sp#t,~(A) = {1}, x < 0 ; sp#.,=(A) = / {1}, m < 0
{1,0}, z = o~ {cpt(r),l}, m E {0, oo}
Consequently, we infer from (4.16) that
(4.15)
(4.16)
spr A = ~,(T \ {t}) U sp~ At~o U {0, 1}. (4.17)
As ~t is an arbitrary function in PCt which is only subject to the restriction (4.13), it follows
from (4.15) and (4.17) that
spz Ate,0 U {0, i} = 7-/(0, 1; v:, v+).
Since 0 and 1 are non-isolated points of 7-/(0, 1; v~-, V+), we obtain in view of the closedness
of spe A,~,0 that
spz A/, o = 7-/(0, 1; g/-, v+). (4.18)
Finally, (4.18) and (4.14)imply (4.11).
To prove (4.12) we consider the weight wt E Av(T ) and the isomorphism
Bt: LP(T,w) -+ LP(T, wt) given by
~(~-) := w(t~), (s#)(r) := f(t~), r e T.

Karlovich, Rarnlrez de Arellano 453
Let J~,o be the ideal of t~ = := s
dear that
wt))/E.(LP(T, wt)) which has the form (3.11). It is
[Bt(1 - ;~*)B~I] '~ - [:~ti]~ 6 J~,o, [BtP ~ = IF:F] '~,
whence
[Bt((1 - ~t)p_ + +l't at,0 = [:~'P+ + -],,0 (4.19)
where the cosets in (4.19) are considered in the quotient algebra/:t,o .- / t,o.
It follows from (4.5) and (4.6) that
Then in virtue of (4.8),
whence due to (4.11),
This together with (4.19) gives (4.12). 9
=,(~,) = =,(~), Z,(~,) = ~,(~).
~,~(p,~,) = v,~(v,,o) =: ~,~,
sp~,o [)~,P+ + P-],~,o = 7/(0, 1; v~-, v+).
Corollary 4.3. Let 1 < p < 0% w 6 Ap(T), ~ = s and let X be given by (4.1)-(4.2).
Then
sp~X = 7/(0,1; ~?, v,+). (4.20)
Proof. Since
we get
(1-A)(X-Ae) = /rsr+(e-r)-Ae)((e-r)(e-s)(e-r)+r-Ae)
{(e- ,)(e- s)(e- ,)+,- e)(,st + (e- r)
whence, by Lemma 4.2,
sp~XU{1} =sps U sp~ ((e -- r)(e -- s)(e -- r) + r)
p ~r ~r
= sp~ bt'P+ + -],,o u sp~ [(1 - f)p_ + P+],.0
spe X U {1} = 7/(0,1; v~-, v+). (4.21)
As 1 is a non-isolated point of 7/(0, 1;v~-,u +) and sp~X is a closed subset of C, (4.21)
implies (4.20). 9
Since the horn 7/(0, 1; vt, u +) does not separate CI i.e., its complement is connected,
it follows from Corollary 4.3 and [20, Corollary 10.18] that for (t, x) 6 T x 1~,
By Theorem 4.1(d), equality (4.22) implies the following.
spz X = sp~,,, X. (4.22)
Corollary 4.4. For every (t, z) 6 T R, the algebra flbt,~ is inverse closed in the algebra
~=s
t,0"

454 Karlovich, Ramirez de Arellano
Theorem 4.5. Let 1 < p < oo, and w E Ap(T).
(a) For every (t, x) E T R and every z E 7-/(0, 1; ut, 1:+), the map cr, given by (4.3)-(4.4)
with r, s defined in (4.1), extends to a Banach algebra homomorphism of ~t,~ into
C 2X2"
(b) For every (t, x) C W R and every A C 92, the coset #t,~(A) E fB~,~ is invertibIe in s
(equivalently, in fBt,.) if and only if
~,(tt,,~(A))7~0 forall zEn(0,1;~,~-,u+).
Proof. It is immediate from Theorem 4.1 and Corollaries 4.3-4.4. []
Now we pass to study the invertibility of cosets #t,~(A) E ff3,,r162 for A E -91 in the
case t e T. According to (3.14) and Lemma 3.1(iii), ~3,,~ = ttt,:r is a commutative
subalgebra of s The maximal ideal space Mt of ff~t,~ consists of four points, namely,
and the Geffand transform
is given on the generators
of ff3~,~ by
Mt := {(t 4- 0, +~), (t + 0,-c~)}, (4.23)
G : ~,,~ -+ C(Mt), B ~ B(t O, 4-cc)
B = (a e PC,, b e C2(f ))
B(t 0, +c~) := a(t 0)b(+c~), B(t O, -~) := a(t 4- 0)b(-eo). (4.24)
Fix a coset B E ~3,,oo. Since sp~,,~ B is a finite set and hence its interior is empty, we get
(see, e.g., [20, Theorem 10.18(b)])
sp~,,~ B = spz~," B for each B C ff3t,:o.
Hence, the algebra ~3t,oo is inverse closed in Z:,~,oo.
Thus we have obtained the following.
Theorem 4.6. For every t C T and every A E 92, the coset #t,~(A) E fBt,~ is invertible in
s (equivalently, in ~t,r162 if and only if
5 Fredholm criterion.
[#,,~(A)](t 4- 0, +c~) r 0, [#t,~(A)](t 4- 0,-oc) r 0. (4.25)
Let 1 < p < c~ and w C Ap(T). We associate the horn 7/(0, 1; ut,u +) with each point
(t, z) C T x R, and the two-point set {0, 1} with each point (t, oo), t E T. The set
is referred to as the horn bundle of LP(T, w). We equip ~ with the discrete topology. Con-
sider the algebra C(ffJ[, C 2x2) of continuous and in general unbounded functions f : gX ~ C 2

Karlovich, Ram/rez de Areltano 455
Given the generator A = aWx(b) of N with a 9 PC, b 9 PCoP(R ) and A 9 T, we
define matrix functions .,4 : ffft --+ C 2x2 by
.4(t,~,~) :--
( a(t+0)(b(~+0)z+b(~-0)0-~))
~(t-0)(b(~+0)-b(~-0))~(.)
if t = A, and
~(t+0)(b(~+0)-b(~-0))~(~) ), (5.2)
a(t-O)(b(m+O)(1-z)+b(m-O)z)
A(t,~,~) :=
( a(t+O)(b~176176176
a(t + 0)(b~(~ + o) - b~(~ - 0))~(;) "~
~(t-0)(b~176
~(t - O)(b-(~ + o)(1 - ~) + b~176 - 0)~) ) '
(5.3)
if t 6 T \ {A}, where b(oo 4- 0) = b~(oo 4- 0) = b(:Foo), and ~(z) is any number with
~(z) = 41 - z).
Theorem 5.1. For every (t, x, z) C 9J[, the mapping A ~ .A(t, x, z) given on the gen-
erators A = aWx(b) of the algebra 9d by formulas (5.2)-(5.3) eztends to a Banach algebra
homomorphism Sym t .... : ~ --+ C2X2. Then the map
where
Sym: 9A--+ C(TJI, C2X~), A~-+ SymA (5.4)
(SymA)(t,z,z) = Sym t .... A = .A(t,x,z), (t,x,z) 9 g)I, (5.5)
is an algebra homomorphism with Ker Sym D ~.
Proof. Fix the generator A = aW~(b) of P.I. Let (t, ~) 9 T R. Then in view of (3.14),
{ [(a(t+O)2t+a(t-O)(1-f(t))(b(x+O)P++b(z-O)P_)]~,o ift=A,
#t,~(A)= [(a(t+O)2t+a(t-O)(1 2t))(b~176176176 iftr
Further, by Theorem 4.5(a), for z 9 7-/(0, 1; u/-, ut+), we get
,~(#,,~(A)) --
( (~(t + o)~ + act - 0)(1 - z))b(~ + 0)
--(a(t + O) -- a(t -- O))zb(~ + O)
if t=A,
(~(t + 0)~ + a(t - 0)(1 [~))b~(~ + 0)
-(a(t + 0) - a(t - 0))zb (~ + 0)
if t#A.
(a(t + O) -- a(t -- 0))(~ -- 1)b(~ - O) '~
(a(t + 0)(1 - z) + a(t - O)z)b(o~ - O) ;
(a(t + o) - a(t - o))(z - z)b~176 - o) "~
(a(t + o)(1 - z) + ~,(t - O)z)bOo(~ - o) ;
(5.6)

456 Karlovich, Ramirez de Arellano
For z 9 C \ {0, 1}, consider the matrices
E, = E; -1 = .
It is checked immediately that (5.6) implies the identity
A(t, x, z) = E~,(#t,~(A))E~ ~ (5.7)
for z9 +)\{0,1} and (t,x) 9
On the other hand, for (t, x) 9 T x R, it follows from (3.14) and Theorem 4.5(a)
that
(r~ =
diag{a(t - O)b(x + 0),
diag{a(t - O)b~176 + 0),
a(t + O)b(~ - 0)},
~(t + 0)6"(x - 0)},
t = )%
t # ~,
(5.8)
diag{a(t + 0)b(x + 0),
o't(#,,.(A)) = diag{a(t + O)b~(x + 0),
a(t - - O)b(x
- - 0)},
t = A,
a(t-O)b~C(x-O)}, t#A.
~om (5.8)-(5.9) a~a (5.2)-(5.3) we get
(o (o
.A(t,x,O) = 1 0 cr0(#t,.(A)) 1 0 '
(5.9)
.a(t, ~, 1) = ~I(~,,.(A)). (5.11)
Finally, we obtain from (5.7), (5.10) and (5.11)in virtue of Lemma 3.3 and Theorem
4.5(a) that for every
(t, ~, z) e m := U {t} rt x n(0,1; ~;-, ~,+),
tET
the mapping A ,+ .A(t, x, z) extends to a Banach algebra homomorphism Sym t .... A of P2
into C ~x2.
Now let t E W and x = c~. For the generator A = aW~(b), it follows from (4.24)
and (5.2)-(5.3) that
~t(t, oo, 0) = diag{a(t + 0)b(+co), a(t - 0)b(-c~)}
= diag{[ttt,~(A)](t + 0, +oo), [#t,=c(A)](t - 0,-c~)},
A(t, oo, 1) = diag{a(t + 0)b(-oo), a(t - 0)b(+oo)}
= diag{[#t,~(A)](t + O,-oo), [#t,oo(A)](t - O, +co)},
(5.12)
(5.13)
where #,,~(A) ~ [#t,~(A)](-, .)is the Gelfand transform a: !B,,~ ~ C(Mt). Consequently,
(5.12)-(5.13) imply that for every
(t, ~,~) E ~\~ = T x {oo} x {o, 1},

Karlovich, Ramirez de Arellano 457
the mapping A ~ .A(t, m, z) also extends to a Banach algebra homomorphism Sym t .... A of
92 into C 2x2. Combining the results obtained for 9l and 9Yr \ 91 we conclude that for every
(t, x, z) E 92t, the mapping
Sym t .... : 9.I -+ C 2 A ~-~ A(t, m, z)
is a Banach algebra homomorphism. Then the map Sym : 9/-+ C(~Y4 C ~ given by (5.4)-
(5.5) is an algebra homomorphism. Obviously, ]C C Ker Sym by virtue of (5.7), (5.10)-(5.11),
(5.12)-(5.13), and (3.15). 9
/,From Theorems 5.1, 4.5(b) and 4.6 we immediately obtain the following Fredholm
criterion.
Theorem 5.2. Let 1 < p < oo and w E Ap(T).
(a) An operator A E 92 is Fredholm on LP(T, w) if and only if
det((Sym A)(t,m,z)) ~ 0 for all (t,m,z) E ffJt.
(b) IrA E 92 is Fredholm on LP(T,w), then its regularizer belongs to 92 also (equivalently,
92/1C is inverse closed in B/K.).
Now let us denote by 91N _> 1) the Banach subalgebra of B~vxN := B(L~v(T, w))
which consists of N x N operator matrices with entries in PA. The algebra 912vxN is generated
by operators A -- aWx(b) where a E PCNxN, b E [PC~(R)]Nx2V, A E T. Their symbols
.A : 9~t --+ C 22vx~~v are defined by (5.2)-(5.3) with N N block entries.
Replacing Theorem 4.1 by its matrix version given in [9, Corollary 3] and [13] (see
also [2, Theorem 13] and [3, Theorem 10.1]), we obtain in the same manner that Theorems
5.1 and 5.2 remain valid under the following natural reformulations.
Theorem 5.3. For every (t, m, z) E flit, the mapping A ~-+ .A(t, x.z) given on generators
A = aW~(b) of the algebra 91N by formulas (5.2)-(5.3) eztends to a Banach algebra homo-
morphism
Symt,~,~ : 9~2vx2v --+ C ~Nx2N.
Then the map
Sym : 91N -+ C(ff)~, C2Nx2N), A ~+ SymA,
where SymA is given by (5.5), is an algebra homomorphism with Ker Sym D K:2v where
]CNxN is the ideal of compact operators in 13NxN.
Theorem 5.4. Let 1 < p < 0% w E Ap(T) and N >_ 1.
(a) An operator A E 92~r is Fredholm on L~v(T,w ) if and only if
det((Sym A)(t,m,z)) r 0 for all (t,x,z) e 937.
(b) The quotient algebra 91~VxN/ICNx/V is reverse closed in BNxiV/ICNxN.

458 Karlovich, Ram/rez de Arellano
6 Index formula.
In this section we define an analogue of the Cauchy index, wind A #, for the symbols .A =
SymA of Fredholm operators A C P2NxN and state an index formula for these operators in
terms of that symbol index (see Theorem 6.5 below). The proof of this index formula will
be given in the next two sections.
First of all note that the family of homomorphisms o'~ defined in Theorem 4.1, in
general, is not uniformly bounded. Consequently we cannot guarantee the uniform bound-
edness of the symbols M = Sym A on ff)~. Nevertheless their determinants are uniformly
bounded.
Writing the symbols A of operators A E ~N in the form
= ( A~ 2
(0.1)
with N N blocks A/j, and using Lemma 3.3 one can prove the following two lemmas by
analogy with Theorems 2 and 3 in [15].
Lemma 6.1. IrA E ~Nxnr, then
where c := NSlt .
sup ]det A(t, z, z)[ < (2N)!(O[]A~[]) 2N, (6.2)
sup [detA~,(t,z,z)l_< (N)!(CI[A'ql) ~v, /= 1,2, (6.3)
Lemma 6.2. If a sequence of operators A ('0 E 9..INxtr converges uniformly to an operator
A E 92N and A ('~) = Sym A('0, then the sequence of functions det .A('q (respectively,
de~ ~), det .A~'~ )) converges uniformly on ~ to det .A (respectively, to det .An, det J[~2).
Now the set ~J~ given by (5.1) we equip with the following topology. Let its neigh-
borhood base consist of the sets u(to, zo, Zo), (to, ~o, zo) E ff)~, which have the form
{(to, Zo)} x Vto,~o if Zo E R, Zo E 7-/(0, 1; v~, ,+);
if zoER, zo=0;
({(t0,~0)} Vt0,0 U ({t0} (z0, z0 +e) 7-/(0, 1; v~,v+))
u(to, zo,Zo)= if ZoER, zo=l;
where to E T, r > 0, T_ -~ to -~ T+ on T, and
if ~o=oo, zo=0;
if zo=OO, zo=l;
~o,~ := 7-/(0, 1; .~,.~) n {z e C: [z - zol < ~},
(6.4)

Karlovich, Rarrdrez de Arellano 459
/ \
9Y~t := {t} ((R x 4(0, 1; ~:, ~t)) U ({~} {0,1})). (6.5)
Then 9~ becomes a compact Hausdorff space.
Write symbols .A of operators A E PXNxN in the form (6.1). If A E 92~r is
Fredholm, then in view of Theorem 5.4 and formulas (5.2)-(5.3) we get
det A(t, ~, z) # 0 for (t, ~, z) E 9X, (6.6)
detJ~,(t,~,z)#0 for (t,z,z) ETxl~x{0,1},ie{l,2}. (6.7)
Hence to every Fredholm operator A @ PgN we can associate the function
det g(t, x, z) det Am(t, c~, 1)
.A(t,x,z) := detAm(t,a,O)detAm(t,a, 1)det.An(t,c%l)' (t,a,z) E 9)I, (6.8)
which is finite-valued on ffJ[ by Lemma 6.1 and (6.7), and does not have zeros on 9)I by (6.6),
(6.7) and (6.3). Moreover, the function (6.8) is uniformly bounded on ~.
Indeed, the set T 1~ x {0, 1} with the following topology compatible with topology
(6.4): a base of open sets consists of the sets
({t} x {o, i}) u x h x {o,1}) u o)},
({t} x (a, fl) x {O, 1}) U {(t, a, 1)}, ((v, 4) x 1~ x {O, 1}) U {(t, oe, 1)},
where --oo < a

460 Karlovich, Ramirez de Arellano
and the matrix functions ai, bi have at most finite sets of discontinuities. Then denoting
c := s
where
i=1
we derive from Theorem 5.4 that
Since
d,(t,,) := ~,(t (t,~) e T (6.11)
i=1
detc 162 detd.r on TxI~. (6.13)
det X(t, ~, z) = det(

Karlovich, Ram/rez de Areltano 461
Since A(t,.,-) e C(ff]~) for all t E T, we derive from (6.15)-(6.16) that A @ ((if)I).
Now let
A = E A~IAI2... A~ (6.17)
i=1
where A~j (i = 1,... ,n; j = 1,... ,r) are of the form (6.10). Then using the linear dilation
.4 of the operator (6.17) defined in [12] and written in the form (6.10), we get that
n
Sym A = E (Sym A~I)(Sym Ai2) 9 9 9 (Sym A~,)
i=1
and Sym 2~ are connected by the identities
detSymA = detSym.4, det Jl~; = det ./L,'~ (i = 1,2), (6.18)
where SymA ( "J)i,i=l, Sym.4
: : ( "/)i,j=l" Then the already proved continuity of the
function (6.8) for the linear dilation .4 implies that the function (6.8) corresponding to the
operator (6.t7) is also continuous on 92.
Finally, let A be an arbitrary Fredholm operator in PlNxN. Then A is the uniform
limit of a sequence A(') of Fredholm operators of the form (6.17) with symbols A('). Since
the functions (6.9) are separated from zero, we obtain from Lemma 6.2 that the sequence
det .A,(')(t, m, z) det ~)(t, co, 1)
Ar ~, z) := det ~)(t, ~, 0) det ~t~;)(t, ~, 1) det.4~)(t, co, 1)
converges uniformly on 9X to A(t, x, z) given by (6.8). But .~(') E C(~), and hence
A 60(g)t) as well. "
Thus Lemma 6.3 and inequalities (6.6)-(6.7) imply that for every Fredholm operator
A E ~[NxN, the function A given by (6.8) is continuous and separated from zero on ~Y~.
Finally, (6.8) and the relation
~tCt, ~, ~) = ~ag{~Ct, ~, ~), .~(t, ~, ~)}, (t, ~, =) e a: 1~ {0, 1},
immediately imply the following assertion.
Lemma 6.4. If A and B are Fredholm operators in 91NxN and C = AB, then
where
~(t, ~, ~) = A(t, ~, ~) ~(t, ~, ~), (t, ~, ~) e ~.
Along with 93I we consider the set
:= (U{t} u (T {co} {o,1}) (6.19)
~, = (~ + ~,+)/2, t e T. (6.20)

462 Karlovich, Ramirez de Arellano
Fix a Fredholm operator A E 9/N It is clear that 0 ~ Jr(92#). Show that ./i(92 #)
is the graph of a bounded, closed and continuous curve A # in the complex plane.
For t E T, define A0(t) := A(t, co, 0). It follows from (6.16) that the limits A0(t 0)
exist at each point t 6 T, and
Ao(t - O) = A(t, co, 1), Ao(t + 0) = .4(t, co, 0). (6.21)
Thus these limits are finite, nonzero, and Ao(t + O) = Ao(t).
For t E T, consider the functions
Lemma 6.3 gives (see (6.11), (6.12), and (6.14)) that for every t E T, the limits At(z 4- O)
exist at each point z E R, they are finite, nonzero, and At(x - O) = At(x) for z E R,
At(co - O) = Ao(t + O), At(co + O) = Ao(t - 0). (6.22)
Denote by J0 the points t E T at which the functions .A(t,-, .) are not constant on
1~ x 7-/(0, 1; u~-, u+). It is clear that the set J0 is at most countable. Denote by Jt (t E T)
the points $ E R at which the functions
At,z: 7/(0, 1; ui-, u+) -~ C, z ~ A(t, z, z)
are not constant. It is clear that the sets .It also are at most countable. For each t E J0 and
each z E Jr, join the one-sided limits At(z 4- O) of the function At at z by the continuous
curve
A~ := {At,,(z) : z e 7/(0, 1; fit, ~t)}, (6.23)
and orient this curve from At(z - O) = .A(t, z, 0) to At(z + O) = (t, z, 1). This construction
gives us the continuous and oriented curve
At # := At(R \ Jr) O {At(co =k 0)} U U A~ (6.24)
joining the points At(co + 0) and At(co - 0) in case t E Jo. In view of (6.22) the curve At #
joins the one-sided limits Ao(t - 0) and Ao(t + 0) of the function A0 at t E J0. Thus, for a
Fredholm operator A E ~NxN, we get the bounded, closed, continuous and oriented curve
in the complex plane. Obviously,
a# := A0(W \ J0) u [J At# (6.25)
tEJo
A #=4(92 # ) and 0~A #.
Theorem 6.5. Let 1 < p < co, w E Ap(T), and N > 1. If an operator A E 92~r162 is
Fredholm on the space L~(T, w), then
Ind A = -wind A # (6.26)
where wind A # denotes the winding number of the curve A # about the origin.
The integer wind A # is a desired analogue of the Cauchy index for the symbol .4 of
a Fredholm operator A C 922VxZr

Karlovich, Rarnirez de Arellano 463
7 Special cases.
First we prove Theorem 6.5 for operators A C ~:nxn and operators in the Banach subalgebras
[~I~]N C ~nxn (A E T), generated by operators of the form
v;~(x+w(b+) +x_w(6_))v,
where b 6 [PC~,(R)]N and X are the characteristic functions of R
Theorem 7.1. Let 1 < p < oo and w E Ap(T). If an operator A E en is Fredholra on
the space L~(T, w), then its indez is calculated by the formula (6.26).
Proof. Following [3, Theorem 10.2], for (t,z) 6 T x C, define the map
by
Sym,,~ : {P+} u {a.r: a e Pen -~ C ~nx~n
Symt,~(P+) = --ZlN (1 z)In , z E (3 \ {0, 1}, (7.1)
a~ag{zIn, (1 - ~)~n}, ~ e {% 1},
where In stands for the N x N identity matrix. Put
Symt,~(aI) = diag{a(t + 0), a(t - 0)}. (7.2)
M := U ({t} ~(0,1;.;,.+)).
tGT
By Theorem 10.2 in [3], for each (t,z) 6 M the map Symt,~ extends to a Banacl~ algebra
homomorphism
Symt,, : Enxn --~ C 2nxzn.
Comparing (7.1)-(7.2) with (5.3) and using the equality (8.8) in [3] we see that for all
A 6 gNxN,
where
{ k;lx(t, o, z)kz , (t, ~) e \ (T {0,1}),
Symt"(A) = Jr(t, 0, z) , (t, z) E T x {0, 1},
/~, = diag {~r~(1 -z)Iiv, -~/(1-z)/zIiv}, z E C\ {0,1},
and ~ - z) denotes any number with = -
imply
Let A E eN be a Fredholm operator. Then the relation (7.3) and Theorem 5.4
(7.3)
det (Symt,z(A)) 0 for all (t, z) e A4. (7.4)
Following [3, Chapter 10.2] (see also [1]), for the symbol
An(t,z) A12(t,z) ) (t,z) EM,
Symt'~(A) = A21(t,z) An(t,z) '

464 Karlovich, Rangrez de Arellano
of A E g~rx2V put
det (Symt,~(A)) (t, z) E ~A, (7.5)
A(t, z) = det (An(t, O)Az~(t, 1))'
where the matrix functions An(', 0) and A22(', 1) are invertible in PCNxN in view of (7.4)
and of the block diagonal structure of Symt,0(A ) and Symt,l(A ). Denote by A0 the points
t E T at which the functions A(t, .) are not constant on "]-/(0,1; u~-, u+).
Denote by A~ # the closed, continuous, and naturally oriented curve resulting from
the essential range of the function Ao := A(., 1) by filling in the curve
{A(t, z): z E n (0, 1; ft, fit)}
between the points Ao(t - O) = A(t, 0) and Ao(t + O) = A(t, 1) for each t E Ao, where ut are
given by (6.20). Obviously, A, # = A(M #) where
M # := U({t} x 7-/(0, 1; f,,ft)),
tET
and 0 ~ A~. By Theorem m.6 in [3] (the m~n steps of its proof are in [1]), we get the
formula
Ind A = -wind A~. (7.6)
Notice that in general case for the validity of (7.6) for every Fredholm operator A E G/Vx/V
we need to replace in [3, Section 10.2] and [1, Section 12] the set of discontinuities of the
function A(.,O) by the set A0 of the points t E F at which the functions A(t,.) are not
constant.
In virtue of (7.3) we have
Since for A C ~nx~r,
det (Symt,,(A)) = det fl,(t, 0, z), (t,z) E M, (7.7)
An(t,z)=An(t,O,z), g2~.(t,z)=A22(t,O,z), (t,z) ETx{0,1}. (7.8)
J~i(t, co,1) = flq,(t, 0, 0), teT, i=1,2,
we conclude from (7.7)-(7.8), (7.5), and (6.8) that
Taking into account that for A E r
we obtain from (7.9) the relation
A(t, z) = A(t, 0, z) for all (t,z) E M. (7.9)
A(97t# ) = fi- Cc[~.Jw({(t, 0)} x 7{ (0,1; ft, f~)))
A # := A(gJt #) = A(Ad #) =: A~. (7.10)
Finally, from (7.6) and (7.10) we get (6.26). ,,

Karlovich, Ramirez de Arellano 465
Theorem 7.2. Let 1 < p < 0% w C Ap(T), and A E T. If an operator A E [~]NxN is
Fredholm on the space L~v(T,w), then its index is calculated by the formula (6.26).
Proof. We will follow the scheme of the proof in [1, Section 12.2] (see also [12] and [15]).
Step 1. Let N = 1. First we prove (6.26) for the Fredholm operator
A = V~I(x+W(b+) + x-W(b-))V~, (7.11)
where b E PC~x (R) and b have only finite sets of discontinuities on R. To this end we
consider the Fredholm operator
It is clear that
B := x+W(b+) + x-W(b-) e B(L~(R, e~)).
Ind A = Ind B. (7.12)
Since A E ~ is Fredholm, Theorem 5.2(a) and formulas (5.2)-(5.3) imply that
Then
b b xeR. (7.13)
b := b+/b_ 9 PCoP(R ).
Obviously, the Fredholmness of B on Lv(R, Qx) implies the Fredholmness of the Wiener-Hopf
operator
W~ := x+W(b)x+I 9 t3(LP(R+, ~)),
and, moreover,
Ind B = Ind W~ (7.14)
According to [5], [6] let b,, denote the set resulting from the essential range of
b : R --+ C by filling in the horns
n(b(z + 0), b(, - 0); v:~(0~), vs
between b(x + 0) and b(z - 0) for z 9 R and the horn
7~(b(-~), b(+~); ~; (e~), ~o+ (a) )
between b(-cr and b(+oo) (note that W(b) = JrbY-1 with b(z) = b(-z) in notation of [5],
[6]). Here
(-u:o(o~), 1 - u+(o~)) := {# 9 R: [z - il-.~ 9 AdR)}.
By Theorem 5.2 in [5], the operator W~ is Predholm on/_?(It+, ex) if and only if 0 ~ b,~,
and
Ind W~ = -wind b~ (7.16)

466 Karlovich, Ramirez de Arellano
where b~ is the closed, continuous, and naturally oriented curve resulting from the essential
range of the function b by filling in the arcs
7-/ (~(~ -0), ~(~ + 0); (~(a)+ ~+~(a))/2, (~(a) + ~+~(a))/2)
between ~(~ -- 0) and ~(~ + 0) for 9 e a and t~e ~c
n (~(+o0), ~(-oo); (~o(a) +"o+(a))/2, ("o(~)+ Vo+ (a))/2)
between b(§ and b(-oo). It is clear that b~ # C b~.
By [22] (see ~so [3, p. 115] and cf. (7.15)), for every t e T,
(-g?,l- v +) = {# e R: IT- tI~w 6 dr(T)}.
Since the operators al + bST 6 B(Lp(T, w)) and
V;~(aI + bST)V~ 1 = (a o fl~l)I + (b o t3~1)SR E B(Lp(R, 8x))
with piecewise continuous coefficients are Fredholm only simultaneously~ we conclude that
where u:~ x = 1/p + o~ ~,+~ = 1/p + fl in view of (4.8). Consequently, using (7.17) and
(6.20) we can rewrite b~ and b~ # in the form
Put
~ = n (b(-oo), b(+o~); ~:. ~&) u U ~ (6(~ + 0),b(~ - 0); ~;, ~+),
~EB.
zEB.
J~4~ # := ({(-A, 0)} x 7/(0, 1; ~_~, ~_~)) U U ({(A, z)} x 7/(0, 1; ~, D~)). (7.19)
For the operator A e !Ux given by (7.11) we get from (6.11), (6.12), (6.14) and (6.15) that
l(~,~,~)= f6_(~+0)~ ~_(~-0)0_ @ ~+(+oo) (~,~,~)e~,
\~+(~+0) + ~+(~ 0) ~_(-oo)'
(~7(~+o)~ ~7(~-o) ) ~-(+~) (-:~,~,~)e~_~,
A(-~,~,0 = \~r(~+0) + ~r(~-0) 0-0 ~+(-oo)'
z6R
~_(+o~) (t,~,~) e U ~-,
A(t,~,z)- ~_(-oo)'
X(t,~,~)- b+(+oo) (t,:~,.) e U ~--
b+(-oo)'
--:k-4~--4),

Karlovich, Ramirez de Arellano 467
Hence for the operator (7.11) we have Jo = {+A} and J_~ = {0} because
It is clear that for all z E R
and
Thus
A(-~,~,z) = ~ b_(+00)/b_(-00), ~ < 0,
[ b+(+00)/b+(-00), 9 > 0.
b+(+00)
A #~,, = 7E(b-l(z - 0), b-l(a: + 0); ~x, ~) b_ (-00)
6_(+00)
A_~,0 = 7-/(b(-00), b(+00); ~_~, ~_~) b+(-00)"
A # := fi.(ff~#) A(A~ #) =: A # (7.20)
It is easily seen that
wind A Xjt~ # = wind b~. (7.21)
Therefore from (7.20), (7.21) and (7.12), (7.14), (7.16) we get (6.26) for the operator (7.11).
Step 2. Now let N > 1 and A E [~]~vxN be given by (7.11) with b E [PC~(R)]NxN,
where again b. have only finite sets of discontinuities on R. Then the Fredholmness of the
operator A e B(L~(T, w)) implies the Fredholmness of the operator
~r := v~(x+w(b) + x_z)v~
where b = b+b --~ and det (b + 0)b - 0)) # 0 for all x E t~. Moreover, in view of the
invertibility of the operator W~(b_),
Ind A = Ind I~d~(b). (7.22)
Obviously, the set A(b) of the points t C F at which b has a jump is finite. By [12,
Lemma 2], the matrix function b can be represented in the form
= fcg (7.23)
where f and g are inverfible matrix functions in [C~(I~)]N and c is an invertible uppertriangular
matrix function in [PCPo~ (R)]~rxN whose diagonal entries c~, ..., c jr are continuous
on A(b). Then in virtue of Lemma 3.1(ii),
~-l(x+W(b)x+I + X_I)V~ = 17V:,(f) V~-I(x+W(c)x+I + X-I)Vz 17VA(g) + K (7.24)
where K E K:NxN. Since for d E [PC~(R)]NxN,
where
Vg~(x+W(d)x+ + x_I)V~ = H~ r (7.25)
Hd= I -- VZ1x+W(d)x_V~ and H~ 1 = (I + V~~x+W(d)x_V~,),

468 Karlovich, Ramlrez de Arellano
we derive from (7.24) that
Ind 17tin(b) = Ind I~x(f) + Ind I~x(e) + Ind I7r (7.26)
An iavertible matrix function h C [C~x(R)]NxN is homotopic to the diagonal matrix
function diag{det h, 1,..., 1} within the invertible matrix functions in [C~ (R)]NxN (see, e.g.,
[18, Appendix VI by B. Bojarski]). This implies that
By step 1,
Ind l~A(f) = IndlYC-x(det f), Ind #~(g) = Ind l/V~(detg). (7.27)
Ind W~(det f) = -wind P#, Ind r162 g) = -wind O #, (7.28)
where/~# and 0 # are the bounded, dosed, continuous and oriented curves corresponding
to the operators/~ = l/~r~(det f) and 0 = I~(detg) of the form (7.11). Obviously,
p# = F#, 0 # = G #
where the curves F# and G # are associated to the operators F = ff'~(I) and G = ff'~(a).
Therefore, we conclude from (7.27) and (7.28) that
it follows that
Ind r = -wind F #, Ind ff'~(g) = -wind G #. (7.29)
Consider the operators ffi(cj) ~ ~, J = 1,..., W. Because due to Theorem 5.4(a),
det Sym I/lr~,(c) # 0 for all (t, m, z) E 93t,
det Sym l~x(cj) # 0 for all (t, m, z) E 93I and all j.
Thus, by Theorem 5.2(a), the operators l/~rA(cj) are Fredholm. Then
N
Ind ~;V-x(e) = ~ Ind I~rx(ci). (7.30)
d=l
Again, by step 1, as lTV~(cd) have the form (7.11),
Ind ~(~j) = -wind Cf (7.31)
where Cf is the bounded, dosed, continuous and oriented curve corresponding to the oper-
ator Cj = I~r~(ej). Obviously, setting C = IJ/~(e) we get
It follows from (7.30)-(7.32) that
N
d=l
wind C f = wiud C*. (7.32)
Ind 17v'~(c) = -wind C #. (7.33)

Karlovich, Ramirez de Arellano 469
Setting t3 = I4rx(b) and using Lemma 6.4 we infer from (7.24) and (7.25) that
where
Therefore
wind Hff + wind B # = wind F # + wind H~ # + wind C # + wind G #
wind Hb # = wind H~ # = 0.
wind B # = wind F # + wind C # + wind G #.
Finally, (7.26), (7.29), (7.33) and (7.34) lead to the equalities
Ind 14rx(b) = -wind F # -wind C # - wind G # = -wind B #.
Setting D = W:,(b_) we derive from (6.14) that
Then, by Lemma 6.4,
whence
:D(t,m,z) = det b_(+oo)/det b_(-oo) for all (t,$,z)e !~.
A(t, z, z) = 13(t, ~, z)~(t, z, z) = 13(t, z, z) det b_ (+cr b_(-cr
(7.34)
(7.35)
wind A # = wind B #. (7.36)
Formulas (7.22), (7.35) and (7.36) give (6.26) for the considered operator A E 9~uxN.
Step 3. Let the Fredholm operator A E [~3a]~rxlv be given by
n
A = E AilAi~.... A~ (7.37)
/=1
with Aii E [fl3x]~rx~v (i = 1,...,n; j = 1,... ,r) of the form (7.11). Then using the linear
dilation .zi of the operator (7.37) we get
and derive from (6.18) that
whence
Ind A = Ind 2~ (7.38)
ACt, = for (t,
wind A # = wind A#. (7.39)
Taking into account that the operator .4 also can be written in the form (7.11) with matrix
functions b:~ satisfying the conditions of step 2, we get
Ind A = -wind A #. (7.40)
Finally, (7.38)-(7.4O) give (6.26) for the operator (7.37).
Step 4. To complete the proof, we remark that any Fredholm operator A C [~X]N
can be approximated by operators of the form (7.37) and that Ind A as well as wind A # are
stable in the operator norm. The latter follows from Lemmas 6.1-6.3 and inequalities (6.6)-
(6.7).,

470 Karlovich, Ramirez de Arellano
8 General case.
This section is devoted to the proof of Theorem 6.5 for any Fredholm operator A E P-L2VxN.
For given A E T, let ,:T;~ be the dosed two-sided ideal of the algebra ~ZCxN generated
by the operator
Hx := iXP+(1 - ~)Z - (1 - iX)P+~I (8.1)
where )~ are the functions in PC~ such that )~(A + 0) = 1 and :~(A - 0) = 0. Obviously,
the symbol 7-lx of Hx does not vanish only on the set {A} x {0} x 7-/(0, 1; v~-, v+).
Let A E 92Nx2v be a Fredholm operator of the form (6.10), that is,
A = ~ aiWx,(bi),
i=1
where a~ E PCZCxN, b~ E [PC~ (R)]2VxN, A~ E T, and matrix functions a~, bl have at most
i
finite sets of discontinuities. To this operator and given operators Ki E .:T~ we associate the
operator
where
B := a+P+ + a_P_ + E K~ E E2v (8.2)
i=1
a. := ~ albi(4-c~) E PCNxN. (8.3)
i=l
Lemma 8.1 IrA E 91~x2V is a Fredholm operator of the form (6.10), then there exist oper-
ators K~ E J~i such that the operator (8.2) becomes Fredholm.
Proof. First let N = 1. Choosing Ki = c~H~, (i = 1, 2,..., m) in (8.2), where ci = coast and
Hx i is given by (8.1), we get
rlz
B := a+P+ + a_P_ + E c'(x'P+(1 - Xi)I - (1 - xl)P+xd) E r (8.4)
i----1
where X, = )~'. By (5.2)-(5.3) and (8.3),
B(t,~,z)=A(t,::,z) if (t,~,z) EgX\O({A,}xR 1;v~,v~)), (8.5)
i=1
B(A,, x, z) = diag{a+(Ai + 0), a - 0)} if (4-~, z) E (0, +oo) 7-/(0, 1; v~, v~), (8.6)
and
B(Ai, 0, Z)
( a+(Ai + 0)z + a_(A, + 0)(1 - z)
(a+(A, -- O) -- a_(A, - O) - ci)~(z)
(a+(Ai+O)-a_(Ai+O)+cl)~(z))
a+(Ai-O)(1-z)+a_(A~-O)z (8.7)
ifz E'P/(0,1;v~,v~), i = 1,2,...,m.
We have
detB(A,,O,z) = a+(A, + 0)a_(A, - 0)z + a+(A,- 0)a_(A, + 0)(1 - z)
+ (c~ + c~[a+(A, + 0) - a+(A, - 0) - a_(Ai + 0) + a_(Ar - 0)]) z(1 - z).
(8.8)

Karlovich, Ram/rez de Arellano 471
Since the operator A is Fredholm, it follows from Theorem 5.2(a) and the relations (8.5)-(8.7)
that
and
detB(t,m,z)r for (t,m,z) E93Ii0({(A,,0)}xT/(0,1;u;,,~)) (8.9)
act ~(a,, O, z) = det ~(a,, oo, 1 - z) # 0 for ~ e (0, 1}.
Then it is clear that for every i = 1,2,... ,ra there exists a constant ~ C C such that
i=1
a+(A, + 0)a_(A~ - 0)z + a+(A~ - 0)a_(A, + 0)(1 - z) + ~,z(1 - z) # 0 (8.10)
for aU ; e 7/(0, 1; v;,, ~). Tamug into account (8.8), (8.10) and solving the equatious
c~ + ~[~+(~, + o) - ~+(~, - o) - a_(~, + o) + ~_(~, - o)] = ~,
we always can find constants c~ E C such that
det B(A,, 0, z) # 0 for all z e 7/(0, 1; v~, v~), i = 1, 2,..., m. (8.11)
By (8.9), (8.11) and Theorem 5.2(a), the operator B ~ 92 with the above chosen constants
ci is Fredholm.
Now let N > 1. Since the operator A is Fredholm, the matrix functions a=~ given
by (8.3) axe invert\hie. Put b = a-la+ and consider a representation b =fcg analogous to
(7.23), where f, g are invertible matrix functions in [C(T)]NxN and c is an invertible upper-
triangular matrix function in PCNxN with a finite set of discontinuities. This representation
implies that
bP+ -6 P- ~- (fP+ -6 P_)(cP+ -6 P_)(I - P_cP+)(gP+ -6 P_)(1-6 P_bP+) (8.12)
where the operators fP+ "6 P- and gP+ -6 P_ are Fredholm, while the operators I - P_cP+
and 1-6 P_bP+ are invertible. The latter gives that the symbols of the operators bP+ -6 P_
and cP+ -6 P_ together with the symbol A of the operator A are invertible matrices for every
(t, m, z) E ff)I \ !)7~A where
~ := 0 ({~,} {o} n(0,1; ~;,, ~D).
i=1
Since the symbol of the operator eP+ -6 P_ is invertible outside of the set !ff~A and
c is an invertibie upper-triangular matrix function in PCNxN, it follows from the already
considered case N = 1 that there exist diagonal matrices c4 E C NxN such that the operator
is Fredholm. Substituting
T := cP+ -6 P_ -6 ~ c~H~,, e ~IVxlV (8.13)
i=l
K, := a-(fP+ + P_)c,H~,,(I - P_cP+)(gP+ -6 P_)(I + P_SP+) E J~,, (8.14)

472 Karlovich, Ramirez de Arellano
in (8.2) we derive from (8.12)-(8.14) that
B ~- a_(fP+ + P_)T(I - P_cP+)(gP+ + P_)(I + P_bP+).
Hence B 6 ~NxN again is a Fredholm operator. 9
Proof of Theorem 6.5. Let A 6 92NxN be a Fredholm operator of the form (6.10).
By Lemma 8.1, there exists a Fredholm operator B 6 ENxN given by (8.2) with Ki E ff~.
Let B (-1) be its regularizer. Then, by Theorem 5.4(b), the operator B (-1) E PdN and
where
=B (-1) B- KiT aiW~i(b =I+ Di
i=1 i=l i-----i
(8.15)
Di :-- B (-1) (aiW;~,(b ~ - Ki). (8.16)
For every A E T, let Zx be the closed two-sided ideal of the algebra [~]NxN
generated by all the operators of the form W~,(b) where b E [C~x(I~)]NxN and b(oo) = 0. It
is easily seen on the basis of Lemma 3.1 that
CZ;~ = Z~, for all C 6 9IN (8.17)
Show that every operator Di belongs to the ideal Z~ C [~]N Indeed, W~,~(b ~ E
Zx, because b ~ E [PC~,(R)]Nx2V and b~177 = 0. Therefore using (8.17) we conclude that
B(-1)aiWx,(b ~ 6 Z~,, as B(-1)a,I 6 i21.N Further, by (8.1) and (3.3),
HA, = x~W~,(X+)(1 - xdZ - (1 - xdw~,(x+)xd
where Xi = 2 x', 2+ E C p (fit) and
#A i
By Lemma 3.1(iii), the operator
= xiW~,(2+)(1 - xdX - (1 - xdw~,(2+)xd
+ x~W~,(x+ - 2+)(1 - xdZ - (1 - xdW~,(x+ - 2+)xd,
(X+ - 2+)(+~ = 0. (8.18)
xiW~,i(2+ )(1 - Xi)I - (1 - xi)Wxi(X+ )XiI
is compact and hence belongs to the idea/Z;~ i. In view of (8.18) the operator
xiW~,(x+ - 2+)(1 - xdZ - (1 - xdW~,(x+ - 2+)xd
also belongs to Ixi. Thus H~ E Ixl and due to (8.17), ff~,i C I~. Hence Ki, B (-1) E I~
and according to (8.16), Di E I~,~ too.
Since D~ 6 Ix, and D~Dj ~- 0 for i # j, it follows from (8.15) that
zrL
A _ B H(I + D,). (8.19)
i=1

Karlovich, Ramirez de Arellano 473
By Theorem 7.1, for the operator B E gNxN we have
Ind B = -wind B #. (8.20)
By Theorem 7.2, for the operators Ci := I + D~ E [~I]NxN we get
Consequently, (8.19)-(8.21) yield
Ind C, = -wind C~. (8.21)
m
Ind A = Ind B + ~ Ind C, = -wind B # - Z wind C? = -wind A #, (8.22)
i=1 i=l
which completes the proof for operators of the form (6.10).
Since every operator A C P2NxN can be approximated by operators of the form
(7.37) with A~j of the form (6.10), we can complete the proof using the linear dilation and
the stability arguments by analogy with steps 3-4 of the proof of Theorem 7.2. 9
References
[1] Bishop, C. J., B6ttcher, A., Karlovich, Yu. I., and Spitkovsky, I., Local spectra and
index of singular integral operators with piecewise continuous coefficients on composed
curves. Math. Nachr. 206 (1999), 5-83.
[2] B6ttcher, A., Gohberg, I., Karlovich, Yu., Krupnik, N., Roch, S., Silbermann, B.,
Spitkovsky, I., Banach algebras generated by N idempotents and applications. Oper-
ator Theory: Advances and Applications 90 (1996), 19-54.
[3] B6ttcher, A. and Karlovich, Yu. I., Carleson Curves, Muckenhoupt Weights, and
Toeplitz Operators. Birkhs Verlag, 1997.
[4J B6ttcher, A. and Silbermann, B., Analysis of Toeplitz Operators. Akademie-Verlag, 1989
and Springer-Verlag, 1990.
[5] BSttcher, A. and Spitkovsky, I. M., Wiener-Hopf integral operators with PC symbols
on spaces with Muckenhoupt weight. Revista Matemdtica 1-beroamerieana 9 (1993), 257-
279.
[6] B6ttcher, A. and Spitkovsky, I. M., Pseudodifferential operators with heavy spectrum.
Integral Equations Operator Theory 19 (1994), 251-269.
[7] Duduchava, R. V., Integral equations of convolution type with discontinuous coefficients.
Soobshch. Akad. Nauk Gruz. SSR 92 (1978), 281-284 [Russian].
[8] Duduchava, R. V., Integral Equations with Fixed Singularities. Teubner Verlagsge-
sellschaft, 1979.
[9] Finck, T., Roch, S., and Silbermann, B., Two projection theorems and symbol calculus
for operators with massive local spectra. Math. Nachr. 162 (1993), 167-185.
[10] Garnett, J.B., Bounded Analytic Functions. Academic Press, 1981.

474 Karlovich, Ramlrez de Arellano
[11] Gohberg, I. and Krupnik, N., Systems of singular integral equations in weight spaces
Lp. Soviet Math. Dokl. 10 (1969), 688-691.
[12] Gohberg, I. and Krupnik, N., Singular integral operators with piecewise continuous
coefficients and their symbols. Math. USSR Izv. 5 (1971), 955-979.
[13] Gohberg, I. and Krupnik, N., Extension theorems for Fredholm and invertibility sym-
bols. Integral Equations Operator Theory 16 (1993), 514-529.
[14] Hunt, R., Muckenhoupt, B., and Wheeden, R., Weighted norm inequalities for the
conjugate function and Hilbert transform. Trans. Amer. Math. Soc. 176 (1973), 227-
251.
[15] Karlovich, A. Yu., On the index of singular integral operators in reflexive Orlicz spaces.
Math. Notes 64 (1999), no. 3-4, 330-341.
[16] Karlovich, Yu. I., The Noether theory of a class of operators of convolution type with
a shift. Soviet Math. Dokl. 36 (1988), 16-22.
[17] Karlovich, Yu. I., C*- algebras of operators of convolution type with discrete groups of
shifts and oscillating coefficients. Soviet Math. Dokl. 38 (1989), 301-307.
[18] Muskhehshvili, N. I., Singular integral equations. Noordhoff, 1953. Extended Russian
edition: Nauka, Moscow, 1968.
[19] Prhssdorf, S., Some classes of singular equations. North-Holland Publ. Comp., 1978.
[20] Rndin, W., Functional Analysis. McGraw-Hill, Inc., 1973.
[21] Schneider, PL., Integral equations with piecewise continuous coefficients in LP-spaces
with weight. J. Integral Equations 9 (1985), 135-152.
[22] Spitkovsky, I. M., Singular integral operators with PC symbols on the spaces with
general weights. J. Funct. Anal.. 105 (1992), 129-143.
Departamento de Matems
CINVESTAV del I.P.N.
Apartado Postal 14-740
07000 M6xico, D.F., MI~XICO
E-mail address: karlovic@math.cinvestav.mx
E-mail address: eramirez@math.cinvestav.mx
MSC 1991: Primary 47G10, 47D30, 47A53
Secondary 47B35, 45E05, 45E10
Submitted: March 28, 2000

Integr. equ. oper. theory 39 (2001) 475-495
0378-620X/01/040475-21 $1.50+0.20/0
9 Birkh~iuser Verlag, Basel, 2001
Integral Equations
and Operator Theory
COMPACT AND FINITE RANK OPERATORS SATISFYING
A HANKEL TYPE EQUATION T2X = XT{
CARMEN H. MANCERA AND PEDRO J. PAUL
Dedicated to the memory of our master and friend V~astimil Ptdk
In 1997, V. Pt~k defined the notion of generalized Hankel operator as follows:
Given two contractions T1 E 23(9~1) and T2 E 23(5-C2), an operator X : ~1 -+ ~2
is said to be a generalized Hankel operator if T2X = XT{ and X satisfies a
boundedness condition that depends on the unitary parts of the minimal iso-
metric dilations of T1 and T2. The purpose behind this kind of generalization
is to study which properties of classical Hankel operators depend on their char-
acteristic intertwining relation rather than on the theory of analytic functions.
Following this spirit, we give appropriate versions of a number of results about
compact and finite rank Hankel operators that hold within Pt~k's generalized
framework. Namely, we extend Adamyan, Arov and Krein's estimates of the
essential norm of a Hankel operator, Hartman's characterization of compact
Hankel operators and Kronecker's characterization of finite rank Hankel oper-
ators.
1. INTRODUCTION
How far may the characteristic intertwining relation for Hankel operators take you?
Toeplitz and Hankel operators on the Hardy space H 2 are two of the most important and
widely studied classes of operators on Hilbert spaces (see [4], [6], [7], [10], [11], [12], [14],
[18], [19], [23], [34] or [35]). These classes are intimately related from its very definition:
Every function r C L~(~?) defines a Toeplitz operator Tr and a Hankel operator He by
Tr : H ~ --~ H 2 and He : H 2 -+ H 2
f --+P+(r f) f --+P-(r f)
where P+ is the orthogonal projection from L2(T) onto H 2 and P_ ----- 1 -P+ is the orthogo-
hal projection onto//2_. In the language of dilation theory, Tr and/arc are compressions of
the multiplication (or Laurent) operator induced by r This function r is called the symbol
of Tr (it is unique) and a symbol of H~ (it is not unique; the kernel of the application
r -+ He is H ~ whose functions are called analytic symbols).

476 Mancera, PaN
Toeplitz and Hankel operators satisfy simple relations that characterize these classes.
The characteristic relation for Hankel operators was given by Nehari in 1957 [18] and
it tells us that a bounded linear map X : H 2 --+ H 2 is a Hankel operator if and only if
XS = S*X where S is the unilateral forward shift in H 2 defined by (Sf)(4) := (. f(~) and
S_ is the unilateral forward shift in H 2_ defined by (S_f)(~) := 7' f(r The characteristic
relation for Toeplitz operators was given by Brown and Halmos in 1963 [7] and it tells us
that a bounded linear map X : H 2 -+ H 2 is a Toeplitz operator if and only if X = S*XS.
It is no wonder that a number of possible generalizations of the notion of a Toeplitz
or Hankel operator have been proposed in the literature and, in this paper, we want
to concentrate on generalizations made by exploiting the properties of the characteristic
relations. Namely, let T1 and T2 be two contractions defined, respectively, on Hilbert spaces
5/1 and 9s and call X : 9s --+ ~2 a generalized Toeplitz operator if X = T2XT~ and call
it a generalized Hankel operator if T2X = XT{. The purpose of this way of extending
the classical notions of Toeplitz and Hankel operators is two-fold: on the one hand, one
obtains new operators that share some of the properties that make these classes important
and, on the other hand, one is able to find out which of these interesting properties of
the classical case depend only on the characteristic relations and not on the machinery of
complex function theory that sometimes is used to prove them. For Toeplitz operators,
this line of research was started, to the best of authors' knowledge, by Douglas [9] and
Sz.-Nagy and Foia~ [32], [33]. In 1988, Ptg~k and Vrbov~ [25], [26] and, later, Pt~k [24]
undertook the problem of obtaining an abstract analogon for Hankel operators, something
that had been previously done by Rosenblum [27], Page [20], [21] and Bonsall and Power
[5] in the particular case when T1 and T2 are backward shifts of arbitrary multiplicity.
In this approach, an essential problem --already identified by Douglas, Sz.-Nagy and
Foia~ in the Toeplitz case-- is to find what operators play the role of symbols. In the
classical case, the symbols are multiplication operators induced by functions from L ~ (%)
or, from another point of view, operators that commute with the unitary bilateral shift
V defined in L2(T) by (Vf)(() := 4" f((). The fact that V and V* are the minimal
isometric dilations of the backward shifts S*_ and S*, respectively, suggests, as it is the
case, that operators intertwining the minimal isometric dilations of T~ and T2 and the
structure of these minimal isometric dilations (including lifting theorems) must be of the
greatest importance for the problem at hand. Ptik and Vrbovi started by considering
common symbols for Toeplitz and Hankel operators. Their research revealed a surprising
fact: a certain boundedness condition must be satisfied by X in order to ensure that the
intertwining relation T2X = XT{ can be lifted. This boundedness condition, that will be
described later, is trivially satisfied in the classical case and, more generally, also when T1
and T2 are backward shifts, so that one of the main virtues of Ptik and Vrbovi's papers
[25] and [26] is to uncover its true meaning in the general approach. More recently, Pts
[24] proposed that it is more natural to consider different --rather than common-- sets

Mancera, Padl 477
of symbols for Toeplitz and Hankel operators. These two approaches, call them (PV) and
(P), yield exactly the same class of generalized Toeplitz operators but not the same class
of generalized Hankel operators. Prof. Pts encouraged us to analyze and clarify the rela-
tionship between (PV) and (P) and we proved in [17] that (P) is more natural and strictly
more general than (PV). In this paper, and already within the (P)-framework that will
be precisely described below, we shall investigate to what extent some well-known results
about classical Hankel operators depend, and in what sense, of the characteristic inter-
twining relation. These results are Adamyan, Arov and Krein's estimates of the essential
norm of a Hankel operator, Hartman's characterization of compact Hankel operators and
Kronecker's characterization of finite rank Hankel operators. We leave for future research
the question whether our results here can be applied to other generalizations of classi-
cal Hankel operators as, e. g., the Cotlar-Sadosky generalized Hankel forms (see [28] and
references therein).
Profl V. Pt~ik, who introduced us to this interesting topic, died while this paper was
still in preparation. During almost fifteen years we had very close contact with him and
his research group. His mathematical ability was quite astonishing; he always pointed out
the right direction to tackle a problem and had very clear ideas of what was relevant or
important to it. We will miss very much the continuous flow of lively and interesting dis-
cussions that we had with him. We also thank our colleague V. Vasyunin (St. Petersburg)
for his helpful remarks.
TERMINOLOGY AND NOTATIONS. Our terminology and notations will be mostly
standard, e.g., given two Hilbert spaces 9fl and 9{2, we shall denote by 2~(9{1, g~2) the
set of all operators (:= bounded linear mappings) from 9fl into 9f2 or simply ~B(~I) if
~1 = 9{2. We denote by X(:K1, 9{2) the subspace of all compact operators in 2~(tK1, 9{2)
and by IIXNessthe norm of an operator X in the Calkin algebra ~B(tK1, ~2)/9r g~2),
that is,
]]Xlless := dist (X, X(Zl, 9f2)) = inf{lIx - KH: K c 9r 9{2)}.
The range X(g~I) of an operator X C 2~(9Q, 9{2) will be denoted by ran(X) and its kernel
by ker(X). However, and although we refer the reader to the excellent books by Bhttcher
and Silbermann [6], Douglas [10], Gohberg and Feldman [11], Gohberg, Goldberg and
Kaashoek [12], Halmos [13], [14], Nikolskii [19], Sz.-Nagy and Foia~ [31] or Young [34], let
us fix some of the (maybe not so standard) notation that will be used in the sequel.
We have already denoted by S, S_ and V the usual forward shifts in H 2, H 2_ and L2(T)
and we shall use the same letters in the vector-valued case. Let T be a contraction defined
on a Hilbert space 9{. We denote by U the minimal isometric dilation of T; this isometry
U is defined on a Hilbert space ~ and we denote by P(9{) the orthogonal projection from
~K onto 9{, by 9{ the orthogonal complement of ~K in ~K, and by 9~ the unitary, or also

478 Mancera, Patil
called residual, part of the Wold decomposition of U (please note that if we had followed
strictly the notation of [31], then we would have written 5(+ and U+ instead of 5( and U).
In what follows, we shall mostly deal with two contractions T1 and T2 defined on respective
Hilbert spaces 9/1 and 9s so we shall write 5(1, 5(2 and so on.
2. PT/kK'S GENERALIZED HANKEL OPERATORS
We start by describing with some detail the notion of a generalized Hankel operator as
finally developed by Pts in [24] after earlier versions made in collaboration with Vrbovs
[25], [26].
An operator Z : 5(1 --+ 5(2 is said to be a Hankel symbol (with respect to T1 and T2) if
U2Z = ZU~. The set of all Hankel symbols is denoted by 9s T2). Hankel symbols are
essentially in ~B(~I, 9~2) because every Hankel symbol Z satisfies Z = P(~2)ZP(~I). This
is the reason why these spaces ~1 and 9~2 are so important in the theory of generalized
Hankel operators. This importance is hiddden in the classical case because if T1 = S* and
T2 = S*, then ~1 = 5(1 = ~2 = 5(2 = L2(T). Note that there are no Hankel symbols if
either ~1 or ~2 is {0}, for instance if either T1 or T2 is a forward shift operator.
Every Hankel symbol Z defines an operator Hz : J/1 -+ 9/2 by Hz := P(9/2)Z]JQ.
This operator Hz satisfies the intertwining relation T2Hz = HzT{ and also a boundedness
condition, that we shall call here PV-boundedness (with respect to T1 and T2): there exists
c > 0 such that for all hi E J/1 and all h2 C 5/2 the following hold
[(Hzhl, h2}[ _< c t[P(9~l)hl[[ [[P(9~2)h2[] 9
The minimum of the constants c appearing above will be denoted by [IHz][pv; note that
[[Z[[ _~ [[Hz[[pv. Operators satisfying this condition were called ~-bounded (with respect
to T1 and T2) by Pts [24] (but the notion of 9~-boundedness used in the previous approach
[25] is slightly different, see [17]). This condition, as we shall precise immediately, plays an
essential role in the searching of a symbol by lifting a Hankel-type intertwining relation.
Note that if T1 and T2 are co-isometries then [/1 and U2 are unitary on, respectively,
5(1 = ~1 and 5(2 = 9~2, so that PV-boundedness holds automatically and IIXIt = [IX[Ipv.
This is exactly what happens in both the classical case and Page's setting, as we shall
see below, and supports our affirmation above that one of the main virtues of Pts and
Vrbovs paper [25] is to uncover its true meaning in the general approach.
We say that an operator X : 5/1 --+ 5/2 is a Hankel operator (with respect to T1 and
T2) if it is PV-bounded and T2X = XT{. The fundamental lifting theorem (given in [26]
and [24]) says that Hankel operators have Hankel symbols: An operator X : 5/1 -+ ~K2 is
a Hankel operator if and only if there is a Hankel symbol Z such that X = Hz in which
case a symbol Z can be found such that [[Z[[ = IIX]lpy. The class of Hankel operators will
be denoted by 9~(T1, T2). PV-boundedness is strictly needed: Pts provides examples of
non-PV-bounded operators satisfying the intertwining relation T2X = XT{.

Mancera, Pafil 479
In the classical case, the set of analytic symbols is H ~176 that can be seen as the set of
symbols such that the associated Hankel operator is zero. So we say that a Hankel symbol
Z is analytic if its Hankel operator Hz = P(9~2)Z]9~I is zero, and the set of all analytic
Hankel symbols is denoted by A~:Z(T1, T2). Thus ~9~Z(TI, T2) is the kernel of the linear
mapping Z E ~KZ(T1, T2) --+ Hz 6 9I:(T1, T2).
The setting studied by Page [20] (see also [5] and [21]) fits into this schema: Let S
be the forward shift operator defined on a vector-valued Hardy space 9/= H2(3K), and
V be the correspondig bilateral shift on L2(?v[). Then the minimal isometric dilation of
T = S* is V* defined on ~K = :R = L2(3K), hence PV-boundedness holds trivially and,
therefore, the class of Hankel operators ~6S(S*, S*) coincides with the class considered by
Page of all operators X : H2(3K) --+ H2(:M) such that S*X = XS. The set of Hankel
symbols is then the set of all operators Z : L2(:h4) --+ L2(th4) such that V*Z = ZV; if 3V[ is
separable then this set can be identfied with L~176 via Z ++ JMr where Me is the
multiplication operator induced by a function r 6 L~176 and J is the unitary flip
operator defined on L2(3K) by (Jf)(4) = f(4-1). Analytic Hankel symbols are then those
operators Y : L2(:h4) --+ L2(3K) such that P(H2(3V[))yIH2(:~[) = 0 and if :h4 is separable
then the set of analytic Hankel symbols identifes with H~176 again via Z ++ JMr
The generalized version of Nehari's theorem follows easily from the fundamental lifting
theorem for Hankel operators and from the fact that [IZII > IIHzllpy (see I17]), and it
says: If X is a Hankel operator and Z E 9{Z(T1, T2) is a Hankel symbol for X then
t l X ] l py = dist(Z, Ag-6Z(T1, T2)) and this infimum is attained.
Finally, let us note, for later use, that taking adjoints produces a 1-1 correspondence
between the corresponding sets of Hankel operators, Hankel symbols and analytic Hankel
symbols when we interchange the roles of T1 and/'2.
3. COMPACT GENERALIZED HANKEL OPERATORS
Hartman's famous characterization [15] (see also [3], [5], [8], [20], [22] or [23]) says that
a classical Hankel operator is compact if and only if a symbol may be chosen for it which
is in the space C of all continuous functions on T. This theorem can be deduced from the
estimate ItHr Iless = dist(r H ~176 +C), which also implies that H ~176 +C is a closed subalgebra
of L~ the closure of the set spanned by H ~176 and the trigonometric polynomials (see
[10], [29] or [30]). The nice extension obtained by Page [20] (see also [5, Thin. 2]) can be
formulated as follows: If T E H2(3V[) is a backward shift operator of finite or ~o multiplicity
then a Hankel operator X E ~(T, T) is compact if and only if it has a symbol in the space
C(9r of compact-valued continuous functions. Within our generalized framework we
show that the "only if" part holds, but have not been able to extend the "if" part in its full
generality; we need additional hypothesis so that can only cover the case when 3/[ has finite
dimension. What we give is, more precisely, an extension of the following reformulation
of Hartman's theorem: A classical Hankel operator is compact if and only if its symbols

480 Mancera, Patil
belong to the closure of the subalgebra spanned by H cr and the trigonometric polynomials.
The role of this subalgebra will be played by the set eA~-Cg(T1, T2) defined by
eAg~g(T1, T2) :-- U U~nA~fS(TI' T2),
n~0
where the closure is taken in the norm operator topology. To our best knowledge, this
subalgebra was introduced by Bonsall and Power [5, p. 448] where it was denoted by A.
We start by extending to our general framework the necessary conditions for a Hankel
operator to be compact obtained by Hartman, Page, and Bonsall and Power.
Let Z E 9{$(TI, T2) be a Hankel symbol. Then U2Z = ZU~ and it follows easily that
for every n 6 N the operator U~Z = ZU~ n is also a Hankel symbol. The corresponding
Hankel operator can be written in any of the following forms
(~) Hzu{. = HzT{ n = Hugz = T~Hz.
To see this, take hi 6 [}(1 and, using the well-known equality P(~g2)U~ : T~P(J{2)
satisfied by the minimal isometric dilation, compute:
Hzu;~hl = P(iK2)ZU{~hl = P(giJZT~hl = HzT{~hl
: T~Hzhl : T~P(9{2)Zhl : P(9{~)U~Zhl = Hu~zhl.
THEOREM 1. Let Z be a Hankel symbol with respect to T1 and T2. Then
dist (Z, CJ[gfS(T1, 2"2)) : nlimcc ]]Hzu{~ ]] Py"
Moreover, if T1 and T2 are co-isometrics such that one of them is completely non-unitary
and Hz is compact then Z E eAg{S(T1, T2).
PROOF. Since [/2 is unitary in its reducing subspace 9{2 and every Hankel symbol Y
satisfies Y = P(9{2)Y, it follows that Y = U~U~nY for every n E N. Bearing this in mind
together with our generalization of Nehari's theorem, we have
lltfzu{ ~ llpv = min{IlZV; ~ - Yll: Y e A[K$(TI, T2)}
= min{]lV~Z - YI[: Y e Af-CS(T1, T2)}
= min{[IU~(Z- V;nY)]]: Y e Agf$(T~,T2)}
: min{lIz - ui YIl: y e Jt S(T ,
: dist (Z, U~J[f'gS(T1, T2)).
It follows from Eq. (~[) that U~J[[K$(T~,T2) c u~(n+~)A~{S(T~,T2). Therefore, the
sequence of distances {dist(Z, U~nAgfS(T1, T2)) : n : 1, 2,... } is a decreasing sequence
that converges to dist (Z, e.f[iK$(T1, T2)), hence
lim [Hzu:~ ,,, : dist(Z, eAg{S(T1, T2))
~--+OO - ~ - v

Mancera, PaN 481
and this is what we wanted to prove first.
For the second part, we assume that T1 and T2 are co-isometries and that/"2 is com-
pletely non-unitary; the proof for the case when T1 is completely non-unitary will follow
by taking adjoints. Then {T~ : n = 1, 2,...} converges to zero in the strong operator
topology and if Hz is compact then {T~Hz : n = 1, 2,... } converges to zero in the norm
topology. By using Eq. (~) and also the fact that the PV-norm equals the usual operator
norm because T1 and T2 are co-isometries, we obtain
dist(Z, eA~Kg(T1, T~)) = fire IIHzu~]l = lim IIHzv{~l] = lira NT~HzII=O.
n--+ c~ BY rt -+ c~ rt -+ c~
Therefore, Z E eA~g(T1, T2). []
To prove that under additional hypothesis the converse is also true, we shall extend
Adamyan, Arov and Krein's estimates of the essential norm of a Hankel operator [1-3]. To
use this approach we have to identify what Hankel symbols play the role of polynomials
in the variable 7.
For each n E N we denote by 3:~'Cn(T1, T2) the set of all Hankel operators X E t}t:(T1, T2)
such that T~X = XT{ ~ = 0. Using again that P(~2)U~ = T~P(~K2), we have
P( ~K2)U~ ZI~K1 = T~ P( gf 2) Zlgf l = T~ X = O.
This implies that given a Hankel operator X E ~(T1, T2) then X E 9~9~n(T1, T2) if and
only if U~Z = ZU~ ~ E Ag-f8(T1, T2) for every symbol Z associated to X.
In the classical case we know that 3:9-Ca(T1, T2) consists of Hankel operators with finite
rank; namely, those whose purely-H_ ~ symbol is a polinomial of degree at most n in 7. In
our case this is not necessarily true and in the next section we shall characterize when all
operators in 9~-C,~ (T1, T2) have finite rank. However, for our purposes in this section it will
be enough to know (Proposition 1 below) that this is the case if T1 or T2 is a backward
shift operator with finite multiplicity.
LEMMA 1. Let Z G ~(T1, T2) be a Hankel symbol and take n E N. Then
min{llHz - X[Ipy : X E ~:~Kn(T1, T2)} = I[Hz~c. I[py.
In particular, if T1 and T2 are co-isometries then
dist(Uz, X (T1, T=)) = IIIr
PROOF. Take X C 9~-{:n(T1, T2)) and let Y be a Hankel symbol for X, that is, X = Hy.
We start by proving that IIHz~]~. I]Py

482 Mancera, Pafil
Using that U~']9/1 = 2"1", that ~1 is U~'-reducing and that U{ I~l is unitary, we have
I((Hz -Hy)T;~hl, h2)l < c [IP(~l)T;nhl]l I[P(9~2)h2ll
= c I]P(9~l)U~nhl[] l]P(9~2)h21[ = c ][P(9~l)h1 [I [IP(22)h21l 9
By taking infima in c we obtain II(Hz - gz)T{nl]PV

Mancera, Pafil 483
THEOREM 2. Let T1 and T2 be co-isometries such that one of them is completely non-
unitary. Assume that all the Hankel operators in 9::ts ( T~ , T2) are compact for every n E N.
Let Z be a Hankel symbol with respect to Tt and T2. Then the following estimates for the
essential norm of the Hankel operator Hz hold:
IIHzrles~ = dist (H~, XX(T1, T~))
= lim dist (Hz, 9:9~(T1, T2))
n--+oo
= lira HTtHzII
= dist (Z, eA~S(T~, T2)).
In particular, the Hankel operator Hz is compact if and only if Z E CJqg~Z(T1, I~).
PROOF. As above, we assume that T2 is completely non-unitary and the proof for the case
when T1 is completely non-unitary will follow by taking adjoints. Then {T~ : n = 1, 2,... }
converges to zero in the strong operator topology. Again, note also that the PV-norm
equals the usual operator norm because/"1 and T2 are co-isometries. By our hypothesis,
we have
~r~f-n(T1, T2) C ~:~(T1, T2) C ~(9('1, ~-('2),
so that we may apply Lemma 1 to deduce that for every n C N the following hold
[]Hz][ess = dist (Hz, 9C(9Q, ~2)) -< dist (Hz, 9g~t:(T1, T2))
[IT~(Hz - K)I[ = [IT~Hz - T~KI].
Since {T~ : n = 1, 2,... } converges to zero in the strong operator topology, it follows that
{T~K : n = 1, 2,... } converges to zero in the norm topology so that
This implies
and, consequently,
lira IIT Hz-T KI] = lira IIT HzI[.
7t --+OO 7~--+ C~
lim IIT;Hztl < IIHz- Kit
[tHzIless = dist (Hz,Xgf(T1, T2)) = lim dist (Hz,9:96n(T1 T2)) = lim tIT~Hzt[.
n'-+CX) ' ~--+(X)
Finally, by Eq. (t) and Theorem 1, we have
Jim ItT~Hzll = lim tIHzu;, II = dist(Z, e~qg-t:Z(T1, T2)),
n'-+OO n--+OO
as it was required. []
Our version for the case studied by Page reads now as follows.

484 Mancera, Pa61
COROLLARY. If T is a backward shift of finite multiplicity then for every Z E 9~(T, T)
the following estimates hold:
[[Hz[[es s = dist(Hz, 9~:(T, T)) = lira [IT~Hzl] = dist (Z, e~tg-6Z(T, T)).
In particular, Hz is compact if and only if Z E eAg-fZ(T, T).
OPEN PROBLEMS. (1) As it was noted already by Bonsall and Power [5, p. 448] the
operators in 9~,~(T1, T2) need not be compact: simply take the projection from H2(?d)
onto the first coordinate when :h~ is infinite-dimensional. We do not know what hypothesis
is the right one to obtain a fully satisfactory "if and only if" characterization of compact
Hankel operators even for the case of co-isometries.
(2) We have A~:Z(T1, T2) playing the role of H ~ and CAg-6Z(T1, T2) playing the role of
H ~ +C so, what should play the role of C? In Page's setting it is C(9C(:~)), in the general
case it might be necessary to figure out new definitions of 9~9{n (T~, T2) and CAg~Z (T1, T2).
4. FINITE RANK GENERALIZED HANKEL OPERATORS
In this section we shall prove the characterization of when 9=9{~(T1, T2) consists on
finite rank operators (Proposition 1), but we shall concentrate mainly in giving appropriate
versions (Theorems 3 and 4) of Kronecker's theorem about finite rank Hankel operators.
In what follows, we shall make an intensive use of a co-isometry that one may associate
to each contraction T; to simplify notation, this co-isometry will be denoted by W rather
that W(T) or something similar. The co-isometry W is defined on the space 9 = P(~)~
by W := (U* IT)*. Note that if T is itself a co-isometry then [P = ~ and W = T. There is
a number of properties of this co-isometry that will be used and that the reader may find
proved in [24] and [25]; in particular, the minimal isometric dilation of W is the unitary
operator U]~ and the orthogonal complement of T in ~ is 9~ O $ = ~ N I (see [25,
1.1], [24, 1.1. and 2.3] or [33]; the reader must be aware of the following fact: In [25] the
operators are defined from 9~2 into ~1 (from ~2 into KI, etc.) so that whenever we use or
quote a definition or a result from that paper, we rewrite it interchanging the subindices
2 and 1).
Pt~k's idea of using this co-isometry is essential in the two step proofs given in [24]:
first for the case when T1 and T2 are co-isometries and then for the general case by using
the corresponding W1 and W~. For Hankel operators and Hanket symbols, the connection
is given in the following lemma. Although some of the items in its statement are proved
and used in [25, 1.4] and [24, 2.5], they never appear explicitely. For this reason, we gave
a complete proof in [17].
LEMMA A. For each Hankel operator X E 9f(T1, T2) there is an operator Xw C ~B(~P1, ~P2)
verifying the following properties:
(1) Xhl = P(gf2)XwP(~l)hl for every hi E 9s and X has finite rank if and only if
Xw has finite rank.

Mancera, Pafil 485
(2) Xw = 0 if and only if X = O; in other words, the operator Xw is the only one
verifying the relation given above.
(3) Xw is a Hankel operator with respect to W1 and W2.
(4) If Y E ~-CS(W~, W2) is a Hankel symbol for Xw then YP(9~I) is a Hankel symbol
for X.
(5) /f Z E 9~S(T1, T~) is a Hankel symbol for X then the restriction ZI~ ~ is a Hankel
symbol for Xw.
(6) The correspondences given in (4) and (5) map ~t~S(T1,T2) onto Ag~$(W~, W:).
PROPOSITION 1. For all n E N, every operator in 3:~n (Tz, T2) has finite rank if and only
if either ker W1 or ker W2 has finite dimension. In particular, if T1 or T2 is a unilateral
backward shift of finite multiplicity then all of the operators in 9~n(T1, T2) have finite
rank.
PROOF. Assume first that T1 and 7"2 are co-isometries. Then Ti = 9~i and Wi = Ti for
i= 1,2.
(~) Assume that ~2 = kerT2 is finite dimensional and the proof for the case
when E1 = kerT1 is finite dimensional will follow by taking adjoints. Let :K2n be the
completely non-unitary part of the Wold decomposition of the isometry T~. As it is well-
known, the completely non-unitary part T~ I~K2n of this isometry is a forward shift operator
and ~2 is its wandering subspace: ~K2n = (~k_>o T~k(s 9 Then for every k C 1~ we have
kerTk = 82 @ T2"(~2) (~'" @ T~k-l(~2),
so that kerT~ is also finite dimensional for every k E N. Now, ifX E 9~-C~(T1,T2) then
T~X : 0. Therefore, ran X C ker T~ and, consequently, X has finite rank.
(3) Assume that both spaces kerT1 and kerT2 are infinite dimensional and
we shall see that we can cook up a non-finite rank operator in X E 9:gQ(T1, T~). Fix
two orthonormal bases {xn : n = 1,2,...} C kerT1 and {Yn : n : 1,2,...} C kerT2 and
consider the mapping X : ~1 -+ ker(T2) C ~2 defined by
O(3
Xhl := E (hi, xn) y~ for hi C 9fl.
rt=l
It is clear that X is a well-defined operator and the proof for the co-isometric case will
be finished as soon as we see that X is a Hankel operator in 9~1(T1, 7"2). But~ clearly,
T~X = 0 and, on the other hand, for hi E ~K1 we have
XT{hl : (T{hl, Xn} yn : (hi, Tlxn> Yn = O.
rt=l r~=l
Since PV-boundedness holds automatically because 7'1 and T2 are co-isometries, it follows
that X 6 9~9-Q(T1, T2).

486 Mancera, Patil
Suppose now that T1 and T2 are arbitrary contractions and apply Lemma A:
Given a Hankel operator X E Js consider the corresponding Xw C Js W2)
such that X = P(IK2)XwP(J~I)[JQ. Then
TrX= TrP( 2)XwP( l)lXl
= P( C2)V XwP( l)l fi [because TrP( C2) =
= P(~C2)U~P(J~2)XwP(J~I)Igil [because ranXw C ~P2 C 9~2]
= P(gi2)P(J~2)U~XwP(9~I)[~I [because 9~2 is U2-reducing]
= P(Ji2)P(T2)U~XwP(NI)IJQ [by [24, 2.3]]
= P(Ji2)W~XwP(9~I)[~1,
where the last equality holds because U2[J~2 is the minimal isometric dilation of W2 and,
therefore, W~I~P 2 = P(~P2)U~'[T2. The unicity establisehd in Lemma A tells us that W~Xw
is the Hankel operator in J-I:(W1,W2) associated to T~X. Consequently, T~X = 0 if
and only if W~Xw = 0 or, in other words, X e :~J-C~(T1,/72) if and only if Xw 9
3:J-C~(W1,W2). Again Lemma A tells us that X has finite rank if and only if Xw has
finite rank. Finally, the co-isometric case proved above tells us that Xw has finite rank if
and only if either ker W1 or ker W2 is finite dimensional. This ends the proof of the general
ease.
Finally, note that if T is a unilateral backward shift of finite multiplicity, this
means precisely that T is a co-isometry, so that T=W, and that T* is a unilateral forward
shift whose wandering subspace ker(T) is finite dimensional (by the way, this is what
happens in Page's setting where ker(T) = :J~). []
We turn now our attention to the task of giving and adequate version for Ptgk's
generalized Hankel operators of the Kronecker's characterization of finite rank classical
Hankel operators [1@ A classical Hankel operator has finite rank 'if and only if it has a
symbol which is a rational function with poles inside the unit disk. We shall proceed in
three steps. Firstly, we shall recall the version given by Pts and Vrbovs [25, 5.2] within
their restricted framework for Hankel operators that was called (PV) in our Introduction.
Then we shall use their theorem to give a version for the case, already within the present
framework, when T1 and T2 are co-isometries, and we shall finish by giving a version for
the general case.
A Hankel operator as defined in [25] is linked to a previously existing Toeplitz
operator with its corresponding symbol. An operator Y : IK1 --+ ~]~2 is said to be a Toeplitz
symbol (with respect to T1 and T2) if Y = U2YU~. The set of all Toeplitz symbols
is denoted by 9~g(T1, T2). These symbols are, as in the Hankel case, operators defined
essentially from :R1 into J~2 because every Toeplitz symbol Y satisfies Y = P(~2)YP(J~I).
Every Toeplitz symbol Y defines two operators T PV 9 23(9Q, 9/2) and H Pv 9 23(9Q, J/~)
by
T PV := P(Jf2)YIgfl and H PV := P(Jf~)YIJQ.

Mancera, PaN 487
These operators satisfy the relations
T Pv = T2TPVT{ and HPVT~ = (U2lgi21)*H Pv.
The operator T PV is the Toeplitz operator associated to Y and it turns out that every
operator X satisfying X = T2XT{ admits a unique Toeplitz symbol Y such that X -- T Pv
(this result, that will be used in the proof of Theorem 3 below, has been proved with
different approaches in [9], [32], [33], [25] and [24]).
Here we are mainly interested in H Pv. If we consider the co-isometry T2 :--
(U21~C2~) *, then Hy PV satisfies the intertwining relation T2H~ v -- HPVT{. It can be
proved [17, Thin. I] that Hy PV, as an operator defined from ~:i into J-C2 ~, is PV-bounded
with respect to T1 and T2; in short, H Pv E ~-C(Tz, T2). A Toeplitz symbol ]z is said to
be analytic if the corresponding H PV is zero and we denote by ~ttg~S(TI, T2) the set of
all analytic Toeplitz symbols. Pt~k and Vrbov~'s version of Kronecker's theorem can be
stated as follows (please note that this is not exactly the statement as it is written in [25,
5.2] but, as it is transparent
need).
in the proof, it is what they prove and what we shall precisely
THEOREM K. Let B E 9~S(T~,T2) be a Toeplitz symbol and let H~ v = P(fff#)B[fffi.
Then the following conditions are equivalent:
(1) ran(H; v) is finite dimensional.
(2) There is a polynomial q of degree n with roots al, a2,..., an E D (that
depends only on HPB v and not on the chosen ToepIitz symbol B) and there is an analytic
Toeplitz symbol A E Af'8(T1, T2) such that B = q( (U~ [:R2))-IA and
dim[(U2 - T2)T~-I(1 - oLIT2) -1... (1 -- anT2)-lAffl] < +oc.
Since the co-isometry T2 := (U2]fff~)* will play in what follows and important
role, let us mention at this point the properties of T2 that will be needed. If the additional
condition Jr2 z- C :R2 holds then, as we have proved in [17, Lemma 2], the minimal isometric
dilation of the co-isometry T2 := (U21fff~-)* is the unitary operator U2 = U~I~:2 whose
domain space ~:2 is the U2-reducing subspace given by ~:2 := (fff2n 9~2)9 ~, where fff2n
is the completely non-unitary part of the Wold decomposition of T~. If, in particular, T2
is a co-isometry, so that K2 = 9~2 D fir#, then we may also write K2 = K2 O ~2u where
2f2u is the unitary part of the Wold decomposition of T~.
THEOREM 3. Assume that T2 is a co-isometry and consider the associated co-isometry
T2 := (U21~2z-) * whose minimal isometric dilation is defined on ~:2 := ~K2 Q ~2u. [or a
Hankel operator X C J-C(T1, T2) the following conditions are equivalent:
(1) X has finite rank.

488 Mancera, Pafil
(2) There is a polynomial q of degree n with roots c~1, o~2,..., O~n ~ D such that
if Z is a Hankel symbol for X then there is an analytic Hankel symbol Y ~ J~$(T~, T2)
such that P(~2)Z = q(U21~:)-~Y,
9 $ ~n-- 1
dlm[P(9(.2)U~T~ (1 - O~lT2)-l(1 - c~2T2) -1 .... (1 - o~n:T2)-IY~I] < +oo,
and ran(P(gf2u)Z) is a finite dimensionalspace that depends only on X and not on the
symbol Z.
PROOF. (1) ~ (2) Let X E ~-f(T1, T2) be a finite rank Hankel operator and let Z be
a Hankel symbol for X so that U2Z = ZU~. Let us see first that P(~2)Z is a Toeplitz
symbol with respect to T1 and T2. Since 2"2 is a co-isometry, we may use the facts mentioned
immediately before stating the theorem to do the following computation:
G (P({2)z) u~ : GP('i

Mancera, Padl 489
Let us now check that Y belongs to JUKZ(T1, T2). On the one hand, we know already that
N *
Y e A~YS(T1, T2) so Y = U2YU~ and Y -- P(-~2)Y. Use that U2 = U~I * 2 is unitary and
that tK2 is U2-reducing to deduce U2Y = YU~, hence Y e ~(S(T1, T2). On the other hand
its corresponding H Pv = P(~2 | )g2-L)YI~I is zero, hence Y~l C ~ and, therefore, its
Hankel operator Hy = P(9/2)YI~l is also zero. This shows that Y C Jt~I:Z(T1, T2).
To finish the proof of this implication we have to see that ran(P(9/2u)Z) is a
finite dimensional space that depends only on X and not on Z. Let X2 be the finite rank
operator defined by X2 := P(~2u)X. Using that X is a Hankel operator and that ~K2~ is
T2-reducing, we have
X2T{ = P(~2u)XT{ = P(~K2u)T2X = T2P(~C2u)X = T2uP(hi2u)X = T2uX2,
where T2u := T2 l~2u is the unitary operator part in the Lander-Wold decomposition of T2
(see [31, Thin. 3.2]). Since T2u is unitary, we may write X2 = (T2u)*X2T{ and this means
that X2 is a Toeplitz operator with respect to T1 and (T2u)*. Let Z2 C 23(5Q, 5r be
the Toeplitz symbol for X2; that is: Z2 = (T2u)*Z2U~ and X2 = P(hi2u)Z215r = Z215r
We use now that 5C2~ is U~-reducing, because ~:2 is U2-reducing and 5/2u = 5r O ~:2, and
also that Z is a Hankel symbol for X, which implies Z = U~ZU~, to obtain
P(g~2u)Z = P(~2u)U~ZU~ = U~P(~2u)ZU~ = (T2u)*P(tK2u)ZU~.
This says that P(:K2u)Z is a Toeplitz symbol with respect to T1 and (Tzu)*. Its Toeplitz
operator is, then,
PV
Tip(~c2,)z) = P(gi2u)Zl:K1 = P(g~2u)P(gi2)ZI~K1 = P(gg2u)X = X2.
The 1-1 relation between Toeplitz symbols and Toeplitz operators yields Z2 = P(~C2u)Z.
In other words, whatever Hankel symbol Z for X we take, the operator P(g/2u)Z is
uniquely defined as the Toeplitz symbol associated to P(9/2u)X. It remains to prove that
Z2 = P(9/2u)Z has finite rank. We start by proving that the subspace ranX2 is T2u-
reducing. Since, as we saw above, T2uX2 = X2T{, we have that ranX2 is T2u-invariant.
Now, T2u is unitary, hence T2u[(ranX2) is an isometry defined on a finite dimensional
space. Therefore, T2u[(ranX2) is unitary. So we have that ranX2 is T2u-invariant and
that T2u[(ranX2) is unitary; it is easy to see that these implies that ran X2 is actually
T2u-reducing. Bearing in mind that X2 = Z2[tK1 and that Z2 is a Toeplitz symbol with
respect to T1 and (T2u)*, it follows that for every n = 0, 1, 2,... the following holds:
Z2U~~I : (T2u)*nZ2~(.1 : (T2u)*nX2~-Cl C (T2u)*n(ranX2) C ranX2.
Finally, the minimality of the dilation U1 means precisely that {U{'2Q : n = 0, 1, 2,... } is
dense in 9(1, and this implies that ran Z2 C ran X2 so that ran Z2 is finite dimensional.

490 Mancera, Pafil
(2) ~ (1) Assume that X is a Hankel operator satisfying condition (2) and let
Z be a Hankel symbol for X. Then
X = P(X2)ZlXl = P(~2u)ZlXl + P(X2~)ZlXl.
Since P(hC%)ZlhCl has finite rank by hypothesis, we have to prove that X~ = P(~C2~)ZJ~I
has also finite rank. We have seen above that P(.~2)Z is a Toeplitz symbol with respect
to T1 and T2 whose corresponding H~:yc2)z ) equals X~ = P(JC2~)X. We shall check that
we may apply Theorem K to dedUce that X1 has finite rank. Condition (2) tells us that
there is an analytic Hankel symbol Y E AJ-f.g(T1, T2) such that P(~;2)Z = q(U2]~2)-IY.
The operator Y verifies
Y = U~YU;. and Y~I C J-C~.
Now, we know that 5C~ is the domain space of T2 whose minimal isometric ditation is the
unitary operator U~I~2 and that ~2 (3 Jf~ = J~2n. Moreover, for every n = 0, 1, 2,... we
have
rT*n~ I
this and the density of {U~C~ : n = 0, 1, 2,... } in JCz yield ran Y c ~C2. Therefore, we
may write
Y = (U~['X2)YU~ and YJ(1 c 5C~
and this means that Y E A3"Z(T1, T2). Consequently,
q((U2)*)-lg = q(g2[~2)-lg = P(J(2)Z.
As we saw in the previous implication, we have that
so the inequality given by condition (2)
(u; - = P(x )u; IX ,
dim[P(X2)U~2P~-a(1 - alT2)-l(1 - a2T2) -1 .... (1 - oenT2)-lY2

Mancera, Pafil 491
REMARK. The main difference between Theorem 3 and Theorem K lies, essentially, in the
appearance of the condition that ran(P(gf2u)Z) must be a finite dimensional space. This
condition does not appear in Theorem K because the role that T2 plays in the definition of
a Hankel operator in the general approach (P) is played in the theory (PV) by :F2 which is a
completely non-unitary co-isometry. To see that condition in action consider the following
simple example: Take the co-isometries TI = 1 G S* and T2 -- 1 | S*_ where 1 stands for
the identity on some infinite dimensional Hilbert space ~f. Then we have
~-C(T1, T2) -- {Ae He : A E 23(9s162 E L~(T)},
9 C$(T1,T2) = {A@M~: A E 23(9s r E L~
A~:S(T1, 2"2) = {0 @ Me: r E H~}.
Clearly, a Hankel operator A @/arc has finite rank if and only if A has finite rank and
He is a classical Hankel operator having finite rank, that is, r is a rational function with
poles in D. This reflects the statement of Theorem 3 because 9/2u = 2( | {0} so that
P(~K2u)(A G Me) = d | 0, and ~:2 = {0} 9 L2(V) so that P(~2)(A @ Me) = 0 | Me.
EXAMPLE. Theorem 3 may be applied within the framework considered by Page [20]
where T(= T1 = T2) is the backward shift operator S* on H2(:h4) and U is the bilateral
shift V* on L2(~4). Then S = T* is completely non-unitary and an easy computation
shows that
~u={O}, ~=s: and 5=V.
Then we obtain that X E 9-C(S*, S*) has finite rank if and only if there is a polynomial q of
degree n with roots al, a2,..., am E D and an analytic Hankel symbol Y E AgeS(T1, T2)
such that q(U2ji2)-IY is a Hankel symbol for X and
dim[P(g2(:h4))YS*_(n-1)(1 - alS*)-l(1 - a2S*) -1 .... (1 - OLnS*)-IyH2(J~)] < +~.
In particular, this latter condition holds automatically if ~ is finite dimensional.
Finally, let us see how Lemma A enables us to go from co-isometries to the
general case.
THEOREM 4. Let :h42 be the unitaw part of the Wold decomposition of U~ IT2 and consider
W2 := P(5-@ V) :R2)U;I(J- @ A ~2). If X is a Hankel operator with respect to T1 and T2
then the following conditions are equivalent:
(1) X has finite rank.
(2) There is a polynomial q of degree n with roots al, a2,..., am E D such that
if Z is a Hankel symbol for X then there is an analytic Hankel symbol Y E Jtgg$(T1,T2)
-1y
such that P(9~2 G 3Vi2)Z = q(U21(9~2 O 9V[2)) ,
dim[P({P2)U;Wff-t(1 - c~1W2)-1(1 - a2W2) -1 .... (1 - a,~W2)-IYJ/1] < +oo,

492 Mancera, Padl
and ran(P(~2)Z) is a finite dimensional space that depends only on X, and not on Z.
PROOF9 We shall make use of Lemma A to reduce this general case to Theorem 3 via
the co-isometries Wl and W2. For the sake of clarity, let us see that W2 is the co-isometry
obtained from W2 with the same procedure used to obtain T: from T2. To prove this,
recall that the minimal isometric dilation of W2 is U21~2 and that R: O ~2 = ~ N ff~2,
hence
(I) ~ (2) By Lemma A, if X E 9-I:(2"i, 2"2) has finite rank then the correspond-
ing Xw E ~((WI, W2) has also finite rank and if Z is a Hankel symbol for X then Zlll
is a Hankel symbol for Xw. Now use Theorem 3 with Xw: On the one hand, the range
of P(3vi2)ZI~I , that coincides with the range of P(3Y[2)Z because Z = ZP(~I), is a finite
dimensional space that depends only on Xw and, therefore, only on X. On the other
hand, there is a polynomial q of degree n with roots ~i, o~2, 9 9 an E ID such that given
ZI~I there is an analytic Hankel symbol YI E JI~fS(WI, W2) such that
and
9 *~n-1 l . . . . .
(,) d,m[P(~2)UIW~ ( ~)-1(1 - ~2~:)-~ (~- ~:)-~Y~] < +~.
Let us see that Y := YtP(9~1) E ~(~1,X2) is the analytic Hankel symbol we need to
complete the proof of the implication. Lemma A tells us that Y E Jt~S(T1,T2) and
inequality ($) together with Y~I = Y1P(~I)~I C YliP1 yield
dim[P(~:)U;~2-~(1 _ ~:)-~(~ _ ~:)-1 .... (~ _ ~)-~yx~] <
d~m[P(~2)UiW~ (~ - ~W~)-~(~ - ~:W~)-~ " (~ - ~:)-~1] < +~.
Finally, since Z = ZP(~), we have
P(~ e ~:)z = p(~: e ~)(zl~)p(l~)
= q(U~l(~: e ~:))-~Y~p(~)
= q(U~[(22 e :~:))-ly.
So that Y is the analytic Hankel symbol we were looking for.
(2) ~ (1) Let X ~ ~t~(T~, T~) be a gankel operator satisfying condition (2).
We shall see that Xw verifies condition (2) of Theorem 3 and, by Lemma A, it will follow
that both Xw and X have finite rank. Let Z~ ~ ~t~$(W~, W:) be a Hankel symbol for

Mancera, PaN 493
Xw. Then, by Lemma A, Z = ZIP(:R1) is a Hankel symbol for X such that Z1 = ZI:R1
and P(:JK2)Z1 = P(N~)(Z[Jq) has finite rank. Take an analytic symbol Y E AJ~g(T1, T2)
such that P(9~2 @ ~2)Z = q(U2[(:R2 @ 1~2))-1Y. Again by Lemma A, the operator
111 = YIR1 is an analytic Hankel symbol with respect to W1 and W2. Restricting the
equality P(:R 2 O :JK2)Z = q(U2I(tR2 G :JY[2))-IY to ~R1 we obtain
Since Y~J-C 1 = YP(~R1)~-( 1 -= Y1PCR1)tK1, the inequality given in condition (2) and the
density of P(tR1)~I in T1 imply
dim[P(T2)U~W~'-I(1 - G1W2)-I(1 - 0~2W2) -1 .... (1 - olnW2)-lyl[])l] < -~-00.
Summarizing: Xw satisfies condition (2) of Theorem 3 above; the polynomial q is the
same as the one for X and the analytic symbol for Z1 = ZI:R1 is Yl = YIJ~I and this is
what we wanted to prove. []
REMARK. It is possible to deduce Proposition 1 from this theorem, but it requires a
longer proof than the more direct one that we have chosen.
OPEN PROBLEMS. (3) We do not know if it is true that X is compact if and only if Xw
is compact. This would enable us to deal with compact Hankel operators with respect to
arbitrary contractions by using, as we we have done for finite rank Hankel operators, our
results from the previous section given for the case of co-isometrics.
(4) It would be interesting to find out more properties of classical Hankel
operators that may be extended, and in what sense, to this generalized setting.
ACKNOWLEDGEMENTS. This research has been partially supported by la Consejeria
de Educacidn y Ciencia de la Junta de Andaluc(a and by la Direccidn General de Inves-
tigaci6n Cienfffica y Tdcnica, project PB97-0706. Some of the results of this paper were
presented at the International Conference on Operator Theory and its Applications to
Scientific and Industrial Problems held in Winnipeg (Canada) in October, 1998.
REFERENCES
1. V. M. Adamyan, D. Z. Arov, M. G. Krein, Infinite Hankel matrices and generalized
problems of Carathdodory-Fej~r and L Schur, Funkcional Anal. i Prilozhen. 2 (1968),
1-17. (Russian)
2. V. M. Adamyan, D. Z. Arov, M. G. Krein, Infinite Hankel block matrices and related
extension problems, Izv. Akad. Nauk. Armyan. SSR Ser. Mat. 6 (1971), 87-112
(Russian); , Amer. Math. Soc. Transl. 111 (1978), 136-156.
3. V. M. Adamyan, D. Z. Arov, M. G. Krein, Analytic properties of Schmidt pairs for a
Hankel operator and the generalized Schur-Takagi problem, Mat. Sbornik 15 (1971),
34-75 (Russian); , Math. USSR Sbornik 15 (1971), 31-72.

494 Mancera, Pail1
4. E. L. Basor, I. Gohberg, Toeplitz Operators and Related Topics, Operator Theory: Adv.
Appl., vol. 71, Birkh/~user-Verlag, Basel, Berlin and Boston, 1994.
5. F. F. Bonsall, S. C. Power, A proof of Hartman's theorem on compact Hankel operators,
Math. Proc. Cambridge Philos. Soc. 78 (1975), 447-450.
6. A. Bhttcher, B. Silbermann, Analysis of Toeplitz Operators, Springer-Verlag, Berlin,
Heidelberg and New York, 1990.
7. A. Brown, P. R. Halmos, Algebraic properties of Toeplitz operators, J. reine angew.
Math. 213 (1963), 89-102.
8. D. N. Clark, On the spectra of bounded, Hermitian, Hankel matrices, Amer. J. Math.
90 (1968), 627-656.
9. R. G. Douglas, On the operator equation S*XT = X and related topics, Acta Sci.
Math. (Szeged) 30 (1969), 19-32.
10. R. G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New
York, 1972.
11. I. Gohberg, I. A. Feldman, Convolution Equations and Projection Methods for their
Solution, Translations of Mathematical Monographs, vol. 41, American Mathematical
Society, Providence, RI, 1974.
12. I. Gohberg, S. Goldberg, M. A. Kaashoek, Classes of Linear Operators I and II, Oper-
ator Theory: Adv. Appl., vol. 49 and 63., Birkh~user Verlag, Basel, Berlin and Boston,
1990 and 1993.
13. P. R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity,
second edition, Chelsea, New York, 1957.
14. P. R. HMmos, A Hilbert Space Problem Book, second edition, Springer-Verlag, Berlin,
Heidelberg and New York, 1982.
15. P. Hartman, On completely continuous Hankel matrices, Proc. Amer. Math. Soc. 9
(1958), 862-866.
16. L. Kronecker, Zur Theorie der Elimination einer Variablen aus zwei algebraischen Gle-
ichungen, Monatsber. K5nigl. Preuss. Akad. Wise. Berlin (1881), 535-600; also in
Werke, vol. 2 (1897), 115-192, Teubner, Leipzig.
17. C. H. Mancera, P. J. Pafil, On Ptdk's generalizations of Hankel operators, Czechoslovak
Math. J. (to appear).
18. Z. Nehari, On bounded bilinearforms, Ann. Math. 65 (1957), 153 162.
19. N. K. Nikolskii, Treatise on the Shift Operator, Springer-Verlag, Berlin, Heidelberg and
New York, 1986.
20. L. Page, Bounded and compact vectorial Hankel operators, Trans. Amer. Math. Soc.
150 (1970), 529-539.
21. L. Page, Compact Hankel operators and the F. and M. Riesz theorem, Pacific J. Math.
56 (1975), 221-223.
22. S. C. Power, Hankel Operators on Hilbert Space, Bull. London. Math. Soe. 12 (1980),
422-442.
23. S. C. Power, Hankel Operators on Hilbert Space, Res. Notes in Math., vol. 64, Pitman,
Boston, London and Melbourne, 1982.
24. V. Pts Factorization of Toeplitz and Hanket operators, Math. Bohemica 122 (1997),
131-145.
25. V. Pt~k, P. Vrbovs Operators of Toeplitz and Hankel type, Acta Sci. Math. (Szeged)
52 (1988), 117-140.
26. V. Pt$k, P. Vrbov~, Lifting intertwining dilations, Integral Equations Operator Theory
11 (1988), 128-147.
27. M. Rosenblum, Self-adjoint Toeplitz operators, Summer Institute of Spectral Theory
and Statistical Mechanics 1965, Broohhaven National Laboratory, Upton, N. Y..
28. C. Sadosky, Lifting of kernels shift-invariant in scattering systems, Holomorphic Spaces
(S. Axler, J. E. McCarthy, D. Sarason, ads.), Math. Sci. Res. Inst. Publ., vol. 33,
Cambridge University Press, Cambridge, Melbourne and New York, 1998, pp. 303-336.

Mancera, Padl 495
29. D. Sarason, Generalized interpolation in H ~176 Trans. Amer. Math. Soc. 127 (1967),
179-203.
30. D. Sarason, Algebras o/functions on the unit circle, Bull. Amer. Math. Soc. 79 (1973),
286-299.
31. B. Sz.-Nagy, C. Foia~, Harmonic Analysis of Operators on Hilbert Space, Akad~miai
Kiad6 and North-Holland, Budapest and Amsterdam, 1970.
32. B. Sz.-Nagy, C. Foia~, An application of dilation theory to hyponormal operators, Acta
Sci. Math. (Szeged) 37 (1975), 155-159.
33. B. Sz.-Nagy, C. Foia~, Toeplitz type operators and hyponormality, Dilation Theory,
Toeplitz Operators and Other Topics, Operator Theory: Adv. Appl., vol. 11, Birkhs
ser-Verlag, Basel, Berlin and Boston, 1983, pp. 371-378.
34. N. Young, An Introduction to Hilbert Space, Cambridge University Press, Cambridge,
1988.
33. K. Zhu, Operator Theory in Function Spaces, Pure Appl. Math., vol. 139, Marcel
Dekker, Basel and New York, 1990.
DEPARTAMENTO DE MATEMATICA APLICADA II,
ESCUELA SUPERIOR DE INGENIEROS,
CAMINO DE LOS DESCUBRIMIENTOS S/N,
41012-SEVILLA, SPAIN.
E-mail addresses: memcera~matina.us.es and piti@cica.es
2000 MATHEMATICS SUBJECT CLASSIFICATION: 47B35.
Submitted: March 30, 2000

Integr. equ. oper. theory 39 (2001) 496-501
0378-620X/01/040496-6 $1.50+0.20/0
9 Birkhfiuser Verlag, Basel, 2001
I Integral Equations
and Operator Theory
On Improving Accuracy for Arcangeli's Method for
Solving Ill-Posed Equations
M.T.Nair I and M.P.Rajan
Let X and Y be Hilbert spaces and T : X -+ Y be a bounded linear operator. It is
well-known that if R(T), range of T, is not closed, then the problem of solving the operator
equation
Tx=y
is ill-posed (cf. Groetsch [3]). One of the widely used procedure for obtaining stable approx-
imations for the least residual norm solution ~ :--- Try for y E D(T t) :-- R(T) + R(T) is
the so called Tikhonov regalarization in which the approximation x~ is obtained by solving
the well-posed equation
(T*T + aI)x~ = T*y.
Here, T t : R(T) + R(T) -+ X denotes the Moore-Penrose generalized inverse of T. When
data y is known only approximately, say ~ in place of y with
t]Y- Yll-< 5
for some known error level 5 > O, then the well-posed equation to be solved is
(T*T + aI)2~ = T*~.
One of the crucial aspect in this procedure is to choose the regularization parameter a in
such a way that
[1~-2,[[-~0 as a--+0 and 5--+0.
It is known that the best rate possible for the error [[xa - ~,~[[ is 0(5 2/a) and it is attained
when 2 9 R(T*T) with an a priori choice a ,.~ 5 2/3. Also it is known that if ~ 9 R((T*T) ~)
1The work of this author is supported by a project grant from National Board for Higher Mathematics,
Department of Atomic Energy, government of India.

Nair, Rajah 497
with 2 = (T*T)~fz then the rate is O(52"/(2~+1)) and it is attained by taking a ,,~ 521(2~+~)
(cf. Groetsch [3]).
Discrepancy principles are often used for choosing the regularization parameter in an a-
posteriori manner. Arcangeli's discrepancy principle is one of the earliest such procedures.
In this method a is chosen so that it satisfies the relation
5
IIT:~,~ - 011 = ~. (1)
It has been a long standing question whether Arcangeli's method yields optimal order
0(52~/(2~+1)) for 2 E R((T*T) ~) with 0 < p _< 1. Though it is settled affirmatively for u = 1
by Nair [5], the problem is still open for the values 0 < u < 1. For these values, the best
known rate is
II :~ - ~,:,11
= O(~="/~)
proved by George and Nair [2]. This result includes, as particular cases, a result for u = 1/2
proved earlier by Groetsch and Schock [4] and the optimal result for p = 1 proved by Nair
[5]. It is to be mentioned that the analysis in [5] and [2] are carried out in a more general
setting of a class of discrepancy principles first considered by Schock [6].
The purpose of this note is to supply a result which improves the above known rate
0(5 2~/3) for the case 0 < v < 1/2 by modifying (1) to
with c > 1. In fact, we prove the following result.
5
IIT~ - 011 = c v~ (2)
THEOREM1 Suppose y E R(T) andS: ~ R((T*T) ~) with O < u < 1. Ira is chosen
according to the discrepancy principle (2), then
and
Note that
I1: - =
u 2u 1
- - >-- whenever 0

498 Nair, Rajan
For the proof of Theorem 1 we shall make use of a following three supporting lemmas.
LEMMA 3 Suppose y 9 R(T) and ~ 9 R((T*T) ~) with ~ = (T*T)~. Then
Proof. It is seen that
I1-~ z,,ll _< IITx

Nair, Rajan 499
LEMMA 4 Suppose y E R(T) and ~ E R((T*T) ~) with 0 < u

500 Nair, Rajan
Similarly, we see that
Proof of Theorem 1.
so that
IIT @-
By Lemma 5 and Lemma 4 with the notations therein, we have
Consequently,
so that
showing that
Also, applying Lemma 5 to Lemma 3,
(c - 1) --~-~ ___ IITx

Nair, Rajah 501
[3] C.W. GR.OETSCH, The Theory of Tikhonov Regularization for Fredholm Equations of
the First Kind, Pitman Publishing, Boston, London, Melbourne, 1984.
[4] C.W. GROETSCH and E. SCHOCK, Asymptotic convergnce rate of Arcangeli's method
for ill-posed problems, Applicable Analysis, 18 (1984) 175-182.
[5] M.T. NAIR, A generalization of Arcangeli's method for ill-posed problems leading to
optimal rates, Integr. Equat. OpeL, 15 (1992) 1042-1046.
[6] E. SCHOCK, Parameter choice by discrepancy principles for the approximate solution
of ill-posed problems, Integr. Equat. Oper. Th., 7 (1984) 895-898.
Department of Mathentics
Indian Institute of Technology Madras
Chennai - 600 036, INDIA
E-Mail : mtnair@acer.iitm.ernet.in
AMS Subject Classifieafion: 65J10, 65J20, 47L10.
Submitted: March 27, 2000