Structure of Hilbert Space Operators 1st Edition Chunlan Jiang

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Structure of Hilbert Space Operators 1st Edition
Chunlan Jiang Digital Instant Download
Author(s): Chunlan Jiang, Zongyao Wang
ISBN(s): 9789812566164, 9812566163
Edition: 1
File Details: PDF, 7.52 MB
Year: 2006
Language: english

s t r u c t u r e of
Operators
Chunlan Jiang Zongyao Wang

Hilbert Space
Operators

Chunlan Jiang
Hebei Normal University, China
Zongyao Wang
East China University of Science and Technology, China
Hilbert Space
Operators
Y ^ World Scientific
NEWJERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI

Published by
World Scientific Publishing Co. Pte. Ltd.
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STRUCTURE OF HILBERT SPACE OPERATORS
Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd.
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Printed in Singapore by World Scientific Printers (S) Pte Ltd

Preface
In the matrix theory of finite dimensional space, the famous Jordan Stan-
dard Theorem sufficiently reveals the internal structure of matrices. Jordan
Theorem indicates that the eigenvalues and the generalized eigenspace of
matrix determine the complete similarity invariants of a matrix. It is ob-
vious that the Jordan block in matrix theory plays a fundamental and
important role. When we consider a complex, separable, infinite dimen-
sional Hilbert space H and use C{7i) to denote the class of linear bounded
operators on H, we face one of the most fundamental problems in opera-
tor theory, that is how to build up a theorem in /3(H) which is similar to
the Jordan Standard Theorem in matrix theory, or how to determine the
complete similarity invariants of the operators. Two operators A and B in
C(H) are said to be similar if there is an invertible operator X, XA is equal
to BX. The complexity of infinite dimensional space makes it impossible to
find generally similarity invariants. The main difficulty behind this is that
it is impossible for people to find a fundamental element in C(H), similar
to Jordan's block, so as to construct a perfect representation theorem. We
appreciate such a mathematical point of view as people's being not powerful
enough to deal with a complicated mathematical problem then that reflects
their lack of sufficient knowledge and understanding of some fundamental
mathematical problems. It is because of the sufficient study of the *-cyclic
self-adjoint (normal) operators that people have set up the perfect spectral
representation theorem for self-adjoint (normal) operators and commuta-
tive C*-algebra. It is also because of the introduction of the concept of
irreducible operators by Halmos, P.R. in 1968 that Voiculescu, D. obtained
the well-known Non-commutative-Weyl-von Neumann Theorem for general
C*-algebra. But irreducibility is only a unitary invariant and can not reveal
the general internal structure of operator algebra and non-self-adjoint op-
V

VI Structure of Hilbert Space Operators
erators. Since the 1970s, some mathematicians have showed their concern
for the problem on Hilbert space operator structure in two aspects. In one
aspect, the mathematicians, such as Foias, C, Ringrose, J.R., Arveson,
W.B., Davidson, K.R. etc. have made great efforts to study the struc-
tures of different classes of operators or operator algebras, such as Toeplitz
operator, weighted shift operator, quasinilpotent operator, triangular and
quasitriangular operators, triangular and quasitriangular algebras etc. In
the other aspect, they have set up the approximate similarity invariants
for general operators by introducing the index theory and fine spectral pic-
ture as tools. One of the most typical achievements, made by Apostol, C ,
Filkow, L.A., Herrero, D.A. and Voiculescu, D. is the theorem of similarity
orbit of operators. This theorem suggests that the fine spectral picture is
the complete similarity invariant as far as the closure of similarity orbit of
operators are concerned. Besides, in the 1970s, Gilfeather, F. and Jiang,
Z.J. proposed the notion of strongly irreducible operator ((SI) operator)
respectively. And Jiang, Z.J. first thought that the (SI) operators could be
viewed as the suitable replacement of Jordan block in £("H). An operator
will be considered strongly irreducible if its commutant contains no non-
trivial idempotent. In the theory of matrix, strongly irreducible operator
is Jordan block up to similarity. Through more than 20 years' research,
the authors and their cooperators have founded the theorems concerning
the unique strongly irreducible decomposition of operators in the sense of
similarity, the spectral picture and compact perturbation of strongly irre-
ducible operators, and have formed a theoretical system of (SI) operators
preliminarily. But, with the research going deeper and deeper, the authors
and their cooperators are badly in need of new ideas and new tools to be
introduced so as to further their research. In the 1980s, Elloitt, G. classified
AF-algebra successfully by using of K-theory language, which stimulated
us to apply the K-theory to the exploration of the internal structure of op-
erators, which features this book. In 1978, Cowen, M.J. and Douglas, R.G.
denned a class of geometrical operators, Cowen-Douglas operator, in terms
of the notion of holomorphic vector bundle. They, for the first time, applied
the complex geometry into the research of operator theory. Cowen, M.J.
and Douglas, R.G. have proved the Clabi Rigidity Theorem on the Grass-
man manifold, defined a new curvature function and indicated that this
curvature is a complete unitary invariant of Cowen-Douglas operators. It is
these perfect results that have inspired us since 1997, to combine K-theory
with complex geometry in order to seek the complete similarity invariants
of Cowen-Douglas operators and their internal structures. Cowen-Douglas

Preface vn
operator is a class of operators with richer contents and contains plenty
of triangular operators, weighted shift operators, the duals of subnormal
operators and hypernormal operators. Its natural geometrical properties
support a very exquisite mathematical structure. Based on some of our
successful research on Cowen-Douglas operators, we have made headway in
the study of other operator classes.
This monograph covers almost all of our own and our cooperators' re-
search findings accumulated since 1998. The book consists of six chapters.
Chapter 1 provides the prerequisites for this book. Chapter 2 explains the
Jordan Standard Theorem again in i^o-group language, and gives readers
a new point of view to understand the complete similarity invariants in the
theory of matrix. And this chapter also helps readers get well-prepared
for the study of operator structure in terms of K-theory in later chapters.
Chapter 3 mainly discusses how to set up the theorem on the approximate
(SI) decomposition of operators by using the (SI) operators as the basic
elements. Meanwhile, to meet the needs of the study of the structure, it
also reports the relationship between (SI) operators and the compact per-
turbation of operators, and proves that each operator is a sum of two (SI)
operators. Chapter 4 describes the unitary invariants and similarity invari-
ants of operators in .Ko-group language by observing the commutants. This
chapter contains the following four aspects: (1) Gives a complete descrip-
tion of the unitary invariants of operators using ifo-group and lists some
properties of lattices of reducing subspaces of operators. (2) Illustrates the
establishment of the relationship between the unique (SI) decomposition
of operator up to similarity and the A"o-group of its commutant, and at
the same time, carefully states the complete unitary invariants and com-
plete similarity invariants, and the uniqueness of (SI) decomposition of
the operator weighted shift and analytic Toeplitz operators using the re-
sults of (1) and (2). (3) Makes a concrete description of the commutant
of (SI) Cowen-Douglas operators by using complex geometry. (4)Discusses
Sobolev disk algebra, the internal structure of the multiplication operators
on it and their commutants by using Sobolev space theory, complex anal-
ysis and the results in (3). Chapter 5 focuses the discussion mainly on
the complete similarity invariants of Cowen-Douglas operator and proves
that the A'o-group is the complete similarity invariant of it. In addition,
our discussion is extended to the other classes of operators which are re-
lated to Cowen-Douglas operators. Chapter 6 concerns some applications
of operator structure theorem, including the determination of i^o-group of
some Banach algebras, the distribution of zeros of analytic functions in the

Vlll Structure of Hilbert Space Operators
unit disk and a sufficient condition for a nilpotent similar to an irreducible
operator.
We would hereby like to give sincere thanks to all the following profes-
sors: Davidson, K.R., Douglas, R.G., Elloitt, G., Gong, G.H., Lin, H.X.,
Yu, G.L., Zheng, D.C., Ge, L.M. etc. For their many years' encourage-
ment and support. We would like to give special thanks to Gong, G.H.,
Yu, G.L. and Ge, L.M., for, since 2000, both of them have enthusiastically
lectured on K-theory and geometry in our seminar, which have enabled us
to make greater progress with our research. We are also grateful to Aca-
demician Gongqing Zhang and professor Zhongqin Xu at Beijing university
and professor Yifeng Sun at Jilin university. They have given us enormous
concern and encouragement since the early days of our research. It is their
encouragement and support that have encouraged us to unshakably finish
the course of research. We also wish to thank Mr. Xianzhou Guo for the
technical expertise with which he typed the manuscript of this monograph.
C.L. Jiang
Z.Y. Wang

Contents
Preface v
1. Background 1
1.1 Banach Algebra 1
1.2 K-Theory of Banach Algebra 3
1.3 The Basic of Complex Geometry 4
1.4 Some Results on Cowen-Douglas Operators 5
1.5 Strongly Irreducible Operators 7
1.6 Compact Perturbation of Operators 9
1.7 Similarity Orbit Theorem 9
1.8 Toeplitz Operator and Sobolev Space 10
2. Jordan Standard Theorem and Ko-Group 13
2.1 Generalized Eigenspace and Minimal Idempotents 13
2.2 Similarity Invariant of Matrix 14
2.3 Remark ! 18
3. Approximate Jordan Theorem of Operators 19
3.1 Sum of Strongly Irreducible Operators 19
3.2 Approximate Jordan Decomposition Theorem 29
3.3 Open Problems 42
3.4 Remark 42
4. Unitary Invariant and Similarity Invariant of Operators 43
4.1 Unitary Invariants of Operators 44
ix

x Structure of Hilbert Space Operators
4.2 Strongly Irreducible Decomposition of Operators and
Similarity Invariant: i^o-Group 57
4.3 (SI) Decompositions of Some Classes of Operators 69
4.4 The Commutant of Cowen-Douglas Operators 80
4.5 The Sobolev Disk Algebra 94
4.6 The Operator Weighted Shift 126
4.7 Open Problem 147
4.8 Remark 147
5. The Similarity Invariant of Cowen-Douglas Operators 149
5.1 The Cowen-Douglas Operators with Index 1 149
5.2 Cowen-Douglas Operators with Index n 154
5.3 The Commutant of Cowen-Douglas Operators 157
5.4 The Commutant of a Classes of Operators 169
5.5 The (57) Representation Theorem of Cowen-Douglas
Operators 176
5.6 Maximal Ideals of The Commutant of Cowen-Douglas
Operators 189
5.7 Some Approximation Theorem 192
5.8 Remark 201
5.9 Open Problem 201
6. Some Other Results About Operator Structure 203
6.1 Ko-Group of Some Banach Algebra 203
6.2 Similarity and Quasisimilarity 206
6.3 Application of Operator Structure Theorem 237
6.4 Remark 239
6.5 Open Problems 239
Bibliography 241
Index 247

Chapter 1
Background
In this chapter, we review briefly some of the facts about operator algebra
and operator theory which will be needed to read this book. Most of the
material can be found in books or papers such as [Admas (1975)], [Apostal,
C, Bercobici, H., Foias, C. and Pearcy, C. (1985)], [Blanckdar, B. (1986)],
[Conway, J.B. (1978)], [Cowen, M.J. and Douglas, R. (1977)], [Douglas,
R.G. (1972)], [Herrero, D.A. (1990)], [Herrero, D.A. (1987)], [Jiang, C.L.
and Wang, Z.Y. (1998)] and [Rudin, W. (1974)].
1.1 Banach Algebra
A Banach algebra is a Banach space A over C which is also an (associative)
algebra over C such that ||ob]|<||a||||6|| for all a, b in A. When A has a unit
e, the spectrum of a a(a) (or (7.4(a) if A needs to be clarified) is the set
{AGC : Ae — a is not invertible in A }. The left spectrum of a ai(a) is the
set {AeC : Ae —a is not left invertible in A }; the right spectrum of a ar(a)
is the set {AGC : Ae — a is not right invertible in A }. The resolvent set of
a p(a) := C<r(a). The left and right resolvent set of a are pi{a) := C<7;(a)
and pr{a) := C<rr(a) respectively. a(a) is a non-empty compact subset of
C and a(a) = cr;(a)Uoy(a). Let / be holomorphic in a neighborhood fi of
a (a) and let c be a finite union of Jordan curves such that indc(X) = 1 for
every A in a(a). Define
f{a)=^-ff{z){ze-a)-l
dz.
Let Hoi (a(a)) denote the set of all functions which are holomorphic in a
neighborhood of 17(a). We have the following theorems.
1

2 Structure of Hilbert Space Operators
Riesz Functional Calculus Let a be an element of a Banach algebra A
with identity, then for every f£Hol(a(a)), f{a) is well defined independent
of the curve c. The mapping f*—>f(a) is an algebra homomorphism and
n n
maps each polynomial p(z) = ^2 CkZk
to c^e + J^ c^ak
.
k=o k=
Spectral Mapping Theorem For feHol(cr{a)), cr(f(a)) = f(cr(a)).
Upper Semi-continuity of the Spectrum Let a be an element of a
Banach algebra A with identity. Given a bounded open set Q, flDo-(a),
there exists S > 0 such that a(b)cQ, provided a — b < 5 and b&A-
Let H be a complex, separable, infinite dimensional Hilbert space and
let £(H) denote the algebra of linear bounded operators on H. For each
Te£(W), a{T),(ri(T), ar(T),p(T),pi(T),pr(T) and f(T) are defined as
above, where f£Hol(a{T)).
Riesz Decomposition Theorem Assume that cr(T) = aiL)o~2, o~Ca2 =
0, where ai,a2 are non-empty compact sets, thenH is the direct sum of two
invariant subspaces Hi and Ti.2 ofT, such that o~(T-ni) = o~i and "Hi is the
range of Riesz idempotent corresponding to o~i(i = 1,2), where T-ni is the
restriction of T onTii-
Let A be an abelian Banach algebra with identity. A multiplicative
linear functional <j> is an algebra homomorphism <f> : A—>C with (f> = 1.
The collection J^ of all maximal ideals of A is a compact Hausdorff space in
the sense of weak-* topology. The Gelfand transform a of a is the function
a : ^2—>C defined by a{tb) = 4>{a).
Gelfand's Theorem If A is an abelian Banach algebra with identity,
a&A and C(^2) is the space of continuous functions on Y^,, then the Gelfand
transform a of a belongs to C(J^) and a(a) = {&(<p) : 4>&Y2}- The mapping
a—>d is a continuous homomorphism of A into C(^2).
Let A be a Banach algebra with identity. A two-sided ideal radA of A
is the Jacobson Radical if it is the intersection of all maximal left (right)
ideals of A- Equivalently, radA = {a : a(ab) = a(ba) = {0} for all b£A}.
A C*-algebra C is a Banach algebra with a conjugation operator * such
that (a*)* = a,(ab)* = b*a*, (aa + fib)* = aa* +~J3b* and ||a*a|| = ||a||2
for all a,bEC and a, /3GC. A *-homomorphism p of a C*-algebra C is a
•-homomorphism from C into C(HP), where Hp is a Hilbert space. If C has
an identity e and p(e) = I, then p is unital; if kerp = {0}, p is faithful. It
is obvious that p is faithful if and only if p is a *-isometric isomorphism

Background 3
from C onto p(C).
Gelfand-Naimark-Segal Theorem Every abstract C*-algebra C with
identity admits a faithful unital *-representation p in C(Ti.p) for a suitable
Hilbert space fip, i.e., C is isometrically ^-isomorphic to a C*-algebra of
operators. Furthermore, ifC is separable, then Hp can be chosen separable.
von-Neumann Double Commutant Theorem Let AcC(H) be a uni-
tal C*-algebra. Then the closure of A in any of weak operator, strong op-
erator and weak-* topologies is the double commutant A", where
A" = {A'}'
and
A'{T) = {TeC(H) :AT = TA for all A<=A}.
We call A'(T) commutant ofT.
1.2 K-Theory of Banach Algebra
.Ko-group. Let A be a Banach algebra with identity, and let e and / be
idempotents in A. e and / are said to be algebraic equivalent, denoted by
e~a /, if there are x, y£A such that xy = e and yx = /; e and / are said to
be similar,denoted by e~/, if there is an invertible z&A such that zez"1
=
/. It is obvious that e ~ a / and e ~ / are equivalent relations. Let M^A)
be the set of all finite matrices over A, Proj(A) be the set of algebraic
equivalent classes of idempotents in A. Set /(A) =
Proj(M'00(A)), then
J(Mn(A)) is isomorphic to J(A). If Pi Q a r e
idempotents in Proj(A),
p~s q if and only if p®r~aq®r for some rGProj(A), then "~3" is called
stable equivalence. KQ{A) is the Grothendieck group generated by J{A)
[B. Blackadar [1]]. The pair (G,G+
) is said to be an ordered group if G is
an abelian group and G+
is a subset of G satisfying
i. G+ + G+CG+;
ii. G+n(~G+) = {0};
iii. G+ - G+ = G.
An ordered relation "<" can be defined in G by x<y if y — x€G+
for
x,y€G.
Isomorphism Theorem Let A, Ai and Ai be Banach algebras and
A = Ai®A2,

then
J(A)* J(At)® f(A2), K0(A)cK0(A1)®K0(A2),
J(Mn(A))~ J(A) and K0(Mn(A))~K0(A),
where "~" means isomorphism.
Let A and B be two Banach algebras and let a be a homomorphism
from A into B. Then there is a homomorphism a* induced by a from
K0(A) into K0(B).
Six-Term Exact Sequence Let A be a unital Banach algebra and let
J be its ideal, then we have the following standard exact sequence
and the following exact cyclic sequence
Ko(J) ^K0(A)^K0(A/J)
d] d[
ffiCA/J) «—ffi(4) «— Kx{J),
where K(B) is the Ki-group of Banach algebra B.
1.3 The Basic of Complex Geometry
Let A be a manifold with a complex structure and let n be a positive in-
teger. (E, 7r) is called a holomorphic vector bundle of rank n over A if n
is a holomorphic map from E onto A such that each fibre Ex = 7r-1
(x) is
isomorphic to Cn
(a;eA) and such that for each zo£A, there exist a neigh-
borhood A of zQ and holomorphic functions e{z), e2{z)^ • • • , en(z) from A
to E such that e{z), e2(z), • • • , en(z) form a basis of Ez = ir~1
(z) for each
zeA. The functions e, e2, • • • ,en are said to be a holomorphic frame for
E on A. The bundle is said to be trivial if A can be assigned to A.
Let E and F be two holomorphic bundles over a complex manifold A. A
map (p from E to F is a bundle map if <p is holomorphic and tp : Ex—>F
is a linear transformation for every AeA.
A Hermitian holomorphic vector bundle E over A is a holomorphic
bundle such that each fibre E is an inner product space. Two Hermitian

Background 5
holomorphic vector bundles E and F over A are said to be equivalent if
there exists an isometric holomorphic bundle map from E onto F.
Let H be a separable complex Hilbert space and let n be a posi-
tive integer. Denote Gr(n,H), the Grassmann manifold, the set of all
n- dimensional subspaces of H. For an open connected subset A of Cfc
,
a map / : A—>Gr(n,H) is said to be holomorphic if at each AoGA
there is a neighborhood A of Ao and n holomorphic W-valued functions
n
^i(z),e2(z),-" >e„(z) such that f(z) = V (e
j(2
)}- I f
/ :
^—>Gr(n,K) is
3=1
a holomorphic map, then an n-dimensional Hermitian holomorphic vector
bundle Ef over A and a map <f> can be induced by /, i.e.,
Ef.= {(x,z)eHxA:xef(z)}
and
(j): Ef—>A,<j){x,z) = zeA.
Given two holomorphic maps / and g : A—>Gr(n,'H), we have two
vector bundles Ef and Eg over A. If there exists a unitary operator U on
H such that g = Uf, then / and g are said to be unitarily equivalent. If
there is an open subset A of A such that Ef& is unitarily equivalent to
•Eg IA! then Ef and Eg are said to be locally unitarily equivalent.
Rigidity Theorem Let A be an open connected subset of Cfc
and let f
and g be holomorphic maps from A to Gr(n, TCj such that
V /(*) = V aw =H
-
zeA zeA
Then f and g are unitarily equivalent if and only if Ef and Eg are locally
unitarily equivalent.
1.4 Some Results on Cowen-Douglas Operators
Let Q be a connected open subset of C, n is a positive integer, the set Bn(Q)
of Cowen-Douglas Operators of index n is the set of operators T&JC(H)
satisfying
(i) Oca(r);
(ii) r a n ( z - T ) := {(z - T)x : x£H} = H for each z€fi;
(iii) V/ fc
er(* -T) = H
(iv) dimker(z — T) = n for each zefi.

It can be proved that if Qo is a nonempty open subset of fi, then
Bn(£i)cBn{yio). For an operator T£Bn(fl), the mapping z—>ker(z — T)
defines a Hermitian holomorphic vector bundle of rank n. Let (ET,TT) de-
note the subbundle of trivial bundle QxH given by
ET := {(z,x)eflxH : xeker(z-T) and n(z,x) = z}.
Let A'(T) be the commutant of T, i.e., A'{T) := {A&C{H) : TA = AT},
then for TeS„(fi), there is a monomorphism TT from ^4'(T) into H™,E JQ)
satisfying TTX = Xker{z_T) for XeA'(T) and zeft, or rT X(z) =
ker(z-T) '•— X(z), where -ff^(j5;T)(^) is the set of all bounded bundle
endomorphisms from E? to ET-
To summarize the above and Section 1.3, we can find a holomorphic
frame (ei(z), • • • ,en(z)) such that
n
ker(z-T) = / ek(z), ZGQ for TeB„(fi).
Fix a zoSfi, denote
Hi = ker(z0 - T ) ,
H2 = ker(z0 - T)2
Qker(z0 - T),
Hm = ker(zQ - T)m
Qker(zQ - T)"1
"1
.
We have:
Theorem CD1 [Cowen, M.J. and Douglas, R. (1977)]
m ...
(i) E ©Wfc = V(e
(*o) :
l<i<«, 0<fc<m - 1};
fc=i
00
r«; E®-Hk = n;
k=l
(Hi) {el- (zo) : l<j<n,0<k<m — 1} is a basis of ker(zo — T)m
,m —
1,2, • • •, where e^ (zo) denote the k-th derivative ofej(z) at z = ZQ.
The following theorems will often be used in the chapters hereafter.
Theorem H [Herrero, D.A. (1990)] IfTeBn(Q), then ap{T*) = 0,
where T* is the adjoint ofT and o~p(T*) is the point spectrum ofT*.
Theorem JW1 [Jiang, C.L. and Wang, Z.Y. (1998)] Let Te6„(fi)
and let Pz be the orthogonal projection from H onto ker(z — T) for z£fl,
then (I — Pz^heriz-T)1
- *s
similar to T.

Background 7
Theorem JW2 [Jiang, C.L. and Wang, Z.Y. (1998)] Let TeC(H).
For given numbers > 0, there exist a positive integer n and Cowen-Douglas
operators {Ai}=1, {bj}?=i+1 such that
F-(®^)©(® s*)||<£.
j=i i=i+i
Theorem JW3 [Jiang, C.L. and Wang, Z.Y. (1998)] Given
TeBi(fl), there exist compact operators K,K2,--- ,Kn,--- with Ki <
7^- such that T + Ki&Bi(Q) and kerTT+K,T+K, = {0},i ^ j , where TA,B is
the Rosenblum operator from C(H) to C(H) given by TA,B(X) = AX — XB
forXeC{H).
1.5 Strongly Irreducible Operators
Operator T is strongly irreducible if there is no nontrivial idempotent in
A'(T) ([Gilfeather, F. (1972)], [Jiang, Z.J. (1979)], [Jiang, Z.J. (1981)]).
Operator T is irreducible if there is no nontrivial orthogonal projection in
A'(T) ([Halmos, P.R. (1968)]). It is obvious that strongly irreducibility
is invariant under similarity while irreducibility is just unitarily invariant.
Denote (57) and (IR) the set of all strongly irreducible operators and
irreducible operators, respectively, on H.
Let K{l-L) be the ideal of compact operators on H and let
•K : £(H)^A(H) := C{H)/1C(H)
be the canonical quotient mapping, A(H) is called the Calkin algebra. The
essential spectrum of operator T is ae (T) = {AsC : A—ir(T) is not invertible
in A{H)} and the Predholm domain of T is pF(T) = Cae(T). It is well
known that
o-e{T) = ale{T)U<rre(T),
where
CTU{T) := aifr(T))
and
are(T) := O-T(TT{T)).

Operator T is a Fredholm operator if 0€PF(T). T is a semi-Fredholm
operator if the range of T, ranT, is closed and either
nulT := dimkerT
or
nulT* := dimkerT*
is finite. In this case the index indT of T is defined by
indT := nulT - nulT*.
The Wolf spectrum aire (T) of T is given by
alre(T) := are(T)nale(T)
and PS-F{T) := Coire(T) is the semi-Fredholm domain of T. The spectral
picture A(T) of T consists of the compact set <7;re(T) and the index ind{T—
A) on the bounded connected components of ps-F(T).
Spectral picture theorem of strongly irreducible operators [Jiang,
C.L. and Wang, Z.Y. (1996b)] Let T be in C(H) with connected spec-
trum cr(T). Then there exists a strongly irreducible operator L satisfying
(i) A(L)=A(T);
(ii) TeS(L)-;
(Hi) If there is another strongly irreducible operator L with A(Li) =
A(T), then LES(L)~, where S(L) is the similarity orbit of L, i.e.,
S(L) := {XTX-1
: Xe£(H) is invertible}
and S{L)~ is the norm closure ofS(L).
Spectral picture theorem of Cowen-Douglas operators [Jiang,
C.L. and Wang, Z.Y. (1998)] Let T be in C(H) with connected a(T)
and a(T)p®_F(T). If pp{T) ^ 0, then there exists a Cowen-Douglas op-
erator A&(S I) such that
A(T) = A(A)
and if there is another operator B&S(T)~, then BGS(A)~, where
P°S-F(T) := {£ps-F(T)na(T) : ind(X - T) = 0}.

Background 9
Commutant theorem of strongly irreducible operators [Fang, J.S.
and Jiang, C.L. (1999)] Operator T is strongly irreducible if and only
if a (A) is connected for each A in A'(T).
The following theorem will be used frequently in this book.
Theorem CD2 [Cowen, M.J. and Douglas, R. (1977)] Each oper-
ator in B(fl) is strongly irreducible.
1.6 Compact Perturbation of Operators
We introduce only two famous theorems on compact perturbation of oper-
ator here.
Brown-Douglas-Fillmore theorem // T and T2 are essentially nor-
mal operators on Ti, then a necessary and sufficient condition that T be
unitarily equivalent to some compact perturbation of T2 is that
o-e(Ti) = o-e(T2)
and
ind{ — T) ~ ind{ — T2)
for ««A^cre(Ti).
An operator T is essentially normal if T*T — TT* is compact.
Voiculescu's theorem Let T£C(H) and p be a unital faithful *-
representation of a separable C* -subalgebra of the Calkin algebra A(H)
containing the canonical images n(T) and n(I) on a separable space Ti.p.
Let A •= p(w(T)) and k be a positive integer. Given e > 0, there exists
KeK,(H), with K < e, such that
T - K^T®A^^T®A^k

where "=" means unitarily equivalent.
1.7 Similarity Orbit Theorem
Complex number A is a normal eigenvalue of T if A is an isolated point
of cr(T) and the dimension of H(X,T), the range of the Riesz idempotent

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RUTH.
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1. SAMUEL.
i 13 2 119
ii 9 1 401
6 2 140

10 1 308
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27 28 2 93
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1 KINGS.
i 21 1 652
ii 5-9 2 644
viii 23 2 38
27 2 124
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xi 13 1 308
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xxii 2 2 387
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1 CHRONICLES.
xxi 1 1 165
2 CHRON.
x 15 1 218
xvii 4 2 387
xix 6 7 2 638
EZRA.
xxxiii 14 15 2 49
NEHEMIAH.
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15 2 215
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21 23 2 48

24 2 48
27 1 681
30 2 505
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1 1 578
1 3 1 58
7 1 93
7 8 1 324
12 1 576
12 1 578
12 2 36
xx 3 2 100
9 1 309
xxii 5 2 108
25 2 437
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4 1 518
4 1 688
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6 1 276
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xlix 6 1 400
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15 2 92
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ciii 17 1 398
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cxxvii 3 1 191
cxxx 3 1 676
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4 2 23
4 1 535
cxxxi 1 2 1 628
cxxxii 7 2 227
11 1 429
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cxxxiii 3 1 407
cxxxv 15 1 100
cxxxvi 25 1 191
cxxxviii 1 2 459
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cxl 13 1 399
cxli 2 2 599
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