Domingo A Herrero Arizona State University
Approximation of Hubert space operators VOLUME I
Pitman Advanced Publishing Program BOSTON
MELBOURNE
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© Domingo A Herrero 1982 First published 1982 AMS Subject Classifications: Primary 47A55, 41A65, 47A60; Secondary 47A15, 47A53, 81C12
British Library Cataloguing in Publication Data
Herrero, Domingo A. Approximation of Hilbert space operators. Vol. 1—(Research notes in mathematics; 72) 1. Hilbert space 2. Operator theory I. Title IL Series 515.7'33
QA329
ISBN 0-273-08579-4
Library of Congress Cataloging in Publication Data
Herrero, Domingo A. Approximation of Hilbert space operators. (Research notes in mathematics; 72— ) Bibliography: v. 1, p. Includes index. 1. Operator theory. 2. Hilbert space. I. Title. II. Series: Research notes in mathematics; 72, etc. 515.7'24 82-10163 QA329.H48 1982 ISBN 0-273-08579-4 (v. 1)
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To Buenos Aires,
on her four—hundred first birthday
'A ml se me hace cuento que elnpezó Buenos Aires,
la juzgo tan eterna como el agua 0
el
aire"
(Jorge Luis Borges)
Contents
1. Stability and approximation
1
1.1 Lower estimated derived from the Riesz-Dunford functional calculus
2
1.2 Lower estimates for the distance to Nk(H)
6
1.3 Lower semicontinuity of the rank 1.4 Stability properties of semi-Fredholm operators
8 9
1.5 On invariance and closure of subsets of L(K)
10
1.6 Notes and remarks
11
2. An aperitif:
approximation problems in finite dimensional
spaces
12
2.1 Closures of similarity orbits in finite dimensional spaces
13
2.1.1 The nilpotent case
15
2.1.2 Proof of Theorem 2.1
16
2.1.3 The lattice
17
2.1.4 Closures of similarity orbits of finite rank operators
19
2.2 The distance from the set of all non—zero orthogonal 20
projections to N(HJ
2.2.1 The limit case
20
2.2.2 On the exact values of 2.2.3 A companion problem:
and
23
the distance from the set of
all non—zero idernpotentstoN(H)
.
2.3 On the distance to
27
2.3.1 A general upper bound 2.3.2 Two
illustrative
25
examples
27
30
2.3.3 An example on approximation of normal operators by nilpotents
33
vii
2.3.4 On the distance to a similarity orbit
35
2.4 On the distance from a compact operator to N(U)
37
2.5 Notes and remarks
38
3. The main tools of approximation 3.1 The Rosenblum operator:
41
X + AX -XB
41
3.1.1 Linear operator equations 3.1.2 Approximate point spectrum of a sum
41
of
commuting
operators 3.1.3 Local one-side resolvents in LIH)
42
44
3.1.4 The left and the right spectra of
50
3.1.5 Rosenblum-Davis—Rosenthal corollary
53
3.1.6 The maximal numerical range of an operator
54
3.1.7 The norm of
56
3.2 Generalized Rota's universal model
58
3.3 Apostol triangular representation
62
3.4 Correction by compact perturbations of the singular behavior of operators
3.5 Apostol—Foia9—Voiculescu's theorem on normal restrictions of compact perturbations of operators
Two
75
3.5.2 Normal restrictions
76
3.5.3 Density of sets of operators with bad properties
78
results
borrowed from the theory of C*_algebras
4.1 Essentially normal operators
79
84 84
4.1.1 Brown-Douglas—Fillmore theorem
84
4.1.2 Berger-Shaw trace inequality
85
4.1.3 Examples of essentially normal operators
89
4.1.4 An application to approximation problems
90
4.2 Matrix models for operators
92
4.3 Spectra of compact perturbations of operators
94
4.4 Voiculescu's theorem
96
4.5 Closures of unitary orbits
99
4.5.1 Operator—valued spectrum and unitary orbits
viii
74
3.5.1 Schatten p—classes
36 Notes and remarks 4.
69
99
4.5.2 Concrete examples of closures of unitary orbits
100
4.5.3 On normal and quasinilpotent restrictions
101
4.6 Irreducible operators
104
4.7 Notes and remarks
106
5. Limits of nilpotent and algebraic operators 5.1 Limits of nilpotent operators
108 108
5.1.1 Normal limits of nilpotents
108
5.1.2 Spectral characterization of NIH)
111
5.2 Closures of
orbits of normal operators with
perfect spectra
113
5.3 Limits of algebraic operators
114
5.4 Normal operators in closures of similarity orbits
115
5.5 Sums of two nilpotents
117
5.6 The Apostol—Salinas approach:
an estimate for the
distance to Nk(H)
5.7 Salinas' pseudonilpotents
120 124
5.8 Limits of nilpotent and algebraic elements of the Calkin algebra
128
5.9 On the spectra of infinite direct sums os operators
130
5.10 Notes and remarks
132
6. Quasitriangularity
135
6.1 Apostol-Morrel simple models
135
6.2 Quasitriangular operators
140
6.2.1 Equivalence between the formal and the relaxed
definitions of quasitriangularity lower estimates for the distance to (QT)
141
6.2.2 Two
142
6.2.3 Spectral characterization of quasitriangularity
145
6.3 Biquasitriangular operators
146
6.3.1 Block-diagonal and quasidiagonal operators
146
6.3.2 Characterizations of biquasitriangularity
147
6.4 On the relative size of the sets (QT), (QT)*, (BQT), [N(HJ+K(H)] and N(H)
153
6.5 A Riesz decomposition theorem for operators with disconnected essential spectrum 6.6 Notes and remarks 7. The structure of a polynomially compact operator
154 157
162
7.1 Reduction to the (essentially) nilpot.ent case
162
7.2 The structure of a polynomially compact operator
164
7.3 Restrictions of nilpotent operators
167
7.4 Operators similar to Jordan operators
171
7.5 A similarity invariant for polynomially compact operators 173 7.6 Nice Jordan operators
177
7.7 Notes and remarks
188
8. Closures of similarity orbits of nilpotent operators 8.1 Universal operators
189 189
8.1.1 Universal quasinilpotent operators
189
8.1.2 Universal compact quasinilpotent operators
194
8.2 Compact perturbations of not nice operators
194
8.3 Quasinilpotents in the Calkin algebra
198
8.3.1 General quasinilpotents
198
8.3.2 Nice elements of the Calkin algebra
204
8.4 Compact perturbations of nice Jordan operators
205
8.4.1 Nice Jordan nilpotents
206
8.4.2 Nilpotents of order 2
211
8.4.3 Quasinilpotent perturbations
212
8.4.4 Universal operators in
215
8.4.5 A general criterion for universality
222
8.5 Separation of isolated points of the essential spectrum affiliated with nilpotents 8.6 Notes and remarks
230 236
REFERENCES
239
INDEX
249
SYMBOLS MW NOTATION
253
x
Preface
The last decade has been fruithful in results on approximation of Hubert space operators, due to a large extent to the impulse given by Paul R. Halmos in his famous survey article "Ten problems in Hilbert space".
The purpose of this monograph (and a second one, by C. Apostol, L. A. Fialkow, D. A. Herrero and D. Voiculescu that will follow and complete the results contained here) is to provide a set of general arguments to deal with approximation problems (in the norm-topology)
of all operators acting on a complex separable infinite dimensional Hubert space that are inrelated to those subsets of the algebra L(f1)
variant under similarities. Many interesting subsets of LUll have this property:
nilpotent
operators; algebraic operators (satisfying a fixed polynomial); polynomially compact operators; triangular, quasitriangular and biquasi— triangular operators; cyclic and multicyclic operators; semi—Fredhoim operators (with fixed given indices); operators whose spectrum is equal to a fixed compact subset of the complex plane cr,
or whose spectra are
contained in a fixed nonempty subset of T; any bilateral ideal of compact operators, etc, etc. The following list illustrates the kinds of problems to be considered here:
a) Given a subset R of LUll
invariant
under similarities, defined
in algebraic, geometric or analytic terms (e.g., the set of all algebraic operators, the set of all operators T such that T3 is compact, the set of all cyclic operators), characterize its norm-closure in "simple terms". Since the spectrum and its different parts are the
most obvious similarity invariants of an operator, these "simple will usually be expressed in terms of properties of the different subsets of the spectra of the operators in the closure of P. b) More generally, obtain a formula for the distance from a given operator to P or, al least, upper and/or lower estimates for this distance.
c) In a surprisingly large number of interesting cases, either P is invariant under compact perturbations, or its closure is contained
in (or equal to) the set of all compact perturbations of R. Compact perturbations will be used as a useful tool for approximation and will be also analyzed with respect to the above mentioned peculiar properties.
d) Analogues to the problems in a) and b) for subsets of the quotient Calkin algebra.
In Chapter I, we shall obtain all the necessary conditions for approximation that can be easily derived from the Riesz—Dunford functional calculus and the well-known stability properties of semi—Fredhoim operators. This chapter is followed by "an apdritif":
the solu-
tion of several approximation problems in finite dimensional spaces,
which only depend on the results of Chapter I and some "handwork" with matr :ces.
It is interesting to observe that, for a large number of approximation problems (but, unfortunately, not for all of them) the "obvious" necessary conditions derived from the results of Chapter I are actually sufficient, but the proofs of their sufficiency are very hard. These
proofs are constructive to a large extent and the "tools" for these constructions are developed in Chapters III and IV: Rosenblum's corollary, Rota's universal model and extensions, Apostol's triangular representation, results on compact perturbations and results "borrowed" from the theory of C*_algebras (the Brown-Douglas-Fillmore theorem, Voicu-
lescu's theorem, results on closures of unitary orbits). Except for those results related to the theory of C*_algebras (Chapter IV), the monograph is essentially self—contained.
Results on approximation of Hilbert space operators really begin in Chapter V with the characterization (due to C. Apostol, C.
and D. Voiculescu) of the closure of the set of all nilpotent operators. The closure of the set of all algebraic operators is then obtained as a corollary and this result is used to characterize biquasitrian— gularity, the closure of the similarity orbit of a normal operator with perfect spectrum and to give several results about the closure of the similarity orbit of an arbitrary operator.
Combining the above results with Apostol-Morrel "simple models", we shall obtain the
theorem on the spectral char
acterization of quasitriangularity. Since algebraic operators and operators that "look like backward shifts" are the simplest examples of triangular operators, this approach to quasitriangularity is, perhaps, more natural than the original one.
Finally, the last two chapters are devoted to a deep analysis of the structure of a polynoxnially compact operator and to the closure of xii
its similarity orbit, respectively.
A large part of the material contained in the last chapter consists of unpublished results of C. Apostol-D. Voiculescu, J. BarrIa-D. A. Herrero and D. A. Herrero on closures of similarity orbits of essentially nilpotent operators.
The author is deeply indebted to Professors Constantin Apostol, I. David Berg, Charles A. Berger, Ronald G. Douglas, Alain Etcheberry, Lawrence A. Fialkow, Ciprian Carl N. Pearcy, Allen L. Shields
and Dan Voiculescu, and to his professors and ex-fellow students from the University of Buenos Aires, Alejandro de Acosta, Mischa Cotlar, Beatriz Margolis, Lázaro Recht, Norberto Salinas and, very especially, to his wife Marta B. Pecuch de Herrero for infinitely many informal discussions and suggestions and, most important, for their friendly support during all these hard years.
The contents of this monograph have been developed in a 0Seminar on Approximation of Hilbert Space Operators" at the University of Georgia (Athens, Georgia, USA), during the academic year 1980—1981. The author wishes to thank the authorities of the University of Georgia for their support and to his colleagues Edward Azoff, Richard Bouldin, Kevin Clancey, Douglas N. Clark and Derming Wang for their assistance during the preparation of the manuscript. Many of their valuable observations have been included here.
Domingo A. Herrero
Teinpe, Arizona
June, 1981
xiii
1 Stability and approximation
As explained in the Preface, in "most" approximation problems related to similarity-invariant sets of operators, the "obvious" necessary conditions for approximation that can be derived from the Riesz— Dunford functional calculus and the stability properties of semi-Fredholm operators turn out to be also sufficient. In a certain sense, this
short first chapter contains all the necessary conditions and the remaining of the monograph (and the second one that will follow [16]) is devoted to explain why these necessary conditions are also sufficient. Throughout this monograph, the word operator will always denote a bounded linear transformation mapping a complex Banach space into another. If X and V are complex Banach spaces, the Banach space of all operators mapping X into V will be denoted by L(X,V). L(H)
for
We
shall write
is a complex separable (H0, H1, H2,..., etc) will
the Banach algebra L(H,H), where
H
Hilbert space. Unless otherwise stated, H
always denote an infinite dimensional Hilbert space. The algebra Luff)
contains
the open subset
GUI) =
W
is invertible}
which will play a very important role
(the linear group of L(H)), here.
A subset P of L(H)
is
called invariant under similaritie8 (or
similarity-invariant) if it is invariant under conjugation by elements of the group G(H), i.e., T
R
=> .S(T)
c
where •S(T)=
W
Gift))
is the similarity orbit of T. If K(fI)
denotes the ideal of all compact operators acting on H and
A(H) = L(H)/K{U)
is the canonical projection of LUll
onto
the
(quotient) Calkin algebra, then the image v(T)=T+K(H) ofT E
Lift)
in
ir:L{H)
÷
A(H) will also be denoted by T.
The reader is referred to [77],
[119] for the general theory of
Hilbert space operators. 1
1.1 Lower estimates derived from the Riesz—Dunford functional calculus of the complex plane Uisa Cauchy
A nonempty bounded open subset
domain if the following conditions are satisfied:
has finitely
(i)
many components, the closures of any two of which are disjoint, and (ii)
of fl is composed of a finite positive number of
the boundary
closed rectifiable Jordan curves, no two of which intersect. In this case, r =
will be assumed to be positively oriented with respect to
in the sense of complex variable theory, i.e., so that 1
1
dA
if
—
T,
/
— 10, if
(the upper bar will always denote closure with respect to the metric topology of the underlying space). Clearly, F is uniquely determined by
(and conversely). We shall say that r is a rectifiable contour.
If all the curves of F are regular analytic Jordan curves, we shall is an analytic Cauchy domain). A, the spectrum If A is a Banach algebra with identity 1 and a of a(a) in of a will be denoted by a(a). The complement say that F is an analytic contour (or fl
the complex plane is the resolvent set of a and the function A (from p(a) into A) is the resolvent of a. It is well—known is an analytic function of A in that domain that it satis-
that
fies the first resolvent equation: (A
Furthermore,
if a, b
A
and A
p
).
(a) np (b), then b)
=
(second
p(a)
resolvent equation).
If a is a nonempty clopen subset of a(a), then there exists an and £2 = 0; in this
analytic Cauchy domain £2 such that a c case
dA
is an idempotent of A commuting with every b in A such that ab = ba. (E(a;a) is the Riesz idempotent corresponding to a.)
The following theorem is just a quantitative form of the classical result on upper semi-continuity of separate parts of the spectrum. The reader is referred to [76], [153], [172], or [173,Chapter XIV] for the basic properties of the Riesz—Dunford functional calculus.
THEOREM 1.1. Let a and b be two elements of the Banach algebra A
2
with identity 1. Assume that the spectrum ala) of a is the disjoint un ion of two compact subsets and such that is nonempty (01 a Cauchy domain such that
ha— bli
II.lldenotes the
(where
norm of A)
for
II
=6 < 1; then
t
Thus,
the
m
= Ia - atli = t6
=
A
<
[0,1].
it
Furthermore, if
1, the second resolvent equation implies that
s <
lI(A—at)
whence
C0,l].
dA
TJ
is a well defined element of A for all t
t
and
c
idempotent =
0
=0.
<
= 0 for all t
so that
all A
0. If
Øand a(b)nD[2
then a(b)n[2
PROOF. Assume that Ila-bil
- (A-a)
=
x
<
let at= (l—t)a+tb, 0 II(A-at)
c [2
l_ (A—a5)
.1 t—s . II(A—a5)
—III
readily follows that t +
Cm—iS)
_21 t—sj
is a continuous mapping from
[0,1] into A. = o(a)nç2
Since
0, it follows that
f
e0 = so that 11e011
1
ity, 11e111
1
dA
t
is idempotent, 0
(Recall that
1).
By continu
and therefore e1 =
dA
This is clearly impossible, unless Recall that if (X,d)
is
0.
0.
0
a metric space and BIXJ
is the
family of all nonempty bounded (closed bounded, respectively) subsets of X, then dfl (A,B) =
where
=
jx
inf{c
>
0:
B c A, A c B},
X: dist[x,A] E},defines a pseudometric in B(X) (a resp.);dH(A,B) is the Hausdorff distance between A
metric in and B, A, B e
The qualitative form of Theorem 1.1 is the following COROLLARY 1.2. (i)
Let a be an element of a Banach algebra
with identity. Assume that a(a) is
subsets %and
°l
such that 01
A
the disjoint union of two compact and that 01 i8
open set [2. Then there exists a constant C
=
contained inabounded
C(a,a1,[2)
> 0
such that 3
for
all b in A satisfying ia—bit < C. Given a in A and > 0, there exists
0 (ii)
<4, i.e.,
provided lia—bli
(Hausdorff PROOF.
the
a(a)
= 0. Take C = (ii) Apply (i) with
a(a) from
mapping a
c
A into
Given 01 and
1.3.
(i)
such that
c
c
c
AE =
If
0 and
=
interior
0
then
is an open subset of
A:
{a
is an open subset of A.
c
If E is
(ii)
subset
8uch that a(b)
follows immediately from Theorem 1.1:
(1)
COROLLARY
0
>
distance) is upper seinicontinuous.
(as above), there exists a Cauchy domain and
6
of
A.
a G6 subset of
then {a A:
In particular, the set {a
6
A:
a(a)
a(a) =
c E} is a
of all quasi—
nilpotent elements of A is a G6 in A. PROOF.
follows from Corollary 1.2(u) and (ii) is an immedi-
(i)
ate consequence of (i).
0
The limit case of Theorem 1.1 yields the following COROLLARY 1.4. Let a, 01 (01 0) and A satisfies the inequality assume that b ia — b
then
be as in Theorem 1.1 and
min{ ii(A—a)11i':
A
0. as in the proof of Theorem 1.1; then Theorem 1.1
PROOF. Define
implies that Since
=
0 and
0
= 0,
lim(t -* 1)
for 0
t
<
1.
it follows from Corollary 1.2(i)
cannot be empty.
that
It
0
is convenient to observe that the result of Corollary l.4(and,
a fortiori, the result of Theorem 1.1 too) is sharp. In fact, we have
L(II) be ½}; then
EXAMPLE 1.5. Let P let
=
A—li
for
such that lip—Bit ½. the other hand, if A = ½, then liP-All = ½ and a(A) = {½} c
lary 1.4, a(B)nc2 On
<
a mon-zero orthogonal projection and ). Thus, by Corol = ½ (A
If T 6
0
all B in L(fI1
L(K), a is a clopen subset of a(T) and E(a;T) is the cor-
responding Riesz idempotent, then the range ran E(a;T) and the kernel ker E(a;T) of E(a;T) are subspaces of H invariant under every a
commuting with T (i.e., hyperinvariant 4
for
T), and
K can be
in
L(K)
written as
the algebraic (not necessarily orthogonal!) direct sum
K
ran
ker E(c;T). (Here and in what follows, subepace will always denote a closed linear manifold of a Banach space.) Furthermore, the spectrum of the restriction E(a;T) of T to ran E(a;T) coincides with a and the spectrum of the restriction Tiker E(a;T) coincides with a(T)\a [173,Chapter XIV]. In what follows, ran E(a;T) will be denoted by K(a;T). If a = (A) is a singleton, we shall simply write K(A;T) (E(A;T)) instead of H((A};T)
(E({A};T), resp). If a = {A} and dim K(A;T) is finite, then A
is called a normal eigenvalue of T; in this case, K(A;T) coincides
for some n 1. The set of all normal eigenvalues of T will be denoted by o0(T). Clearly, a0(T) is contained in the point with
of T (i.e., the set of all eigenvalues of T). The easen
spectrum
tial spectrum of T, i.e., the spectrum of
COROLLARY 1.6. Let A, B
L(K);
in A(K) will be denoted by
then
Assume that a is a nonempty (clopen) subset of 0(A) and let
(i)
(a Cauchy domain) be a neighborhood of a such that
(iii)
0
is a neighborhood of a such that
for
PROOF.
If
dim ff(a;A) ). If a is a nonempty clopen subset of ae(A) and the Cauchy
(ii) furthermore, dim K(a;A) = dim K(a';B) domain
=0.
then
A
all B in Luff) such that
<
(0
= 0, then ae(B) A€
(i) and (iii) follow immediately from Theorem 1.1, applied
and to A = A(KJ, respectively. (ii) This follows from the proof of Theorem 1.1. Observe that, if (0 At= (1-t)A+tB, then the continuity of the mapping t + implies that the idempotents E(a;A) = and E(a';B) to A = L(Il)
the same (finite or infinite) rank.D
=
Until now, we have only applied the arguments of functional calcu lus to a very particular class of functions analytic in a neighborhood of the spectrum a(a) of an element of the Banach algebra A; namely, the characteristic function of a suitable neighborhood of a clopen sub set of a(a). Analogous results hold in a much more general setting; namely,
PROPOSITION 1.7. Let a be an element of the Banach algebra A with identity 1 and let f be an analytic function defined in a neighborhood well—de— > 0, > 0 such there exists of a(a). Given
that f(b) is
5
fined for all b
in
A
satisfying
a—
bil < 6 and, moreover,
iif(a) —f(b)iI PROOF.
a.
<
c
Let
be a Cauchy domain such that c(a) c By Corollary 1.2(i), there exists 6] > 0 such that a(b)
c
for all b Clearly, f(b) is well—defined for all
<
integral
these b, by means of the
f(A)
f(b) =
The second resolvent equation implies that max{
:
A
l+maxf
<
c
:
A
c
and
Iif(a)—f(b)
.max{ I
(l+ii(A—a) _l)I)
provided ia—bit <
It
then
:
(for some
I
.fl a —
A
.
bit,
<
follows that, if Ia —bit < 6 f(b) is well—defined and IIf(a)
for a suitably chosen 6,
0 <
—f(b)II <
6
11
The following particular case is especially important for our purposes.
COROLLARY 1.8. Let a
A,
let
be an
isolated point of o(a)
and
let f(A)
for
I
,
= 10
in some neighborhood of in some neighborhood of o(a)\{u},
some k > 1.
a sequence of elements in A such that i8 well-defined for all n large enough and
Let (n÷oo).
Then f(a)
=0 if and
only if lim(n
co)
0.
1.2 Lower estimates for the distance to
A: Let Nk(A) = {a ments of order at most k 4k = let !4(A)
= uk:l
denote the set of all nilpotent ele-
1,2,...)
of the Banach algebra A, and
Nk(A) be the set of all nilpotents of A. In order to
simplify the notation, the set of all nilpotents (of order at most k) resp), or simply by N if K is in L(H) will be denoted by N(f1) understood.
It is a trivial consequence of Proposition 1.7 that, given a non—
6
A: p(a) = 0) is closed in A. In Ak particular, if we choose pk(A) = (k = then we obtain constant polynomial p,the set {a
1,2,...),
COROLLARY 1.9. 14(A)
(ii)
Nk(A) is closed in A for all k = 1,2 subset of A.
(i)
is an F
The following result provides a partial answer to the problem of estimating the distance from a given operator to PROPOSITION 1.10. Bkx = 0,
(i)
Let A,
B e L(I1).
x in K and some k
for some
1,
If jlAkxll
= lxii
= 1, but
then
IIA-Bil (ii)
If
liAkxII
= lxii = 1,
1, and max{IiAIl,IlBiI}
(iii)
for some k
but Bkx = 0 for
= M, then IIA—Bil
Let a, b 1, then
some x in K and some k
l/(kM).
(a Banach algebra). If
A
= 1,
Ilakil
but bk = 0,
lia-bil
(iv) If M,
liakIl =1,
then Ila—bil
PROOF. that hlBil
(i)
but bk = 0,
for some k
1, and
=
l/(kM). Let hAil
= M and IIA-Bhi
= 6.
It is completely apparent
M+6, and therefore
1 = 11Akx11 = hl(Ak_Bk)xhI
Mkjl(M+6)J
lAhIkjhlIA_Bii.1IBIP =
l[(l.)k_l](6/M)
so that (M+6)k
=
Mk+l. Hence,
=
(M+o)k_Mk,
6
If we assume that IIBII = M and IIA-Bhl = 6, then
hAil
P1+6 and we
arrive at the same inequality, whence the result follows. The remaining statements follow by the same argument. REMARXS. 1.11.
(i)
0
Since, in Proposition 1.10(u) and (iv), M
cannot be smaller than 1 and (by the mean-value theorem) 11(2kM)
for all k
<
1, the estimates of Ci) and (ii)
l/(kM), ((iii) and (iv)) are of
the same order. (ii)
The argument of the proof of Proposition 1.10(i) also ap= lixil = 1, but p(B)x = 0, for some
plies to the case when nomial p, p(A) =
J. Since liip(A)—p(B)]xiI = 1, IIA--Bli cannot
7
be "too small", where the words "too small" have a concrete numerical expression in terms of p and A . The same applies, of course, to (ii), (iii) and (iv).
1.3 Lower semicontinuity of the rank PROPOSITION 1.12. Let A s
and let {An}n2i be a sequence of
L(H)
operators such that IIA_Anil -* 0 (n +
rank A
(i)
urn inf(n +
with dim
co)
rank
for some
If lAX!!
(ii)
d, then IIAxH
then
and all x in a subspace H with dim all x in a subspace >
dlxii for
0
d. (iii)
If An
PROOF.
(i)
K(H) for
all n > n0, then A
If urn inf(n +
co)
K(H)
then
rank
there is nothing
to prove.
Assume that urn inf(n +
ca)
rank
d <
oo•
Passing,
to a subsequence, we can directly assume that rank
if necessary,
for all n= 1,
If rank A d+l, then there exist d+l linearly independent is a linearly independ vectors y1, y2,..., H such that ent set. Clearly, V = linear spanfyl,y2,...,yd÷l} has dimension d+1 It is easily seen {0} for each n = 1,2 and therefore Ynker that is a bounded sequence and there exists y s V and a subse= 0, so that quence such that lAy II = 1 but lim(j co)IiAn 2
=
J
cannot converge to A, even in the strong operator topology, a
{A 3
contradiction. Therefore, rank A < d. (ii)
0 and Observe that our hypothesis implies that + 0 (n + cc). Now the result follows from an elementary
ysis of the spectral decompositions of the hermitian operators A*A and (see, e.g., 1117]). The third statement is trivial.
C
It is convenient to recall that A e L(H) is compact if and only if ran A does not contain an infinite dimensional subspace 171],[96]. We shall use the following non—standard notation: If A K(H) is not a finite rank operator, then rank A will be will be an equivalent way to say that defined as w—• ThUS, rank T = T is not compact. The different possible ranks will be linearly order-
ed by
8
1.4 Stability properties of senii—Fredholm operators
Recall that T L(tI) is a semi-Fredholm operator if ran T is cbs ed and either nul T = dim ker T or nul T* = dim ker T* = dim fl/ran T is finite (where TA denotes the adjoint of T in L(H)). In this case, the index of T is defined by
md T = nul T-nul TA. The following theorem resumes the main properties of the semi— Fredholm operators. The reader is referred to [106] and [l53,Chapter IV] for details. THEOREM 1.13. Let T e
There exists a constant
(ii) x
.i. ker T and 11T*yii
chosen
as
611y11
for
(iii) There exists tS = (T+AA) *
cS
all y
>
0
for
such that 11TX1I
j. ker
T;
TA; moreover,
tS
land
>0 such that ifA £ L(Il),I1AII
ind(T+AA) = md
nul(T+XA)
lxi <
nul T and nul
nul T*moreover,nul(T+XA) and nul (T+XA) *are constant forD
(iv) In particular, if that IT_Anli +
0
(n
urn sup(n
(v)
all
can be
min{X
then T+XA is semi-F'rsdholm and
nul T
be a semi-Fredholm operator; then
L(fl)
T* is also a semi-Fredhoirn operator and md T* = —md
(i)
If K
-*
=) K(tlJ,
{An}ni
then
oo),
X
<
is a sequence of operators such
is eemi-Fredholm for all n > no and
nul Any nul lim sup(n then T4-K is serni-Fredhoim and
co)
nul md
ind(T+K) =
T.
If B is another semi-Fredhoim operator and md T+ind B is {oo,_=} or {—=,=}), then TB is a (i.e., {ind T,ind B} semi-Fredholm operator and md TB = md T+ind B. In particular, Tk is (vi)
well-defined
aemi-Fredhoim and md Tk =
k(ind
T),for all k > 1.
A semi—Fredholm operator T is a Fredholm operator if =.
<
md T <
The well—known Atkinson's theorem asserts that T is Fredholm if and
only if
is invertible in A{tfl[119], [153]. Hence, A-T is Fredholm}
=
=
(the Fredhoim domain of T)
is an open subset
of
The left fright) spectrum of an element a of a Banach algebra will be denoted by =
Thus, cie(T) =
trum) and cire(T) =
(cirCa), resp) and its complement
=
is the left (right, resp)resolvent set of a. where
(right
that the intersection aire(T) = re(T) the Wolf
A
=
(left
essential spec
essential spectrum). It is well—known (some authors call
spectrum of T) contains the boundary
of ae(T)
9
Its
and therefore, it is a nonempty (compact) subset of cc. complement where and Pre(T) = reCT) = e(T) (T), coincides with ps_F(T) = 1A c Q: A—T is semi—Fredhoim), the semiFredhoirn domcLin of T.
The following results are an immediate consequence of Theorem 1. and its proof (see [l53,chapter IV]).
13
L(U).
COROLLARY 1.14. Let T
Ps_F(T)
is
the disjoint union of the (possibly empty) open
89t8
P:_F(T) and
ps_F(T)
= {A
then
cc:
ps_F(T),
semi-Fredholm with ind(A—T)
A—T is
(ii) For each h,
(v)
If h
=h},
a lower semicontinuous
+0
and
co)
(n +
=
0,then
cc\a(T) of T, and
is a bounded set.
0, then
If the minimal index of A—T, min.ind(A-T)
is
the resolvent set p(T)
ps_F(T) contains (iv)
where
=
ps_F(T),
A
min{nul(A-T)
defined by
is
,nul(A_T)*},
the function A + min.ind(A—T) is constant on every component of except for an at most denumerable subset P_F(T) without limis a point of Furthermore, if it points in ps_F(T) ps_F(T) and ps_F(T) in the same component as but A ps_F(T), then then
ps_F(T),
A
.
4'
min.ind(A-T). (nul(A_T)* psF(T) and nul(X—T) < >
(vi) A
4'
If
A
P:_F(T) <=> ker(A—T) c (ker(X_T)* c
<
resp), then
ran(A_T)*m, reep.).
ps_F(T) is the set of singular points of the semi—Fredhoim domain ps_F(T) of T; pS_F(T) = PS_F(T)\P_F(T) is the set of regular points.
It
is completely apparent that
is open and contains p(T). On
the other hand, it is easily seen that p_F(T) contains o0(T). InChap ter III we shall return to the analysis of these sets.
1.5 On invariance and closures of subsets of L(H) A subset R of L(H)
10
is
called invariant
under unitary equivalence
T
where U(H) = {U
L(K):
R
U
=> 11(T) c
is unitary} and 11(T)
{UTU*:
U e U(K)} is
the unitary orbit of T. It is completely apparent that 11(T) c S(T). We shall establish without proofs some very elementary facts that will be frequently used in the future.
PROPOSITION 1.15. If R is a subset of L(H) invariant either under similarities, or under unitary equivalence, or under compact perturbations (i.e., R+K(H)
=
R), then R
has the same property.
1.6 Notes and remarks
Theorem 1.1 is just the quantitative version of [l53,Theorem 3.16, p.212] (see also [177], or [].32,Theorem 1]). Corollaries 1.3(u) and 1.9(u) are two elementary observations due to S. Grabiner [108] and 0. A. Herrero [132], respectively. Proposition 1.10 is a mild improvement of a result due to D. A. }Ierrero 1150,Lemma 4.3] (see also [44, Lemma 4.1]). The notion of "rank T,= o-" for a compact operator T, not
of finite rank, was introduced by
Barria and D. A. Herrero in [44]
in connection with the analysis of the similarity orbit of a nilpotent operator (See also Chapter VIII). The notion of "minimal index" is due to C. Apostol [10]. The fact that the singular points of are isolated points of this set was discovered by I. C. Gohberg and M. G. Krein [107). In the above mentioned article, C. Apostol proved that
PF(T) is, precisely, the set of points of discontinuity of the funcinto the orthogonal projection of K onto tion that maps ker(X—T)
(see also t25,Lemma 1.6 and
Corollary
1.7]). This result will
be analyzed in Section 3.3.
11
2 An aperitif: approximation problems in finite dimensional spaces
In
this chapter we shall analyze several intrinsically finite di-
mensional problems, as well as infinite dimensional ones which can be solved through an essentially finite dimensional approach or by an ar-
gument in Which the (finite or infinite) dimension of the underlying Hubert space plays absolutely no role. It will be convenient to introduce some notation: H will always
If
denote a complex separable Hubert space of dimension d, 0 d (with its canonical inner prod0 d < then we shall also write uct) instead of H. LUll, A
If A, B
B (A
B) will mean that A and B are similar
(unitarily equivalent, resp.). A slm B will be used as an alternative
way to indicate that B e S(A), i.e., that IIB_WnAWn
+ 0
(n +
oo)for
a suitable sequence {Wn}ni in GUll. If A sTm B and B sTm A (equivalently, S(A) = S(B)), then we shall say that A and B are asymptotically similar. (In symbols: A#B.) It is completely apparent that sTm is a reflexive and transitive relation and that # is, indeed, an equiv alence relation in L(H) .(Use Proposition 1.15. It is well—known and and = are also equivalence relations.) trivial that If A E L(111) and B
bert spaces (in symbols:
are isomorphic Hil— L(H2), where H1 and ), then A#B will beunderstood as"up
The to a unitary mapping U from H2 onto H1', i.e., S(A)=S(UBU*) same observation applies to the other relations. induces a partial order < in the quotient set The relation .
L(H)/#,
defined
by:
L(H):
Let [A] = {T S(B)
of
T#A};
[B] <
if A stm B.(Equivalently,
c S(A).)
Given a (finite or denumerable) uniformly bounded family operators such that A L(HV) for all v in r, we shall denote by acting in the usual
the direct sum of'the operators
r
evl. < =) is a
fashion on the orthogonal direct sum H = e., if x =
•VF
Av)X = 12
[A]
=
AvXv• Clearly,
of the spaces live
vector of H, then
= sup(v
F)
IlAvil <
i.
= {l,2,...,n},
If r
•...
L(if) for all v in r and card r = a (0
A E
note the operator
of
sum
we shall also write acting on
cx
.
If
A
e'), then
de
(orthogonal direct
=
a copies of if).
If M is a subspace of if, then M1 =
is the orthogonal comple-
ment of M in if.
Given f, g E (føg)x =
<x,g>f,
FUll
if, f@g e L(tf) is the rank one operator defined by where <.,.> denotes the inner product of H. if,
=
j
= l,2,...,n;
n =
is the ideal of all finite rank operators acting on H. Let {el,e2?...,ek} be the canonical orthonormal basis (ONB) of be the operator defined by
and let
(2.1)
=
0,1,2,...; q0 is the 0 operator acting on the trivial space {0}, is the 0 operator acting on the one-dimensional Hubert space and admits the matrix representation
(k =
010
.
.00
001... 00 000... 00 (kxk)
000.. .01 000... 00 with respect to the canonical ONB,
for
k =
2,3,...).
These operators will play a very important role throughout this monograph.
Finally, sp(a) = diu8 of a
A (a
max{IAI:
A €o(a)} will denote the speotrai ra-
Banach algebra).
2.1 Closures of similarity orbits in finite dimensional spaces
As remarked in the introduction, for many approximation problems the "obvious" necessary conditions derived from the results of Chapter I turn out to be sufficient too. Here is a concrete example of this situation:
THEOREM 2.1. Let T
and let p(X) =
'tjlm
(A—Aj)
,' Aj1
13
for i
be its minimal (monic) polynomial; then the closure of the
j)
similarity orbit of T is equal to S(T)
rank q(T) for all qlp},
rank q(A)
(A e
where
denotes a monic polynomial q dividing p. Furthermore, if L L # T if and only if rank q(T) for all if and only if L T.
2.2.
COROLLARY
rank q(L) =
then the following are equivalent:
Let T
(i) S(T) is maximal with respect to inclusion (equivalently, [T3 is a maximal element of
T
(ii)
and
as in Theorem 2. 1,
= d (where the
j
=
have the same meantng as
Theorem 2.2);
T is a cyclic operator;
(iv)
=
S(T)
(v)
{A
s
L(cEd):
=
a(T)
and dim Il(A;A) =
dim U(A;T)
for all X in cj(T)}. COROLLARY 2.3. Let T
is minimal (equivalently, [TI is a minimal element);
S(T)
(i)
then the following are equivalent:
1,
k) (defined as in Theorem 2.1) is equal to 1 for all j
(ii)
2,...,m,
T is similar to a normal operator;
(iii)
S(T) is closed in
(iv) (v)
S(L)
T
U(X;L) = dim tI(A;T)
Let
T, A s
for all L in
and assume that IIA_WnTWn
suitable sequence {wn}n:i of operators in =
0(T) and dim
such that a(L)
(A
lii
+ 0
(n
=) for
a
Clearly, q(A) and
are well—defined (for all n =
1,2,...)
for all
qip and (by Proposition 1.7) 0
Since rank
=
rank
q(T)
(n +
(for all n =
rank
lows from Proposition 1.12(i) that rank q(A)
1,2,...),
it fol-
q(T).
Hence the conditions of Theorem 2.1 are necessary. The sufficiency of these conditions will be proved in several steps. The second statement of the the first one:
It is obvious that S(L)
0(T) and rank q(L) = rank q(T) for all
is a trivial consequence of =
S(T)
if and only if 0(L)
(Use Proposition 1.15).
On the other hand, a simple analysis of the Jordan forms of L and T
shows that rank q(L) = rank q(T) for all qjp if and only if L and T 14
are similar. (In particular, this implies that they have the same spectrum.)
2.1.1 The nilpotent case
LEMMA 2.4. If 1
then
Given
0
<
c
<
1,2,...,m), Wf1=
be the canonical ONB and
1, define and Wf.= f.(j
computations show that W
k
Straightforward
is invertible,
=W
Q
by
= 2,3,...,k).
(l/c)f1 and W1f.= f.(j
...,m),
=
=
= 2,3,...,k)
1,2,
and
Q+cf1øf2,
2,3,...,m),
where Qf1= 0, Qe1= f1, Qe.= e1_1(i = (j
sim
PROOF. Let {e11e2,...,e} and and respectively, so that
of
=
k-li
m
Qf2= 0 and
= 3,4,...,k). It is immediate that Q
and that Q
= lim(c
+ 0)
Q(un-
less otherwise stated, lim must always be understood as a limit in the
fl
norm-topology) .
LEMMA
2.5. Let
be a nilpotent of order m; then
T
S(T)= {A
rank A3
rank T3 for
l,2,...,m}.
j
PROOF. We have already observed that the condition "rank A3
rank
= l,2,...,m"
for j
is necessary. (Observe that the minimal pol
ynomial of T is p(X) = Am)
rank T3 for j
Assume that rank A3
= 1,2,.. .,m.
Clearly, we can
directly assume that P and A are Jordan forms, i.e.,
(r2)
(T1) P =
q
eq >
0).
(Tk) •. .
.Wq
and nk
A=
•. .
2
1
h
We shall proceed by induction on m(T,A) =
(rank T3-rank A3).
n implies The case m(T,A) = 1 is trivial. Assume that m(T1,A1) that T1 sTm A1, whenever A1 and T1 are nilpotent operators acting finite dimensional space and satisfy rank A13 rank T13 for all j = be nilpotent operators such that rank A3 1,2,..., and let A, T rank P3 for all j
= 1,2,...,
and m(T,A) =
If T and A have a common Jordan block = rank rank rank
n+l. then P = 0
for all j
A=
= 1,2,..., 15
m(T11A1)
(n+l)-r
n and, by induction, T1 sTm A1. A fortiori, T=q
•T1 sTm
A have no common Jordan blocks and if r < mum index such that then T has the form
r-l
1 is the mini-
r
(where n0=O and q0 acts on a {O}—space, if r
1) and we have
rank A5 < rank TS for ar-i + 1
mh — 1.
S
Indeed, if rank AS= rank T5 and if a5 Ct5, resp.) denotes the number of Jordan blocks of A (T, resp.) with order of nilpotency great er than or equal to s, then it is obvious that a5 > and this yields the contradiction rank
=
a5+rank AS > t5+rank
Setting T1=
TS =
rank
_10T', we can check that rank r—l
rank
r
and rank T13= rank T3—l for
T3 for j = -1, so that
rank
rank
<
3
for j = 1,2
mhl
mhl
On the other hand, m(T1,A) < m(T,A) - (rank T -rank T1 = m(T,A) -l = n, and consequently, T1 sTm A. If r > 1, then T slm T1 by ni—i ni is similar to temma 2.4. If r=]., then )
)
0)
and, letti,iig c +
A, and
T1. In either case, Tstm T1
0, we conclude that T
therefore A €S(T).
U
2.1.2 Proof of Theorem 2.1
Assume that T, A T is p, p(A)
=
(0
(A—A.)
d < co), the
minimal polynomial of
A., if I
and rank q(A)
j)
rank
q(T) for all
We want to show that T sTm A. Clearly, we can directly assume (without loss of generality) that T and A are unitarily equivalent to their Jordan forms; let T where = e.1 and is
= •jl
a Jordan nilpotent acting on the subspace
0 <
<
d,
j
= 1,2,
..,m. It is easily seen that H(Ar;T) = tjr= ker(Ar_T) r = ran and
= ran(Xr_T) r=ker Pr(T)i where Pr(T) = (r = 1,2,...,m)., whence we readily conclude that dim fl(Ar;A) = dim
for all r = l,2,...,m and (since d
dim that a(A) = c(T) = = Furthermore, since rank(Ar_A)5pr(A) rank(Ar_T)5pr(T) for all 5 = l,2,...,kr, r = l,2,...,m, it is not difficult to see that 16
r+Rr)t where Rr is a Jordan nilpotent acting on Hr and satisfying the conditions A
rank (R)5
rank
s =
l,2,...,kr,r=l,2,...,m.
By Lemma 2.5, r =
l,2,...,m,
for all
sTm Hr and, a fortiori, Ar+Qr slm whence it readily follows that T
stm j1
=
A.
The proof of Theorem 2.1 is complete now.
0
d-rank q(T), the conditions "rank can be replaced by the conditions "nul
REMARK 2.6. Since nul q(T) = rank q(T) for all
q(A) q(A)
>
nul
q(T) for all qlp"•
PROOF OF COROLLARIES 2.2 AND 2.3. It is easily seen that T has
the form of Corollary 2.2(u) if and only if S(T)
is maximal;
(v) =>
(i) is a trivial implication and (ii) => (v) follows from Theorem 2.1. Now Corollary 2.2 follows from the well—known algebraic fact that (ii), (iii) and (iv) are equivalent. Similarly, S(T)
is minimal if and only if k.= 1 for all j =
1,2,
...,m, if and only if T is similar to a normal operator acting on Q (v) =>
(i)
is a trivial implication and (ii) => (v)
d
follows from Theo-
rem 2.1. On the other hand, it easily follows from Theorem 2.1 and its then the minimal polynomial proof that if 1 for all j =
l,2,...,m,
of A
S(T)
is necessarily equal to p and that A
T, i.e., S(T)
=
S(T) is a closed subset of Conversely, if 2(T) is closed and T stm A, then A S(T) = 2(T) T. A fortiori, A sTm T, whence it readily follows and therefore A The proof that [A] = [T] = S(T) is a minimal element of of Corollary 2.3 is complete now.
0
2.1.3 The lattice
The partially ordered set (poset)
is
(L(H)/#,<)
never a lattice
(unless dim H = 0). However, it contains many interesting lattices. Con = is a finite dimensional space and let sider the case when H (T.)
T = then rank
•j=l
c
L(W
d
);
Tj+l+2Tj+2+...+(n_J)rn and this implies that
0,
for j =
l2,...,n.
(2.2)
Let Ed be the set of all (d+l)-tuples (m0,mi,m21...,md> of non— 17
negative integers such that and
for all j
0
>
= l,2,...,d—l.
Observe that these two conditions imply that m. > =
0, and
= l,2,...,d—l.
for j
unless m.
It follows that 0
= 0,1,2,. ..,d.
d—j for all j
= (mO,ml,...,md)
Let
(2.3)
and
be two elements of
=
then we shall write u
if m.
u v
=
for all j
= 0,1,2,. ..,d,
It is easily seen that
is a poset,
d =
...
>
= 0
and 2
<
so that
v
C
It is completely apparent that
least upper bound (l.u.b.) of
and
v p' is the
with respect to the partial or-
der . Since Ed is finite, every subset of Ed has a 1.u.b.. In particular, ii
is
A
=
1.u.b.tv
Ed:
v
u'}
ji and v
<
the (unique) greatest lower bound of
and i'. It readily follows that (Ed,) is a finite lattice with supremum
(d,d—l,d—2,...,2,1,0) and infimum (d,0,0,...,0,0,0). Given
where
,m ,...,m ) = (m 01 d
for j
I
d
,
define
by
T
= l,2,...,d-1
and Td=mdl; (2.2) and
(2.3) guarantee that the mapping ii
[Tn]
(2.4)
is a bijection from Ed onto N(IE )/# and, moreover, that rank T for j = 0,1,2,.. .,d. Combining these observations with Theorem 2.1, we
obtain
THEOREM 2.7. The mapping (2.4) defines an order-preserving bijecis a fiIn particular, onto
tion from
nite lattice with supremum
18
and infimum tO]
(0 =
2.1.4 Closures of similarity orbits of finite rank operators Let i-i
be an infinite dimensional Hilbert space and let T, A
F(H); then T and A are algebraic operators with nul P = nul
A =
= and
there exists a finite dimensional subspace H(T,A) reducing both, A and T, such that TIH(T,A)' = = 0 (H(T,A) can always be defined so that dim H (T,A)
<
Assume that A
2
rank T +
S(T);
2
rank A).
then we can prove exactly as in the finite
rank q(T) and, by using Proposition
dimensional case that rank q(A)
nul
1.12(u), that nul q(A)
q(T) for all qip, where p is the minimal
polynomial of T. Conversely, if A satisfies those conditions, then it is not difficult to check that AIH(T,A) satisfies the same conditions with respect to TIH(T,A) and therefore, by Theorem 2.1, TIH(T,A) sTm AIH(T,A). A fortiori, T = TIH(T,A)eO
hence, we have
A =
COROLLARY 2.8. Let T L(H} be a (necessarily algebraic) finite rank operator with minimal polynomial p, then
L(H):
rank q(A)
>
nul
q(T)
for all qlp}.
then L#T if and only if L
Let L
REMARK
2.9.
Since H is infinite dimensional, the conditioiis "rank
rank q(T) and nul q(A)
q(A)
T.
still imply that o(A) =
o(T).
nul
q(T) for all qip" of Corollary 2.8
However, the following example shows
that those two conditions cannot be replaced by "o(A) =
rank q(T) for all
q(A)
rank 2 and let A be an orthogonal projection of rank 1; then
{O,l},
and rank
Let T be an orthogonal projection of =
and rank q(A)
rank q(T) for all qip (p(X) =X(X—l) ),but A cannot belong to S(T), because 1 = dim ti(l;A) dim H(l;T) = 2. Re=
versing the roles of A and T, we see that conditions of Corollary 2.8 cannot be replaced by "o(A) = 0(T) and nul q(A) nul q(T) for all Let WF(H) ftJ(H)nF(H) and NFk(H) = Since H is infinite dimensional, UF1(H)
C NF2(H)
C
..
HFk(H)
c
{T
NF(tfl:
NFk+l(H)
rank T
k—1}.
c
is an infinite chain (all the inclusions are proper:
no two sets in
this chain coincide), and this chain naturally induces a chain of lattices
19
(wF2(H)/#,<) a...
(NF1(H)/*,<)
c (WFk(H)/#,<J
where
(as can be easily checked by using Theorem 2.7) is order—isomorphic with (Zk,); moreover, the above chain is in correspondence with a chain of natural order—preserving inclusions between the lattices
t
(
...
c
c ...).
c
c
Let (E,) be the set union of the Ek'S with the induced partial order, i.e., if for all k E, then p'
p'
and we define
in
if
u'
holds
in 0
COROLLARY 2.10. (NF(K)/#,<) is a lattice with infimum [0] and no
maximal element. This lattice is order-isomorphic with (E,) REMARK 2.11. However (as we shall see in Chapter VIII),
(t1F(I-1J/#,
<) does have a supremuiu in (L(tl)/#,<).
2.2
The distance from the set of all non—zero orthogonal projections to
2.2.1
N(K)
The limit case Let
P(IIJ be
the set of all non-zero orthogonal projections in
L(H). What is the exact value of the distance =
from P(U) to N(U)
inf{IIP—Qil:
P(I1), Q
P
A—li
(2.5)
(H an infinite dimensional space)?
According to Example 1.5, a(Q)n{A:
N(H)}
½}
> ½ for all Q in L(I-I)
such
that
0.
Since o(Q) = {O} for every nilpotent Q, the above inequality ½. It will be holds, in particular, for all Q in NIH). Hence, shown that
= ½; more precisely, if =
inf{!!P—QIi:
P
=
inf{I1PQII:
P
P1K),
Nk(14)}
Q
and Q
then it is completely apparent that ½
6k < 1 and
(&k)
is non-increasing with k. We have the following THEOREM 2.12. ½ <
½+sin
(m+l)'
integral part of (k—l)/2), for all k In particular , 20
lim(k
oo)
âk =
where in
= f.(k1)/2]
3.
lim(k
øo)
=
= ½.
1= the
We shall need some auxiliary results.
2.13.
LEMMA r+s = m+n.
Let 1
< in
<
n
and let m <
<
r
<
n,
s be Buch that
is an ONB of
If
resp.),
T =
and Lr=
a unit-
L
(T, L E
ary operator V s
L(U
r
ea
S )
such that
lIT — V*LVII = s(m), where s(m) =2sin
IIVTV* — LII
i
for m+1
r, and
for
(m+i)
s.
m+i j
PROOF. Define
1
t t
in,
ni+i
t
5,
m+l—r
ern+l_t,
(cos 2(rn+l))ern÷l_t+ (sin
=
(—sin 2(m+i))em+l_t+ (cos
=
1
0,
t
<
and s—i
R
tn—i
is also an
It is completely apparent that
generates the same two-dimensional sub—
the pair
ONB
t
space as the pair {em+i_t,ft} (for each t such that 1 < rn), T R. Now a straightforward computation shows that and L = L = VRV*, where V is the unitary operator defined by m+l—r = 1
t 0
t<
s
and
On
the
other
hand,
m-t-i-r t < -1,
0, (T—R)g0 = (l—cos =
(cos 2(m+l) (sin
(T_R)kt
(sin
=
0,
=
(sin
=
(cos
and
1
2(m+l)
m+l 2(rn+l)
1)kt+i,
1
t
t s, t m-l,
21
vf. . .} denotes the closed linear apan of L...}), these two—dimensional subspaces are pa: wise orthogonal and (T_R)kt for 1 t < rn-i, we conclude i Since
(1
=
t
m;
that IT—Ru =
.2 max{[sin 2(m+l) max
a(sin
+ (cos
2(m+l)
2(rn+i)
—1)
]
—1) +bcos
2(m+l)
I}
1aI2+1bl2=i =
+ (cos
Since IIVTV*_LII
_l)2]½= 2(m+l)
= IIT_V*LVH = IT—Ru, we
2 sin
(m+l)
=
s(m).
are done.
U
The above proof admits a very simple geometric description: action of q + e e + ÷ e + e + 0) can be described Ce r r r—i r—2 2 1 by an arrow of length r, and the action of T by two arrows oi lengths r and s, as follows m
...
r
Similarly,
S
R can be indicated as
n
T maps ern+i to em, the pair
of "parallel" vectors (j =
onto the pair
l,2,...,m—l)
and the pair {ei,fm} onto
On the other hand, R maps em+l to f1, which is equal to the result of a "slight twist" of the vector e1 in the two-dimensional space V{em,fi}, the pair {em+i..jifj} onto a "slight twist" of the "parallel pair" Ci = 1,2,. ..,m—l)ai a "slight twist" (in the reverse sense) of
to
The accumulation of all these "slight twists" after m+l steps wi
and f1 to 0. This fact will be described by the
map em to ing scheme
m
(T÷R) Conversely, if we take
of
-
as the original ONI
and modify R to obtain T, then the corresponding
scheme will be
(R÷T)
—
)m n
m
In this case, the twist will move g0 to km to 0 and k1 to 22
In what follows, we shall freely use these schemes instead of ana lytic descriptions in the proofs.
.4
COROLLARY 2.14. If k
3, ),
e
1(k—1)/2],
m
1
p =
k—rn
-l
then U9
pj
the unitary operator defined by V e.=
p
)3e.
(where
and U
=
p
where —w
p
tivep—th root of 1) and there exists a unitary operator V
is
is a primik L(ct
such
)
s(m).
that
PROOF. It is straightforward to check that
VP
(Observe that the three operators are unitary and
=
(_1)P•)
=
Represent
by a "curled arrow" and modify the action of the operator in the subspace V{el,e2,...,em,em+l,ekm,ekm+l,...,ek} as in Lemma 2.13
in
-÷
We conclude
as in the above lemma that there exists an operator
= s(m).
such that
Lmp
P
CkJ
0
PROOF OF THEOREM 2.12. With the notation of Corollary 2.14: Let be the orthogonal projection of ak onto the one-dimensional where m = [(k—1)/21 and r = k—rn; then
kernel of
½-l-½ s(m) = ½+ sin Hence,
½+sin P
Since set of
(n+l)
readily follows that and IQU
Q
c
N(Q:k):
1Q11
3/2+sin (2.6)
3/2+sin
is a compact sub
it is easily seen that the infimuxn in (2.6) is
actually attained for some pair
in this set. Thus, by our
previous remarks (Example 1.5), >
½.
2.2.2 On the exact values of PROPOSITION 2.15. If
2
h = dim
H
=, then 23
N2(K)} = /2/2.
P(H), Q
P
inffllP—QII:
Furthermore, the above infimum i8
actually
attained by the opera-
tors
10
½—½
0h—2 c P(H1 and
0 0
½
—½
0h_2
where the 2 x 2 matrices act on a two-dimensional 8ubspace
of Il'
and
0h—2 denotes the 0 operator acting on (H2)1.
PROOF. It is straightforward to check that N2(H)
and
and
hand, if Q
= 1, or
=
Assume that I(P—Q)xII Let z
=
<
<
ran
P(H), then
N2(H) and P
either Q = 0
0.
Q
/2/2. If x
<
(/2/2)IIxjI
ker Q, y i. ker /2. We have 1 =
/2/2.
=
the other
On
and
ker
ker
Q and y i
Il(P_Q)*yiI
IP*yII =
<
Q, then ITPxTI
(/2/2)IIyII.
(i.e.,
Pz = z) be a unit vector and write z=x+y, x
Q. Since
it follows that IxIHIylI
P
IPxIl+flPylj
ljPx+PyII
a contradiction. Hence,
(/2/2)
<
1,
,/2/2.
0
It readily follows that 62=12= ,/2/2. This is the only known val
2).
ue of 6k or 1k (k
If k = 3,
111 P3= 1/3 1 1 1
011 and
111 then P3
=
2/3
0
0
2
=
1
2/3[q3+(q3)
],
000
(P is a rank one projection), Q3
and 11P3—Q3Ilis
< 2/3. equal to 2/3. Thus, ½ 13 < This result is certainly better than the poor estimate 63 ½ + = 3/2 given by Theorem 2.12 and the same is true for the k-disin 1/k
mensional analogues
and
of P3 and Q3, respectively, for small values
of k.
However, there is some numerical evidence that the estimate for all k 20. is much better than CONJECTURE 2.16.
= 1k
for all k
2,
and ½TT
24
< lim(k + Ca) k(nk_½)
<
2ir.
has a limit
2.2.3 A companion problem:
The distance from the set of all non—zero
idempotents to W(K)
Clearly, P(H) is not invariant under similarities. The smallest set invariant under similarities containing P(I1)
is the set of all non zero idempotents acting on H, ECU) = P P1K), W G(H)} = We shall see later that if and R2 are two L(H): {E
subsets of LIft)
,
where
is an infinite dimensional space, then R1nR2
ft
is "very large", in general; namely, when
contains an operator
is not an algebraic element of AIR), for j = 1,2.(See of the second monograph.) Very little is known about
such that Chapter
RJ
dist[R1,R2] = inf{11R1—R211: for the case when one of the sets
R.,
j
= i,2}
or R2 is a subset of the algebra-
ic operators (or, more generally, operators whose images are algebraic elements of the Calkin algebra). In particular, what is the value of ri'
=
dist[E(H),N(I-f)]
for the
or
case when ft is infinite dimensional? What is the answer if ft =
(and dim H = 'a)
N(H) is replaced by Mk(tl)
for
some k > 2?
Clearly, if k
=
k
),W(a )I
and
=
then 0
for
<
<
distl.E(ft),Nk(U)] all k
2.
= 1/k,
PROPOSITION 2.17. (i) and
all
PROOF.
1/k and (i)
but IIE—Qil
> 1/k
for
all E
in
(k 2).
Q
0
(ii)
(ft infinite dimensional),
0.
=
be the canonical ONB of
Let
e.øe.
and
(n =
and let
1,2,...).
=
straight-
Wkfl = forward computations show that.Pk is a rank one projection, Wk,fl is ia
vertible, Ekn= Wk,nPkWkn1= (n
E
k,n
—
— =
k
)
n
k-i k-i
i
n
-r
er+jøerfl
ereerll+(1/n)
l/k+1/n, 25
1/k.
we see that
On the other hand, if F
is an ONB, then {l}
and
c
a (F) c {O,l} and
trace F so that
rank F
=
=
1,
1/k for (at least) one value of j, 1 < j k. so ), we can find an ONB strictly upper triangular with respect to that basis, i.e., = 0 for 1 j i k. In particular, we have
Given F as above and Q that
max
IF—Oil
so that IF—Oil
= max
l<(F—Q)f.,f.>I
=
1/k
for all j
>
=
>
1/k,
1/k.
1/k, i.e.,
k, it easily fol1/k or = 0 for 1 < j < I k and 1,2,.. .,k; i.e., F admits an upper triangu-
Furthermore, since =
lows that, either hF—Oh
it
0
for 1
i
j
lar matrix with 1/k in all the diagonal entries with respect to the But the second possibility implies that a(F) {1/k}, a ONB = 1/k for all F contradiction (recall that k 2). Hence, IIF-Qhl > k k inE(cC) andallQinN(cE).
Since the second statement is a trivial consequence of the first one, we are done.
Clearly,
o
the trace argument cannot be applied in the case when H
is infinite dimensional, but the above result suggests the following CONJECTURE 2.18.
and all Q
in
= 1/k,
=
Nk(HJ
(k
but hIE-Qil
> 1/k
for all E in
2).
The last result of this section says that the above conjecture
true
at least for k = 2.
PROPOSITION 2.19. If 2
h = dim H H
=,
then
EU!), Q
= ½,
but this infimum cannot be attained for any pair (E,Q), E
E(H).,Q
N2(H). PROOF. If E = 1, then it is clear that lIE—Oil = hhl—Qhl = 1
for every quasinilpotent Q.
sp(l—Q)
Let 1 M ran E
E= be the matrix of E
E(tfl\{1}
ran EI(ran E)1 and let 26
00 (ranE)'
with
respect to the decomposition U =
= tc be the matrix of Q
L(H)
(with
Dj
respect to the same decomposition).
It is immediate that Q
if and only if
A2+BC = D2+CB = AB+BD = CA+DC = 0.
Assume
that
c {A:
Qil < ½; then Ill — ½} and 0(D) c {A:
E—
1—Al
½ and
All
½}.
Al
IIDII < ½ and therefore Thus, by the spectral
mapping theorem, 0(A2) = a(—BC)
c
{X:
A
¼}
and
= o(—CB) c {A:
Al
¼}.
Since a(—BC)\{O} = c(—CB)\{O} (see, e.g., [119], [153], [l72])and A is invertible (recall that Ill—A II o(-BC) = cy(A2)
{¼}
c
½ <
1 ),
it readily follows that
a(—CB) = 0(D2) c {O,¼} and cJ(A)
Ill—All > sp(l—A) = ½. Assume that IIE—Qil = ½. Since a(A) = {½}
0(D)
= {½}
{0,½}. Hence, IIE—Qil
+0
of unit vectors in ran E such that
sequence
(n
oo). On the other hand, A2 = -BC and A invertible imply that llcxIl
+
Elixil
for
some c
IIE—Qil
>
a
there exists a
=
0 and for all x in ran E, so that
>
lint
sup(n +
lint
sup(n + co){lI(l_A)x112+lI_Cx112}½
(¼+c2)½
co)
> ½,
contradiction. Hence IIE—QIl
On
>
½
all Q
in
the other hand, it readily follows from Proposition 2.17 that inf{lIE—Qlj:
2.3
for E
E(H), Q
N2(H)}
0
½.
On the distance to Nk(H)
2.3.1 A general upper bound
k
LEMMA 2.20. Suppose that T 2 and some €. 0 < e 1. Let
1 and llTkll L(H), 11TH (T*T)½ = f[0,1]A dE
n for some
(spectral
decompo
Bition) and let P = E([0,b/c]); then I!f(l_p)T(l_P)]k_]ll
PROOF. Clearly, IITPII
< (k—lh/c.
and
27
lI(T*T)½Tk_lxII2
=
12
= IIP(T*T) ½Tk 1x+1-P (T*T)½TklxjI2 (l—P) (T*TY½Tk_lxIl2 =
II(T*T)½(l_P)Tk_lxIl2
elI(1—P)T1x112,
>
<
for all x in H. Hence, On the other hand,
so
that
IUl_P)T]k_lIt (l_P)T]kij+...
{II(1—P)TPT[ <
THEOREM 2.21. Tf T
and
some c,
0 <
1,
c
Il(l—P)T
L(fI), 11Th
U
and
1
hhTkhl
for some k
2
then (2.7)
dist[T,Nk(H)]
where qk(E) ia a continuous, positive, non-decreasing function defined
such Moreover,
on (0,1]
=
0.
=
0)
that
(2C)½
can be inductively defined by
Ic+4k_l((k_l)/c)2}½,
and
for k = 3,4
2.20
PROOF. Let P be defined as in Lenuna
T=
and
E1T1 ranP
let
E2 T2 ker P
be the matrix of T with respect to the decomposition H = ran Poker P. If k = 2, define 0 T1
T' Since
PT(l—P)
0
=
0
-
E10 2
and ker P
i.
ran
P,
it
E
0
IIT—T'hl Assume
00
kerPO,
A
that k >
ranPO 2
follows from Lenuna 2.20 and its proof that 2 H 0 00 21½ 1
E2 T2
and
3
0
+
IT2
II
0
T2
and (2.7) holds for j for j =
Clearly, 28
=
= hI(l—P)T(l—P)
1
<
k—l,
(c+c)½
with
(2c)½.
= (2c)½
3,4,..,k—l. and, by Lemma 2.20, IIT2k)1I
Thus, by our inductive construction, we can find (ran P) such that Define
Nk_l
E
0
=
0
It is easily seen that T'
Nk(tI) and a formal repetition of our
previous argument (for the case when k = 2) shows that IT—T'tI
=
<
(def) k.
COROLLARY 2.22. Let polynomial and let exi8ta
IS
=
p(A)
L(H):
0 8u0h that, if 11Th
>
i
if
= =
1 and lIp(T)
II
0). Given IS,
c
j)
be a
> 0, there
then
< £.
1, then the result follows from Theorem 2.21. As> 0 such that the m open disks
PROOF. If m =
sume that m > 2. There exists
of radius
centered at
A2,..., Xmi respectively, are > 0 be such that if lp(A) j < then A is
pairwise disjoint. Let
contained in one of these disks. Let dA.
=
We shall show that for small ltp(T)hI.
p().)
is a bounded function of hlp(T)II, 11Th,
Observe that p(A) -p(T) =
(A-T)q(X,T) for a polynomi
al q in the variables A, T. Thus
1—
=
Now, for A such that
small
if IIp(T)Ih
where
<
is.
=
is
we have that
Let E=p(A) 1p(T); then
is small since
bounded by a funtion of 11Th and p(A). Thus the
are bounded (j
=]
2,...,m). On the other hand,
= 11th
dxli
IIJII.c0
=
Since each
small (provided hlp(T)Il
use Theorem 2.21 to perturb
is small)
we may
T
a nilpotent of order at most k3. If II
=
ran
define
with respect to this decom-
29
position (j
= 1,2,... ,m)
and
=
T'
Then p(T') = 0 and
IIT—T'Il
provided
< 6 for some 6
lIp(T)
>
0
j m}
1 <
P.11:
<
small enough.
be as in Theorem 2.21 and let 1, be the 0 < < k = 2,3,...; and define = then Theorem 2.21 implies that, if k = 2, then dist[T,N2(H)] < L 3 k = 3, then dist[T,N (H)] and dist[T,W (fiji = O(c 3 k k2k for all k 2. = Let T, k and
(i.e.,
k—th root of
if
)
These results can be slightly improved if, instead of (k 3) in Lemma 2.20. With this choice of P, it is possible to show that can actually be defined
we chooseP=E([0,{2c2/(k-2)}1"3])
so that
0(fl(9k/8)(2/3) k
=
dist[T,Nk(H)] (k =
3,4,...),
but these results seem to be very far from the best pos
sible ones.
defined
CONJECTURE 2.23. There exists a continuous function on [0,1], such that iJ,(n)
>
0
on (0,1], ip(0) = 0 and dist[T,Nk(ti)]
for all T in L(H) such that 11Th
functions
1
and
lITkII
(i.e.,
<
the
can be replaced by a single one).
2.3.2 Two illustrative- examples
The rough argument of the proof of Proposition 1.10 might suggest that those estimates are very poor. However, Lemma 2.13 shows that the 'very poor" lower estimates given by Proposition 1.10 are actually the best possible except, perhaps, for a constant factor independent of k. Indeed, if T and L have the form of that lemma, then nul Thus, if W (U) is an invertible (unitary, resp.) op> nul erator, we can always find a unit vector x = x(W) = 1, but (WTW-l)m+1X = 0 that
(=
(
21/(m+l) —1 (dist[L,U(T)] > l/(m+l)
resp.) such
= 0, resp.), whence
((ii), resp.) that dist[L,
it readily follows from Proposition 1.10(i)
S(T)]
x(U),
,
resp.)
An even more surprising example can be constructed on the same lines. We shall need the following auxiliary result (With the notation of Lemma 2.13):
the 30
COROLLARY 2.24. (i) Let bilateral shift defined by
be
=
an OND ef H and let U be
for all n
Let
=
let A
and
=
(B e
L(II_),
reep.) be the forward (backward, resp.) unilateral shift defined by for all ri 0 (Bg_1 = 0 and for all < —1). = =
let
Finally,
L(Gm). Then there exists a uniUmOH
tary operator V•ti +
that
such
= s(m),
(ii) If k tor Wkm:
3
>
and 1
m <
<
(=)
(am)
k1 and
=
[(k—1)/2], there exists a unitary opera
such that = sCm).
PROOF.
For suitably defined A'
Ci)
A and B'
B, we have
whence we obtain (—=,m)
——
—
:
(0,oo)
- (A'wB')fl
= sCm) and the unitary operator Vm can be chosen so =
that
=
(ii) Since r=k—2m
and
= _km•
1, there exists U'
U such that
(co)
+
k
k
k m (the fact that r
1
k
m guarantees that we can consistently apply the arm
=
gument of Lemma 2.13 to each step.), with whence the result follows.
0
EXANPLE 2.25. Let S be a unilateral shift of multiplicity one and let T
L(ti)
all k >
1,
be unitarily equivalent to s( =
dist[T,Nk(tl)
— 1
for all k
that
= {A: j
XI
<
then
llTkjT
= 1 for
1) and
4 sin ir/([
(k—l)/2]+l)
<
8ir/k,
3.
1 and = 1 for all k PROOF. It is not difficult to see that for all X in the open unit disk D = {X:
< 1),
whence it follows that aCT) =02,re(T) = D. Qk=0, it follows from Proposition l.10(iii) Since 21/k_1 3, m= (k =1,2,3,...). On the other hand, if k that flT—QI[ Aj
Let Q
31
[(k-l)/2] and the operators are defined with respect to a suitable ONB of if, it is not difficult to infer from Corollary 2.24(i) and (ii) that (co)
—
(oo)
Til
(co)
—
—
=
_S,S*Ij
= 2s(m)
(k—l)/2]+l)} <
= 4{sin
Our next example shows that the infinite dimensional aiupliations of finite dimensional examples can produce certain surprises. (An oper ator T in LUll
called an ampliation if T
L(H0), 1
A
EXAMPLE
(h)
is
dim
2.26.
oo.)
Let k > h
kh
(A, B
AGi
3; then the operators A =
and B =
)) satisfy
dist[B,S(A)] = 1;
(2.8)
however,
4{sin ir/([ (k—l)/2]+l)
— 1
+sin
(2.9)
,r/([(h—l)/2]+l)} <
PROOF. Observe that
nul A = k >
nul
B = h. Thus, distEB,S(A)]
fling of this section.) On the other hand, A
S(A),
that 0
1.
at the begin
(Use Proposition 1.10(i) as in our previous observations
0, so
sA for all
whence we obtain (2.8).
The lower estimate of (2.9) follows from Proposition l.lO(iii)
= 1.)
=
and
B'
The upper estimate follows from
For suitably chosenT
the proof of Example 2.25:
A'
B, we have
IIB'
— A'
—
—
+Sjfl
A'
<
4{sin
IT/([(h—l)/2J+1)}
it/([
(k—i)/2]+l)
<
[1
is infinite dimensional, then denotes the (where NUll: IIQ1I 1) "looks like" = {Q set of all rational numbers), in the following sense: Observe that Example 2.26 suggests that, if II
1
W+iW = uk_i (Qk+iWk), where "large" nowhere dense subset of A
and (W+iW) C,
32
and
N
1
(H)
n k},
is a
0
(k
0o)
(in the complex plane) coincides with the much larger set 1
= Uk=i
(H)nNk(K)].
A
CONJECTURE 2.27. some constant C >
N3-(1-f)}
Q
C/k
for
independent of k.
0
An affirmative answer to this conjecture would provide some heuristic explanation to the wild structure of
(see Chapter V).
2.3.3 An example on approximation of normal operators by nilpotents Let k
50, p =
>
and r =
=
be the operator defined by
with respect to
=
where
of
the ONB
and let
n
j
for
j
for
(Roughly speaking: grow from
n =
<
k—i.
is a truncated weighted shift; the weights
to 1 through p steps of length r and then go down from 1
to 0 through p steps of length r, so that the upper step has length 2r with weights equal to 1, i.e., a. =
1
for r(p—1)
<
j
r(p+1).)
In the first modification, we shall "ignore" the
1,2,
...,r(p-l) and r(p+l)+l,r(p+l)+2,... ,k and apply Corollary 2.14 to the
It is easily seen that
subspace
we can modify
in order to obtain an operator
where
=
is a unitary operator acting on a si.thspace
Ur+i
of dimension r+1 and there exists an orthonormal system is an ONB of
such that for j r-2,
for h =
(r(p—l),r(p+l)], Tjer(pl)
for h =
(l_fl)er(p÷l)+l and
is a primitive (r+l)—th root of
1); furthermore,
1,2,...,
r,r+l,...,2r =
s(r—i) Let T1 be the operator obtained from Tj by replacing each weight equal to 1 by 1—ri and
then IIQk_RiH Now we can apply the same argument to R1 in order to obtain an op
where U2r is a unitary operator acting on
erator
a subspace of dimension 2r, whose eigenvalues are equal to minus the 2r 2r-th roots of 1, T2e. =
Qke.
for
all j
/ (r(p—2)
for h = 1,2,...,r—2,
,r(p+2)], T2er(p_2, (l_2fl)er(p+2)+lI
is an orthonormal system that spans a subspace orthogonal to the span of the vector ({ej}
and this second modification only affects the vectors 33
in the subspace spanned by ({e.}.
so that <
s(r—l)+n,
etc.
An easy inductive argument shows that after p-i steps we finally obtain an operator
Lk= such that IIQk—L,,jI
< s(r—l)+n
On the other hand, if 1
k
for all k >
<
<
50.
50, then
> 1. Thus, we have the
For each k
1 there exists a normal opera
following
PROPOSITION 2.28. (i) tor
such that liLkil
1
and
<
5(ir/k)½.
(ii) If H is infinite dimensional, there exists a normal operator M such that o(M) = D, where D = {A: < 1) and distrM,Nk(H)] < 5(ir/k)½ for all k = 1,2 In particular, M N(H).
PROOF.
(i)
If k > 50, define
as above. If 1
<
k
50, take Lk
=
(ii)
Let {Am}ffl°°l be an enumeration of all thosepoints A in D such that both Am! and (arg Am)/ir are rational numbers (arg 0 is de-
fined equal to 0) and let M be a diagonal normal operator with eigenvalues A1,A ,...,X ,... such that nul(A —M) ==for all m = 2 m m (co) e., N =
Lk)
1,2,...,
Given k, it is easy tosee thatMcanbe written as N whence it readily follows that
dist[M,Mk(K)] A fortiori, N
NIH)
(sup
AmI)IIL1K_QkI!
i.
Am
<
0
.
The result of Proposition 2.28(i) is, in a certain sense, the is normal and there exist
best possible. Observe that if Nk and (i)) a(Nk)
> 0 such that
<
then (by corollary 1.6
is a connected set containing the origin. If the points
of a(Nk) are more or less evenly distributed in a connected neighborhood £2 of the origin with smooth boundary (namely, a(Nk)
will include £2 and therefore k7rEk2
£2 =
D), then
m2(0), where m2 denotes
Ek
the planar Lebesgue measure. Hence, >
(On the other hand, if ek is too small, ed, a contradiction.)
=
then
cannot be connect
o(Nk)E k
CONJECTURE 2.29. There exists a constant c > 0 (independent of k) 34
Ck½ for every normal operator N
such that such that IINII =
(k = 1,2,...)
1
The following result provides some extra support to this conjecture. Observe that if A is hermitian and 0 A 1, then the paints of a(A) are not evenly distributed in any set of positive measure.
2e+ire2 independently of k, and 2c+
(More precisely, m2(a(A))
0.)
0, as c +
PROPOSITION 2.30. If A >
(1/2/k),
1, and 1
A
0
a(A)., then
k = 1,2
PROOF. Assume that IIA—Qil c for some Q Q = H+iJ (Cartesian decomposition); then IA—Hil = IRe(A—Q)II < IIA—Qil e and trace (H) = trace (Re Q) = Re trace (Q) = Re 0 = 0.
On the other hand, it is easily seen that cy(A)
(Use Corol
lary 1.6(i)), so that
1+(1—2e)+(1—4c)+...+(1—2nc),
trace (A)
integral part of (l/2e)). It is clear that
where n = n >
1/2c—1. Hence, trace (A)
(n+1)
(n+l)
>
>
l/2c
and
j = (n+l)—(n+1)ne = (n+l) (1—ne)
—
>
l/4e.
Let A = f A dE and H = 5 A dF (spectral decompositions). If a and rank F((a—c',=)) < rank E([a,co)), then it is easily e'> e
[0,1],
seen that there exists a unit vector x
ran
E([a,=))eran F((a—e',oo))
and therefore
l<(A—H)x,x>j
IA—fill
>
ct—(cz—e)
=c,
(2.10)
a contradiction.
rank
Hence, rank F([a—c,w))
rank
F([ a,"))
E( U a—c,"))' for all a
0 = trace (H) Hence, c
>
E([ct,oo)) and, by symmetry, rank
1/2/k.
>
trace
[0,1].
(A)
It folows that
>
By a compactness argument (exactly as in the
proof of Theorem 2.12), we conclude that Qk = min{IIA—QlI:
=o, UQII
2}
>
1/2,/k.fl
2.3.4 On the distance to a similarity orbit
be a cyclic operator with minimal polynomial p, and T is i j); then
Let T e p(A) =
ilar to the Jordan form
Let
J— A
L((t
)
(A.+q1,). 3
-I
be an operator with spectrum 35
and dint
Define
.(Cleariy,
=
and '
=
=
= Bd...h+2 = an upper triangular
a1,
=
a;
8d =
then A
8h1+h2_a2,
...
,
admits a representation as
matrix of the form B1 a12
a13
B2 a23
.
.
aid
.
.
a suitable ONB of
It is not difficult to con-
clude from Theorem 2.1 that U] a12 a13
.
.
.
aid
p2 P3
.
.
0
Moreover, the same result applies to any upper triangular representation of A. Hence, we have COROLLARY 2.31. Let A and T be as above; then dist[A,.S(T)]
mm
max
p. —
ceE(k)
(2.11)
Bci•tls
where 1(k) denotes the set of all permutations of k elements.
Unfortunately, the estimate (2.11) is very poor, in general. Name
ly,
if
clic,
is cy-
and 0k have the form of Proposition 2.28(i), then sim 0k
(by Theorem 2.1) and <
However, sp(Lk) =
1
(—1
ci(Lk)) and
+ 0 =
{0},
(k
co).
so that the only
information that we can obtain from Corollary 2.31 is that distULk,
1. PROBLEM 2.32. Find a formula for 36
dist[A,S(T)] (A, T c
We shall close this section with a partial answer to this problem.
COROLLARY 2.33. If T=ek_l rank A
and A=,knl
rank A j for j. =
rank operators, rank T j = r+l
(Tk)
for some r > 2,
l,2,...,r
are finite
and rank T
r+1
<
then
—l
dist[A,S(T)) <
2
s([(r—l)/2]).
PROOF. The lower estimate follows from Proposition 1.10(i). In order to obtain the upper estimate, we can directly assume (for some d, 0 < d <
without loss of generality that A, T
Then, our hypotheses and formula (2.2) imply that
=
for
j
1,2,
=
Since r > 1, this means, in particular, that T
r—l, but Tr <
and A have exactly the same number of direct summands, which is equal to T =
After eliminating all common direct summands, we can directly assume (without loss of generality) that T =
[(r-l)/2]
Let in = (A
u
(T))
q •
and
•k=r+1
and mkrk—O for k=r+l,r+2,...,n. and let n = d-Tm; then r
r
blocks of length r
—
flu,
in
(where r <
n and n' is the largest Jordan block of A),
flu
flu'
s(m).
whence we obtain
s(m). Combining these
Similarly, we have two inequalities, we obtain
dist[A,S(T)1
dist[A,U(T)] +
<
s(m).
2
o
2.4 On the distance from a compact operator to N(H} be an ONB
Let
of
Clearly, sp(F) = k
ONB for
tors
•k
2 and let
arid let F :
= 1 <
j
d}. Embed
in
be an
ii
such that f. belongs to the subspace H. spanned by the vec It follows from Theorem 2.12 that e1,e2, (j =
l,2,...,d).
37
..
•
-
can be chosen in such a way that
½ +
<
Define
2it/k.
G1 A12 A13
.
•
G2 A23
.
.
Aid
. .
G=
e
0
Gd
where
and
then
1
IFSO((k_l)d) —Gil =
1
<
Let K
K(H)
(H infinite
< d}
j
d}+2ir/k=sp(F)/2+2ir/k.
j
dimensional) and let c
Then there exists a finite rank operator F
0
>
be given.
such that IlK-
Fil
<
C/3.
Moreover, by the upper semicontinuity of the spectrum (Corollary 1.2), F
can be chosen so that sp(F) < sp(K)+c/3. Since F = d
H, dim H C
FIH, let
M
F(fl),
E
there
exists a finite dimensional subspace H
1, such that H
C
e
C
L(H)
Then G
for some k
G and G IN1 C
C
=
above
0.
C
IIK_FCIHIIF'C_GcII ½
Since
C
c/3 and define G e tIN) as
in such a way that G IN C C kd = 0 and F (H), G
C
IIKGII
0. Let F =
=
H1
C
be a subspace of dimension kd containing II
large enough to guarantee that 2ii/k <
and G
and F
reduces F C
of
<
< C/3+½ sp(F)+c/3
sp(K)+C.
can be chosen arbitrarily small, we obtain the following
upper bound:
PROPOSITION 2.34. If K
Kill)
(H an infinite dimensional space),
then the distance from K to the set of all finite rank nilpotent opera tars cannot exceed ½
sp(K). In particular, every compact quasinilpo—
tent operator can be uniformly approximated by finite rank nilpotents.
2.5 Notes and remarks
The problem of characterizing the closure of a similarity orbit in simple terms was raised by D. A. lierrero in [139]. This reference (for a in a Banach
contains all the basic properties of the sets S(a)
algebra A), the notion of asymptotic similarity, several properties of 38
the poset (A/#,<), the analysis of several relevant examples, etc. The results include the proof of the implications (ii)
>
(iv)
(i)
>
of
Corollary 2.3 (in the above mentioned more general setting [139,Propo— sition 1])
Theorem 2.1 and Corollary 2.8 are due to J. Barrfa and D. A. He— is a lat-
rrero [43,Theorem 1.1], who also proved that (F(I-f)iiN(I-f)/#,<) tice
dim
(0 <
for
to). The concrete model
Il
this lattice (Theo-
rem 2.7) is an unpublished result of C. Apostol and D. Voiculescu[34].
In [129], J. H. Hedlund tried to dist[P(H),N(H)] and showed that 1/4 <
value of ri
<
p2=62/2/2
(Proposition
2.15). He also analyzed the norm of the difference
for sever-
al values of n < 100 and suggested that these operators could be used to prove that
+
½
(k +
communication to the author.)
(D. A. Herrero [132,Corollary 9], by using infinite dimensional arguments; 2) The solution of the problem evolved as follows: does converge to ½,
as k +
CD.
1)
A. Herrero [133,Section 7(d)], modify
½+[1+(k_1)½]/2k
ing the previous argument);
½
½+1/2/k (N. Sali
nas, [l8l,Lemma 3.3], by using an argument due to P. R. Halmos [125]); 4)
>
½ and
and
<
½+sin
<
½+(8 log k)/k (D. A. Herrero, [149,Proposition 6.5]) ii/([
(k—l)/2]+l)
CD. A. Herrero, [150,Corollary 5.2]).
P. R. Halmos and L. J. Wallen called an operator T in L(H)
k
er partial isometry if
1
a pow-
and prov
ed that T has this property if and only if T
where S is a unilateral shift of multiplicity one and adjoint S* and V is a unitary operator [126]. Lemma 2.13 and its consequences (Corollaries 2.14, Sections 2.3.2 and 2.3.3, Corollary 2.33) are contained in the article [150], by D. A. Herrero, essentially devoted to obtain upper and lower estimates for the distance dist[A,U(T)] from a power partial isometry A to the unitary orbit of another power partial isometry T. However, as remarked in [150], the technique described by Lemma 2.13 is due to I. D. Berg [481 (see also [46fl;this technique has
also been exploited by C. Apostol and D. Voiculescu in a differentcontext (unpublished results:
Corollary 2.24(i) is due to these authors.)
Proposition 2.28(u) is the key result of the article [132], where it has been obtained by a very different argument. Proposition 2.30 is an unpublished observation of D. A. Herrero. According to Proposition 2.30 and {149,Corollary 6.1], if A c
O
1
and 1
o(A), then 1/2/k <
<
10/log
k,
for all k large enough.
The results of Section 2.2.3 are unpublished observations of D. 39
A. flerrero (Rowever, the fact that
=
J.
dependently proved by B. Aupetit and
i/k
was also
in [200,Exaxnple 2.4]).
It is worth to remark that the proof of Proposition 2.17(i) actually yields the following stronger result:
PROPOSITION 2.35. For each k infCIlT—QII:
T
£
c(T)
2
Q E
N(&C)} = 1/k4
but this infimum is never attained.
The results of Section 2.3.1 are due to S. L. Campbell and R.
Gellar
[64].
That every compact quasinilpotent operator is the norm limit of finite rank nilpotents is an observation of 9. G. Douglas [l23,Problein 7] and
the
improvement given here (Proposition 2.34) is due to D. A. Herrero E149,Proposition 6.6].
40
3 The main tools of approximation
In
this chapter and the following one, we shall analyze the main
"tools" of approximation. Some of these "tools" developed within the framework of Operator Theory: The Rosenbium equation AX — XB = C, Rota's universal model and its generalizations, the triangular representation of C. Apostol and the theorem of C. Apostol, C. cu On
and D. Voicules-
normal restrictions of small compact perturbations of operators. These, as well as some of their consequences, will be developed
here. Two other important results (the Brown—Douglas—Fillmore theorem and Voiculescu's "non—commutative Weyl—von Neumann theorem"), from the theory of C*_algebras, will be analyzed in Chapter IV.
3.1 The Rosenblumoperator:
X +
AX-XB
3.1.1 Linear operator equations
LetAbe a Banach algebra with identity. The mapping a +
where
is defined by La(b) = ab (b e A) is called the regular left representation of A. It is well-known that this mapping is an isometLa C L(AJ
ric isomorphism from A into L(A).
Similarly, the regular right representation (defined by a + Ra L(AJ, where Ra(b) =ba) is an isometric anti-i8ornorphism (i.e., ab +
e
=o(a) for all a in A. =ac—cb, i.e., Given a, b A, Tab L(A) is defined by TabLaRb• It is easily seen that La and Rb commute and that RbRa)• In particular, o(La)
T(aA) (A,
=
'ab
—
U), so that =
"'(a—A)(b—A)"
iflf{IIL(aU)ll+IIR(blJ)lI)
=
LEMNA
3.1.
Let X be a complex Banach space and let A, B c L(X) be
41
that AB=BA; then
two operators SUCh
cj(A+B)
c
ci(A) +a(B) ={cz+8:
a(AB) c o(A)cy(B) = PROOF. Let 8
B. Clearly, 1 (p 1
0(B),
8
c
0(B))
cy(A),
a
0(A),
C
L(X) be a maximal abelian algebra containing A and belongs to 8 for all A / 0(A)
and
resp.).
Let
be the set of all multiplicative linear functionals on 8; then the Gelfand theory implies that a(A+B) ={4)(A+B):
M8)
4
={4)(A)+4)(B):
4)
M8) = o(A) +a(B) and ={4)(AB):
c f4)(A): COROLLARY
3.2.
{4)(A)4)(B):
M8} =
•
M8}.{4)(B) :
•
0(Tb)
cj(a)
M8}
•
M8} = a(A) .o(B) .
4)
I]
—a(b).
it
follows from Lemma 3.1 that 0(Tab) = PROOF. Since LaR0 = RbL, a(La_Rb) c 0(La) —ct(Rb). On the other hand, a(La) = 0(a) and a(R,0) = c,(b), whence the result follows. fl We shall see later (Corollary 3.20) that when A = L(U)
this inclu
sion is actually an equality.
3.1.2 Approximate point spectrum of a sum
of
L(X) is the set
The approximate point spectrum of A = {A
C:
commuting operators
A-A is not bounded below)
and = {A
A—A is not onto)
is the approximate defect 8pectrum of A. It is completely apparent that a
Furthermore, if A
(A)
a
c
(A)
c
a
(A) and a
(A)
denotes the Banach space adjoint of A, then
it is not difficult to see that
a(A)
and
Hubert space, a(A) = where = {A: A Li) for each ii a
42
ar(A*)* c
and cYd(A) =
a(A)
=
LEMMA 3.3. Given any Banach space X, there is an isometric imbedding of )( into a larger Banach space X', and a mapping A -* A' into
of L(X)
L(X') which is an isometric isomorphiern such that every A' is an
extension
=a(A') =0(A).
of A and o(A')
PROOF. Let
be the Banach space of all bounded sequences of = sup
elements of X with the norm 0
let
oo)}. It is
(n
seen that
is a
subspace of
and the imbedding of
We define X' =
(= the coset of {Xn})i where Xn = x for all
X into X' by x +
n=
Clear
1,2
ly, this mapping is an isometric isomorphism of X into X'.
Similarly, given A in L(X), we define A'
L(X') by A'[{xn}] =
A' defines an isometric isomorphism from L(X) into
then A L(X')
If
of unit vec-
then there exists a sequence
A
+0
tors in X such that that
(n -* Go).
It is not difficult to check
whence it follows that c71.(A)
=
On the other hand, if
c o(A') c for all x
c
in X, then it can be easily seen that
cfl[
{xn}]Ij
for
e X'. Hence, c(A') c a(A) and therefore
all
=
L(X) and AB Go BA,
THEOREM 3.4. If A, B E
then
Ii)
(ii)
c
(AB)
c a
(B)
(A) .a 71
71
71
a(A+B); then, by Lemma 3.3, A and A'B' =B'A'. It readily follows that the subspace
PROOF.
(1) Let A
M = ker(A'+B'—A) is invariant under A' and B', and B' IN = (A—A') jM.
Thus, if a
a (A'IM)
then
A—a
a
A = c€+(X—a)
Take A
(ii)
a(A' M)+a (B' IN) c
0(); then N
has
Go
(A')+a (B')
ker(A'B'-A)
11
71
0, then A' IN
and therefore
IN)
a(A)+a(B).
is invariant under A'
(A') or a (B') contains 0, so 0
and B'. If A = 0 then a If A
(B 71
11
the two sided inverse A
—l
a (A)a (B). 71
71
B' JN. Choose a in
aa(B'IM). But this implies that a e a(A'!W) c giving in turn c(B'IN) c
then A/a cy(A'), A/a
A
a(A/cz)
c a(A').c(B') = a(A).a(B).
0
The duality between approximate point spectrum and approximate de fect spectrum mentioned above, gives the following
43
L(X) and AB =
COROLLARY 3.5. If A, B
then
C
(i)
(AB)
c
(A)
(B)
L(fl); then
LEMMA 3.6. Let K be a Hubert space and let A (i)
c(LA) =
(ii)
c145(RA)
clL(A).
=
c(RA)
=
=
(A) = (A); then there exists a sequence of unit vectors such that + 0 (n follows =).
PROOF.
(i)
Let
e G
A
It
that ii(A_LA) = cannot be bounded below.
then there exists a sequence
in Luff) such that = 1 for all n = 1,2,..., and a'). For each n, choose y SI such that 1 2
=
It
1; then follows that If
÷ 0
(XnYn)
so that
c elLA).
c(LA),
Conversely, if A
+ 0,
=
(n +
0
and IIXnY
(n -'
11 = lxii
oo), whence we conclude that A
=
is onto, then there exists B inL(f1) such that (A_RA)B =
= 1, so that A / Hence, aÔ(RA) Conversely, if A—A is left invertible and C(A-A) = 1, then
B(A—A)
is right invertible, Rc = R(A_A)RC = RC(A_A) = R1 = 1, i.e., whence we conclude that oa(RA) c ar(RA) c Hence, The second statement follows by the same arguments. El
From Theorem 3.4, Corollary 3.5 and Lemma 3.6, we immediately obtain the following
COROLLARY 3.7. Let A, B
L(K);
then
C
(i)
(ii)
C
We shall see later (Section 3.1.4) that these inclusions are actu ally equalities.
3.1.3 Local one—side Let T
function
L(K) and let çì be an open subset of L(K} satisfying the equations
A continuous
-..
(A—T)F(A)
will
resolvents in L(H)
1
(A
and F(A) —F(ii) =
(p—A)F(A)F(p)
be called a right resolvent for T (defined on
it
(X,i.j
is complete-
ly apparent that such a function F is an analytic function defined on 44
Qwith derivative F'(A) = —F2(A). Similarly, if is an open subset of tinuous function satisfying G(A) (A—T)
on
$; clearly, G is analytic on
Moreover, if = $*), then G(A) =
olvent for T* (defined on vent
is
acori
and G(A)—G(1i) =
(p—A)
and (A
+ LIfE)
then G is called a left resolvent for T (defined
(A, p
G(A)G(p)
1
is
F(A)
a right res—
is a left resol—
for T (defined on For any
fR
r (A,T) =
(A_T)*L
(A—T)
[ (p—T) * (p—T)]
(p ,T) =
It
we shall put
p
A
—l
(3.1)
(p—T) *.
is easily seen that Rr(.,T):pr(T) + L(II)
is continuous on this domain and satisfies the
+ LIfE))
equation
1,
(R
(.,T)) does not satisfy, in general, the resolvent equation, i.e.,
it is not a right (left, resp.) resolvent for T.
THEOREM 3.8. Let T olvent for T set S c
defined
> Lift). Given pF(T)npr(T) except
0,
there
exists a right ree—
for an at most denumerable on which does not accumulate in pr(T)j such that S c
distIA,apr(T)]
= {A
<
c}.
Applying the above theorem to T*, we obtain the following dual result.
COROLLARY 3.9. Let T olvent for T
defined
LIfE). Given
0, there exists a left rea
>
on pF(T)npt(T) except for an at most denumerable such that 5' is in-
which does not accumulate in
set S' c
cluded in
The T
LifE)
proof of Theorem 3.8 will be given and N is a subspace of fl, then
of T to N, i.e., the restriction of
in a series of lemmas. If
will denote the compression denotes the or-
to N, where
thogonal projection of El onto N. If N is invariant under T, then TN =
TIN. LEMMA
such that 0
3.10. Let T
LIfE) and let M be an invariant subspace of T Then ran TM! is closed if and only if ran T is
closed.
PROOF. Since 0 jection. Thus, if we write
Pk
(TTM)*
will be a finite rank pro-
45
= {z we have N c N1, dim (M1eN)
N1:
and, consequently, ran T is closed if
<
and only if T(M+W) is closed in K. This reduces our problem to showing that
T(M+N) is closed
is closed.
<=>
Observe that Pker(TIM)*PMTN =
implies that PMTM c TM, whence
0
we obtain that T(M+N) c TM+PMITN On the other hand, given y
TMM+TM.LN = TM+TMiJ4.
and z
N
,
N, we can choose y0
N
such
that TMYO=_PMTZ, whence we obtain TMY+TM z = T(y+y0+z) where the sum in the right side T(M+N) is orthogonal and TM is closed.
It readily follows that T(M+M) is closed if and only if
closed range and
T
let N be an invariant subspace of T such that
(TM)
=
M
and 0 belongs
Then 0
to pie(TM ).
it follows
PROOF. Since T* has closed range and 0
from Lemma 3.10 that TM* has closed range. Moreover, since TM has dense pi(TM*); equivalently, 0
range, TM* must be injective, so that 0
Pr(T& LEMMA
U
3.12.
Let T
be an operator with closed range and let
LU-I)
E M be an invariant subepace of T such that ker T c N and 0 ker(A—T) cM} is a neighborhood and the set {A Then 0
of the origin.
PROOF. Since ker T c N and TM is right invertible, 0 cannot be an
eigenvalue of TMI. Using Lemma 3.10 we see that ran TM! is closed and
pi(TMI).
therefore 0
For each
0 =
=
is injective, it follows that
Since
ker(X—T), we have
in the open set pi(TMI) and
=
0, i.e.,
N. We
conclude that pi(TM LEMMA
let
46
)
c
c
3.13. Let T
natural numbers; then
ker(X—T)
c ap(T)(Ai
M}.
U
if i
1)
and
= ker
PROOF. The inclusion 'c' is obvious and the converse inclusion is trivial for n = 1. We shall proceed by induction over n. Assume that we have
ker 11k=lUkT) and let An+1€cY (T)\{X1,A2,..,A} and Given x
ker
be a natural number. fl+l
we have that
,
ker
and, by our inductive assumption, we can find (k = l,2,...,n) such that (Afl+l_T) — J — k
n+l
A
n+l
Define
x =
—i—i
= l,2,...,n, and Xk.
Using the relations
,m—l k
m n+l
I
-1
3.
=
—i—i J
k = l,2,...,n,n+l, whence the inclusion
ker follows.
0
LEMMA 3.14. Let T H be such that the set of ze L(I1} and let x roes of the function A ÷ Pke(X T)X3 has an accumulation point then Pker(X_T)X = 0, for all A in the component of Pr(T) containing the
point
Let Pker(AT)X = O} and assume that = has an accumulation point in 0, we can find p, lim(k + such that sequence {uk'k:l c = II. PROOF.
Since
and
a
Observe that A + = lr;T) (A'-T) (where Rr(X;T) is dethen fined by (3.1)) is continuous at p. Thus, if M = ker(ij—T)
c M
and therefore ran(u-TM) is closed. But ran(p—TM) is obvi-
ously dense; hence p
pr(T&
is a neighborker(A—T) c M}. By Lemma 3.12, % con hood of p. Since x M we see that Pker(A_T)X=O for all A and the = tradicting the assumption that p 3A. This implies that Let
,
proof is complete.
0
47
LEMMA 3.15. Let T
L(H)I x for all k =
the component
H and
be such that
e
then Pker(AT)X=O for all A in
of pr(T) containing
PROOF. Let M =
that
Since C-T is surjective, we see is so that (k =
0,1,2,...),
c
dense.
On the other hand, follows that ç Since x
M
it
,
c M. Hence,
is closed and it
follows from Lemma 3.12 that the set
and therefore Pker(A_T)X E
ker(A-T) c N) is a neighborhood of
By Lemma 3.14, Pker(A_T)X = 0 for all A in
0
(A
U
PROPOSITION 3.16. Let T £ L(H) and let a be a compact subset of
then there exists x in H such that
Pr(T)fl0p(T).
Pker(AT)X
0 for all A
a.
be the components of pr(T) having nonempty inwe tersection with a and set For each i Ci = choose and ker(p1—T), 0. Then it is possible to find PROOF. Let
1,2,...,h),
= 1,2,...,h)
non—zero complex numbers aj(i satisfies Let
= {A
0,
1 =
such that the vector y =
1,2,. ..,h.
If A=Ø, we take x=y. If
a:
we can apply Lemmas 3.14 and 3.15 in order to show that set,
=
{Ak}knll and that there exists natural numbers
that
m y
mi-i
k)1
k
such }1,
{ker(Ak—T)
By Lemma 3.13, y
0,
is a finite
k =
l,2,..,n.
On the other hand, since
{kerflk_l(Ak_T)
is closed, we can find x
H such that y
(Ak_T)mk}* x.
sible:
p =
,
H and therefore y =
n
(for all k =
if p
(for
a; then x = (1j_T)*Z for m (p_T)*{jjk_l(Ak_T) }* x. But this isimpos
= 0 for some p
Assume that Pm,.
some z
some j, 1
j
1,2,...,n),
we have
n), then y
case, this contradicts our assumptions on y. LEMMA 3.17. Let T that nul(A—T)
1,
A
tIll), and let
For any c
>
0; if m +]
In
either El
be a component of pr(T) such
0 there exists a right resolvent
R of T on U1 except for an at most denumerable set S1. which does not and satisfies S1 c accumulate in U1] PROOF. By Lemmas 3.14 and 3.15 and Proposition 3.16, there exists 48
H such that 0 for all A an at most denumerable subset which does not accumulate
a vector y that S1
(Take a = {A
e},
cz1:
where S1 is
and such in Proposition
3.16.) Fix A0
and let z be a non-zero vector in ker(A-T). If R0 is a fixed right inverse of A0—T, then any right inverse of A0-T is of the form where t K. Choose (Since 0, it readily follows 0, so that t is well—defined.); then
that
=
=
= 1R*y,x>
for all x in H, so that y i
ran
= 0
Rt.
the following identity holds:
For A
((A_A0)Rt+l).
(A—T)Rt =
This shows that A ÷
is a mapping from
into the compo
nent of PF(Rt) which contains the point at infinity. Also, for A
the Fredhoim index of (A—A0)Rt+l is zero. Suppose {0} for some A1 Then it follows from the above identity for some x 0, (Ai_T)Rtx = 0. In this case, Rtx ker(A1—T) and since the
last
y
ran Rt and
space is one—dimensional, ker(A1—T) c ran Rt. This contradicts 0.
It readily follows that the operator (A—A0)Rt+l is invertible for The operator valued function
all A in
RtC(X_A0)Rt+l]1
R(A) =
This completes the proof. D
is a right resolvent for T in
LEMMA 3.18. Let T
except
F of T on
L(H) and let For any e
that nul(AT) =n, A
0,
>
be a component of Pr(T) such there exists a right resolvent which does not
for an at most denumerable set
accumulate on 0n' and satisfies
c
PROOF. We proceed by induction on n. The result is clear if n = 0, is a component of the resolvent set p(T) of T (and F(A) = and the case n = 1 is contained in the preceding lemma.
for then 0 A
Suppose the result has been obtained in the case n = k—l. Let 0k be a follows from Pro component of pr(T) such that nul(A—T) = k, A
It
position 3.16 that for any c
>
0
there exists a vector y c H for which
where St is an at most denumerable
0, for all A set which does not accumulate in
Let
M
viously, M
A
=
is
ker(A-T)nfy}1,
for A
and satisfies S' c [aokJE. 0
k
,
and let M =
.
Ob—
invariant under T and relative to the decomposition H =
49
T, A
T=
0
It (A_TMi)
is easy to establish that A_TM is onto for A
is
It
onto for A
and clearly
follows that for A
nul(A—TM) = =1. By Lemma 3.17 and our inductive hypothesis TM has a right resolvent R(A) on and TM! has a right resolvent G(A) on where S" is a (possibly empty) finite or denumerable subset of which does not accumulate on and satisfies S" c
k-l
and
Define -R(A)AG(A)
R(A) F(A)
A
G(A)
=
(with respect to the above decomposition), where Sk = S'US". It is eas
ily seen that F(A) is a right resolvent for T on
Sk
Sk does not have any accumulation point in
and Sk
c
u PROOF OF THEOREM 3.8. clearly, it suffices to define a right resolvent F on each component of except for an at most denumerable subset S with the desired properties. Let be a component of pr(T)
0,
such that nul(A—T)
If
A
only possible definition for F is F(A) = restricted to nent of If
c},
=0, then
(A-T)1=
p(T) and the
the resolvent of T
is true, in particular, for the unbounded compoIf 1
=, then = {A
<
is a bounded component of
intersects the compact set then we define F on
subset of
where S(c1)
is an at most denumerable
which does not accumulate in Q and satisfies
0, then we can use the same arguments as in that lemma in order to construct a right resolvent F on
by using Lemma 3.18. If
is an at most denumerable subset which does not accumulate in (the condition S(c)
c
c
is
(an)
trivially satisfied in this case).
It is completely apparent that this defines a right resolvent for where S = u{S(c2):
T
is a component of pr(T)\p(T)} is an
at most denumerable subset of
with the desired properties.fl
3.1.4 The left and the right spectra of THEOREM 3.19. Let A,
(i)
(ii) 50
then
B
Ur(A) —
0r(TAB)
= =
=
(A)
—
or(B)
PROOF.
By Corollary 3.7 and our observations at the beginning
(i)
of Section 3.1.2,
Assume that p C Or(A) L(Hj, there exists X C
(i.e., p can be written as p =a—13,
L(tI)
=C. Since
(T)II
that
c
and that
and
where a c
and
c cJr(A)
is onto. Then, given C
such that
(X)
m
for all T in L(f1). In partic
ular, X=X(C) can be chosen so that (m/2)IjXjI
Since
a
= (A-ct)X
TA_ct
is closed, there is a constant m > 0 such
IICII.
we can find unit vectors x, y in H
cirCA) and
such that lI(A_ct)*xII < m/4 and II(B-8)yIl < m/4. Clearly, we can find C
O such that =
lid!; for
this C and X=X(C) chosen as above, we
have 0
<
= =
=
IF'- II(B—6) y
C
< II
fl}
c
that p / Or(A) equivalently, cir(A_p)nci2(B) =0. and GL(B) are nonempty,compact, and disjoint, there ex-
Since
ists Cauchy domains c
and Q such that cir(A_p)
and
0.
It
C
C
follows from [1], [2]
that there exists an analytic function RA_p(A) defined on that
c
(see also [185])
such
c PF(AU), then we can
1 on this domain. (If
choose RA_p(A) as the right resolvent constructed in Theorem 3.8.);sirn
ilarly, there exists an analytic function LB(A) defined on
(If
c PF(B), then we can choose LB(A) as the left resolvent given by 1 on this domain. Clearly, RA_P and LB are analytic in a neighborhood of
Corollary 3.9.) such that LB(A) (B-A)
be a component of
Since
=
it
follows from Cauchy's theo
LB(A) dA = 0.
rem that
A fortiori,
fLB(A) Let
=
and
=
dA = 0.
It is not difficult to check
is the (positively oriented) boundary of a Cauchy domain conoriented) boundary of a Cau— is the taining cy(A), while Hence, by Cauchy's c p such that chy domain that
theorem and the properties of the analytic right inverse (Observe that RA_P(A) =
[X—(A—p)]1 for all A I
we have dA = 0, 51
and
RAM(A)dA
I
Given Y in L(K),
let I
= X = j-L
cf,(Y)
1.
=
=
RA_p(A)YLB(A)dA.
It is completely apparent that L(H)
defines a bounded linear mapping from itself. We shall verify that is a right inverse for
into
Indeed, (X)
=
(A-jj)X-XB =
=
= =
=
on
= 1
i.e.,
Hence,
L(I1).
and the proof of (i)
c cr(A)
is complete.
(ii) By Corollary 3.7 and our observations at the beginning of Section 3.1.2, some
and
c
Assume that p E aL(A) _ar(B), i.e., p = a—B for some a E c9,(A) and This means that, given c > 0, we can find unit vectors
x, y in H such that Il(A—a)xII Ikoyll =
<
<
Since
£ can
=
II
=
(A—a)x Oy—x®(B—B) *yII
=
<
then xOy
II
(A—a) (x®y) —
(xOy)
(B—B) II
l(A—a)xII.IjylHlxIl. II(B—8)
<
2c.
be chosen arbitrarily small, we conclude that
not bounded below, i.e., p
=
c and II(B_8)*yl!
1 and
(xøy)II
is
c
It readily follows that
c
be three and Now assume that ai(A_P)nar(B) = 0 and let •A' C c c c and Cauchy domains such that = 0. By [1], [2](see also [185], Theorem 3.8 and Corollary 3.9), and an analytic there exist an analytic function RB(A) defined on function LA_p(X) defined on in
and LA_p(A) (A—A)
such that (B_A)RB(X)
1 for all A in
for all A
Proceeding as in
the last part of the proof of (i), we can easily check that 4(Y)
defines
=
a left inverse of
LA(X)YRB(A)dX It readily follows that
c
U
From Theorem 3.19 and its proof, we obtain 52
COROLLARY 3.20 (Rosenbium's theorem). If A, B (i) = a(A) —c(B);
/ a(A)
(ii) If
= 0
or B =
0
COROLLARY 3.21. If A (i)
ar(LA) =
(ii)
of(LA) =
dA
T
in Theorem 3.19 and using Lemma 3.6, we
then
LUl),
ao(LA) =
c(LA)
such
and
= 0
By taking A = obtain
then
—a(B), then there exists a Cauchy domain
c
that o(A-i.i)
L(H),
=
=
=
0(A).
c(RA)
=
3.1.5 Rosenbium-Davis—Rosenthal corollary The following is the most important consequence of Theorem 3.19 for the purposes of approximation. COROLLARY 3.22. Let
and U2 be two Hubert spaces, let
A E
and assume that cir(A)nci(B) = 0. Then
L(H2,H1)
L(H2) and C
B
the operators
AC and
OB (acting on H = PROOF.
H1$H2.) are
Assume that K1
similar.
and H2 are infinite dimensional spaces;
then we can identify them via a unitary mapping of H1 onto H2, i.e., we can directly assume that H1 = K2 = K0 and H = H(2). By Theorem 3.19
is
Ci),
=
onto and therefore there exists X
L(H0) such that AX-XB
-C. Then
lx lx A0 OB 01 = 01 OB
AC
lx and
01
is invertible
Hence,
1x-l 01
1—x =
0
1
1X1AC
A acts on a finite dimensional space and B acts on an infinite dimensional space, define a = IIAIHIBIft1 and consider the operators aSA It follows from the first part B and of
the proof that 53
,AC
cxOO =
OB
C
(,x$A)$B =
OOB
W Assume that W
A C
0
=
B
0
W is invertible, where W11
=
21 22 L(H1eH2), W12 e L(t11@H2,H2) and
L(I12,H1$H2),
we OBJ A C
L(f1
2
), w22
and
=
I
then f
A C
0
1
'
-W
'AC (o
AC 1.
so that ciW12-W12(A$B)
ctW
0.
= 0 and
Since c
=
0,
it
readily follows from Corollary 320 that W12=0 and W21=0. Hence W11 and W22 are invertible operators, and (consider the (2,2)—entry of the above 2 x 2 matrix!)
AC
1
0 B
AC i.e.,
0 B
w22 =
is similar to ASB.
The cases when 112 is finite dimensional and 111 is infinite dimensional or both, 111 and 112 are finite dimensional spaces can be similar
ly analyzed to reach the same conclusion:
AC OB 3.1.6
—AeB.
0
The maximal numerical range of an operator It was observed in Section 3.1.1
min{IIA—AIHIB—Ali:
T}. In order to complement our previous results about
A
it will
be shown that this inequality is actually an equality. The concept of maximal numerical range of an operator plays a central role here. ReL(H) is the set defined by call that the numerical range of T W(T) =
{:
x
II,
lxii = 1).
The classical Toeplitz—Hausdorff theorem asserts that W(T) is a convex set [l19,Problem 166]. The maximal numerical range of T is the set W0(T)
54
{A
+ A, where
=1
and
-* 11Th).
LEMMA 3.23. W0(T) is a
closed, convex subset of W(T).
PROOF. Everything but convexity is obvious. Let A, p eW0(T). Without loss of generality, we can assume that
=1. Let fx11}111,
for
be two sequences in H such that
1,
-*
÷ 1, where
÷
and
A
all
(n ÷ '°). Consider T11P11
1.1
is the projection of H ontO V{X11,y11} and let
be apoiñt
Then for each n, it is possible, by the Toeplitz-Bausdorff theorem, to choose Bn such that of the segment [A,p] joining A and ji.
=
8
<
ufl = 1,
and
n
where
=
land
therefore maxfla II < 1) for all n sufficiently large. We have
1
=
Observe that j
M=(l_02)½
(otherwise Ijctx11+
-
=
>
Since 2 0
Ux11112
— 211'rx11lj
+ IIT*Tx11112
1-
-
=
and, similarly, jI(l_T*T)ynIj +
(n
n
'°). Therefore,
L(tl), the
If A
(n +
-* 0
0
(n +
+ land
we conclude that
W0(T) is a convex set.
operator
U
is the inner derivation of L(H)
induced by A.
3.24. Let PROOF. Note that
LEM?.IA
+
orthogonal to
for
a11 ÷ p
(n
is
a
e
+ 0 (n +
unit vector
then
where
co), the
proof is complete.
PROPOSITION 3.25. Let T (i)
c
all n=l,2,...,
where
p. Set TX
Set
hI(TVn_VnT)Xnh!=21811F
Since
IIAII=l}. Since p
A
H such that
there exist
411Th and
2(11T112_1p12)½.
W0(T), then
e
c
L(I1);
[]
then the following are equivalent
0
(ii)
(iii) (iv) PROOF.
11Th2 +
At2
11Th
(i) =>
IIT-4-Ah12 for all A all A
for
(ii)
It
follows from Lemma 3.24 that, if 0 c W0(T),
Therefore,
then
(ii) => (i)
If
then there exist
e U and
A11
L(H) 55
and + 211Th (n + so). It readily follows that + UTI! and 11Th. Passing, if necessary, to a subsequence, we can directly assume + that and Since +1 + y (n ÷ such that
it
is completely apparent that ii, y
is
and H
+ to 0, as n +
=
o, as
-*-
n nfl +
n +
y = lim(n Since both
+
=
(Indeed, it follows as in the last step of the
.
÷ 0, as n + co.) Thus,
proof of Lemma 3.24 that
that 0
tends
Thus, we have
= — + nfl nn n n nfl
where
W0 (T). Moreover, since
uniformly convex, the norm of
OO)=_lim(fl ÷
belong to W0(T), it follows from Lemma 3.23
and
W0(T).
(i) =>
(iii)
If 0
W0(T), then there exists a sequence
e
of unit vectors such that Ii(T+A)x
for
+
+
112
+ 11Th2 +
all A
IIT+Ail Ci) Assume that hiTil By rotating T, we may assume that le W0(T) (iv)
=>
c/2)
= 1,
11Th. If
If x
x /
M, lixhI=l, let
> 0.
Since (iii) =>
Thus, hiT-nil
(iv)
c
A
but
e
lxii
M <11Th!.
M, then
—
0 j W0(T).
0. Let
>
M
<
Tx= (a+ib)x+y, where y
= (a—n)2+b2#11y112 =jITxhl2+(112
because a >
for all x€M};
and let
Let ii = <
<
.L
x;
then
11TH2
< 11Th, contrary to the
hypothesis.
is a trivial implication, we are done.fl
then
THEOREM 3.26. Let T
PROOF. We have already observed that
min{hhT—Alh: 16Th
A
e
2
By an elementary argument of compactness (observe that hIT-All I
+
=),
A
=,
U}. as
it is easily seen that the above infimum is actually attainthis infimum is a minimum. But liT-pit E a:,
ed at some point
iI(T—p)+Ahi for all A
Proposition
3.25:
i.e.,
a: implies (iv) =>
that IISTII_
(ii).)
3.1.7 The norm of the operator
56
Al2
116
(T—p)11 =2 hIT—phi.
(Use 0
LEMMA 3.27. Let T E L{K). The mapping A
-'
W0(T+A)is upper sent-
continuous.
= 1.
PROOF. We can assume, without loss of generality, that 11Th
Suppose that Re W0(T) le for y
a and let
lxii
> 0. Let N =
a+c}; then M < 1. It is clear that IIT+AhI H, ilyil
= 1 and Re
<
1,
a+c, we see that N2+21A1+1A12
hl(T+A)Y112
Thus, for
=
However,
>
(l—M2)/4, it follows that Re W0(T+A) < a+c.
By Lemma 3.23, W0(T+X0)
is
closed and convex (for fixed A,,
Hence, W0(T+A0) coincides with the intersection of all the open halfplanes containing it. Thus, we can find finitely many open halfplanes S1, S2,..., Sm such that c
By the first part of the proof, we can find
5
=
tS(c,A0)
>
0 such
that
"5=1
provided
A I
Hence,
<
-
W0(T+A) is an upper semicontinuous mapping.
A
U
We define the normalized maximal numerical range WN(T) of an oper From Lemma 3.27 we ob0, to be the set W0 (T/IITII). L(fl), T
ator T
tain the following COROLLARY 3.28. If IIT+AU
0 for all A,
then the map A
WN(T+A)
is upper semicontinuous.
LEMMA
3.29.
Let A, B
L(H) be two non—zero
then the
following are equivalent (i)
(ii)
IIAIF4IIBII.
WN(A)nWN(_B)
0.
PROOF. The proof is very similar to that of Proposition 3.25 Ci)
=> (ii), and so we shall only sketch a portion. Let A and c > 0; then there exist x, y s =
and =
ft
0
such that lixib =
= 1 and
< c. Since
/IIAII =
it is possible to define an operator U of norm 1+c" which sends x to y and -By/IIBII to Ax,IIAII(where E', c"
of the proof is virtually unchanged.
are small if E is small). The rest 0
57
LEMMA 3.30. Assume that A, B e ar multiple of the identity. Then min{IIA—AII+IIB—AIj:
if and only if
L(Hl and neither A nor B is a scal
IIA—pII+ltB—pII
A
0.
0. Then IFrMII = IITA_U,B_pII it is obvious that UTABIT cannot be larger than = IIA-ulI+IIB—pII. Since we see that the condition is necessary. A
PROOF. Assume that
In order to prove the sufficiency, we can directly assume that p = 0. Thus, given A e and c > 0, there exist unit vectors x, y in II such that After some algebraic transfor >
where K
mations, we find that le is a constant independent of A and c.
Assume that
=
0. Then, dH[WN(A),WN(_B)J =
6
0, and
>
(by upper semicontinuity; Corollary 3.28) dH[WN(A+A),WN(_B_X)] >
6/2,
for A small. Thus, by convexity and upper semicontinuity, any choice
of x, y which satisfies the above conditions, must satisfy the inequal , 6/4 for A small. But thenwe ity
fOr a suitable choice of argA are led to the inequality IA!6/8 small, which is impossible. Thus, ii=0 was not minimal, which and completes the proof. THEOREM
3.31.
El
Let A, B IltABII
L(H); then
=
A
A or B is a A €G}. multiple of the identity, the rest of the proof is trivial. Let p E a:
be any point such that the above minimum is attained at p. By Lemmas 3. 29 and 3.30,
IITABTI
IIA—PIHIB—Pil.
The proof is complete now. REMARK
El
3.32. It is completely apparent from Proposition 3.25(iii)
a: such that ITT—ph = min{IIT—ATT: A However, simple examples show that, in general, the p of Lemma 3.
that there exists exactly one p 30 is not unique.
3.2 Generalized Rota's universal model Let T
L(tI)..Since a(T) is a compact set, it has a fundamental
system of open neighborhoods which are analytic Cauchy domains (i.e.,
Cauchy domains whose boundaries consists of pairwise disjoint regular 58
analytic Jordan curves; see definition in Section 1.1).
where
is an analytic Cauchy domain containing a(T), and let L2(r) be the Hubert space of (equivalent classes of) complex Let I' =
[1
functions on r which are square integrable with respect to (1/21T)-times
the arc—length measure on I'; M(r) will stand for the "multiplication
by A" operator acting on L2(r). The subspace H2(r) spanned by the rational functions with poles outside 17 is invariant under By M(r) we shall denote the restriction of MCI') to H2(r) and its compression to L2(r)eH2(r), respectively, i.e. M(r)
=
0
(3.2)
2
M_(r) L CreH Cr)
with respect to the above decomposition. (Here and in what follows, we write A11 A12
.
.
.
H1
A21 A22
.
.
.
H2 (3.3)
Akl
.
.
1tk
as an alternative way to indicate that A admits such a k x k operator matrix decomposition with respect to the orthogonal direct sum
where
i,j
1
k.
H
=
Such a decomposition is clearly
unique.)
It is well—known [361, [73], [74,Sections 9] that H2(F) is a reproducing kernel space and that it can be realized as a space of anafurthermore, it can be easily checked lytic functions defined on that
=
cU4_(r))
=
[2, and ind(A—M+(r))
ind(A_M_(I'))*=_l for all A in [2,
(3.4)
M(r) is normal and iIr(M(r))II
for each
=Ur(M+(r))H
=jjr(M_(r))ll =
max{Ir(A)
A e
[21
rational function r with poles outside [2.
The filbert space completion H2(r)8H of the algebraic tensor prod uct of H2(r) and H can be regarded as a space of analytic H—valued
functions defined on [2, or as a space of (equivalent classes of) weak-
ly measurable square integrable functions on r, in which case it will 59
also be denoted by H2(r';H). ThEOREM 3.33. Let T e L(HJ
and
be as above, and let R = ran
(M+(r)øl—leT); then R is a sub8pace of H2(r;H) invariant under M+(r)øl, the restriction of to R is similar to and the compression of to is similar to T.
PROOF. Since N+(r)øl-løT is equal to multiplication by A-T in B2(r;H) and X-T is invertible on I', it is clear that R is closed. More
is implemented lOT. All we have to do is find an operator L:H2(F;H) + H with range II over, the similarity of M+(r)øl and M+(r)eiIl?
and kernel R such that H)
the element Lf
=
ToL. We shall define for f
H by 1
Lf =
f(A) dx.
It is easily seen that Lo(M+(r)@l) = Lo(lOT) = ToL, so that R c ker L. On the other hand, if fX(A) is
equal
x (x
H), then
L
to H.
Let
such that the
be an analytic Cauchy domain containing
to r are restrictions of functions analytic in a neighborhood of dense in H2(r) (By Runge's approximation theorem, it suffices to choose
contains a component of that every bounded component of and let e0(A) 1, e0 H2(r). Let S LU32(r;H)) be the operator defined by Sf = f-eøLf. Since Lf=L(e®Lf), it follows that ran S=ker L. Thus, in order to complete so
the proof, it will be sufficient to show that S(f IF)
E
R for every K-
valued function f analytic in a neighborhood of in a neighborhood of Because of the analyticity of we have Lf
It
follows that for (S(flr))
f(A) dx.
=
e —
=
=
a1
L]fx)
dA
f(x) dx.
Since
f(x) dA
is an analytic function of S(flr)
it readily follows that
=
f(A)
The proof is complete now. REMARK
60
3.34. According
to the proof of Theorem 3.33,
dx)
R.
0
the diagram
H
2
T' = CM (F)el) +
(r;H)
LIRI[ T
is commutative, i.e., T =
Since Lf=x (where
H), it follows that LJR1
x, x
is
bounded below by
lxii =
l} = ([length(r)]/2ir)½.
On the other hand, the Cauchy-Schwartz inequality implies that
([length(r)]/21T)½
IILU
A
fl.
A
Let r be a rational function with poles outside B
(3.5)
and let
R
M+(r)e1. = T;
then =
=
llr(Tt)ll
max{Ir(A)l
:
A
whence we obtain the following COROLLARY 3.35. (i) of o(T),
hood
operator
some
T'
Given T
P, i.e.1 llr(T')ll
for
LUI) and a bounded open neighbor-
is a spectral set (in the sense of von Neumann) for
maxflr(X)j:
A
e
all rational functions r with poles outside $. (ii) In particulars if a(T) = {0} and E > 01 then there exists T ouch that IIT'Il
LEMMA
3.36.
<
c and T sTm 0.
Let H = H1H2$...
and assume that A
L(H) admits
an upper triangular operator matrix with respect to this decomposition, i.e.'
61
A11 A12 A13
.
A22 A23
.
A33
•
.Al,k_lAlk
H1
.A20k_iA2k
H2
.A3k_lA3k
H3
0
Uk_i
then
A stm
Ak_l,k_1$Akk.
.
PROOF. Let
composition). k+l •n •n
(with respect to the same de-
.
is invertible with inverse V and a straightforward computation shows that
Clearly, W Ic
,
0
(r)el
M
L2(r)oHeH2(r;H) B
M+(r)@1IR
z1
where
=
0
T'
0
0
Zel,
2
vation with Lemma
R
3.36,
H2(r;H)eR
Z2
M_(r)ø]. L2(r)øHeH2(r;H)
M+(r)øl and T'
—
with
L(H) and let =
Apostol's Given T
stm and
M(r)øi sTm
3.3
be an analytic Cauchy do-
then
M(r)e].
M(r) stm
T. Combining this obser
we obtain
COROLLARY 3.37. Let T main containing aCT)
LI
112(r;H)
Ze].
+ 0
=
00).
(3.2) and Theorem 3.33 that
It follows from M(r)ø]. =
(n
stm
triangular representation
L(H), let Hr(T) = V{ker(A—T):
A
e
let
V{ker(A_T)*: A and let H0(T) be the orthogonal complement of Hr(T)+Hi(T)• Denote the compression of T to 0(T), Hi(T) and 00(T) by Tn Ti and T0, respectively. Recall that A L(H) is a trian=
gular operator
62
if it admits an upper
triangular
matrix; i.e.,
a11 a12 a13 a22 a23 a
(3.6) o
with
respect to a suitable ONB.
Apostol's triangular representation and its basic properties are established in the following THEOREM 3.38.
(i)
Ur(T) is orthogonal to UL(T), so that (3.7)
H = tl(T) and Hr(T)WIlo(T) are invariant under T,
(ii)
so that T ad-
mit8 a 3 x 3 upper triangular operator matrix representation Tr *
T= with
0
T0*
o
0
(3.8)
respect to the above decompostion, where Tr
(iii)
=
TIHr(T)
is
a triangular operator, a(Tr) = 0p..(Tr) =
O(Tr) is a
re(Tr) intersects the set 0
*
and min.ind(A_Tr)
(iv)
is
perfect
set, every component of
ap(Tr)and
= 0 for all A s
PS_F(Tr))
the adjoint in L(Ht(T))
so that
=
of
or(Ti) = set, every component of re(Tt*) intersects = 0 and 0, so that
P_F(Tr)
=
and
the triangular operator o(Tt) is a perfect
the 0
and
set
for
all A
c
Furthermore,
is a continuous function for A
(v)
E
pS_F(T) and discontinuous for A (vi) ps_F(T) C
(vii)
c
1)e_F(T) ca(T0);
(viii) (ix) (x)
T —
a0(T)
c
{A11A2,. where TA acts on
If
A
is a
=
finite subset of P_F(T),
then
a finite dimensional subepace, o(TA)
= A
and A c We shall need several auxiliary results. 3.39. If
B
A
e 0(B) and II(A—B)xU
for
all x e
63
H, then IIA—Bli > M for every triangular operator A.
PROOF. Assume that A has an upper triangular matrix of the form
(3.6) with respect to the ONB {en}nl. Let be the orthogonal projec tion of H onto V{e1,e2,... and let y be a unit vector orthogonal (so that (A_B)*y = 0); clearly,
to ran(A-B)
Given c, <
0
c for all n
= APE.
< 1, there exists n0=n (E) such that
<
IPn(A_B)*yjl
can be regarded as an operator acting
n0.
on the finite dimensional space Mn= ran Pa. It follows from the finite dimensionality of Mn that this subspace contains a unit vector such that I[Pn(X_B)*Pn]*znII = IIPn(A_B)PnznII < We have and lim inf(n
=)
urn inf(n
oo)
lirn inf(n
oo)
>
urn inf(n
oo)
>
N-c.
IIA—Bil
II(l_Pn)
(A—B)PzIj sup(n
=)
Since c can be chosen arbitrarily small, we conclude that 11A
M.
BIl
0
COROLLARY 3.40. If A i8 a triangular operator with matrix (3.6)
with respect to the ONB {en}nl, then (1)
0(A)
(ii)
=
Every clopen subset of
inter8ecte d(A) = (ann}ni and
every component of ci(A) intersects d(A); (iii)
Every isolated point of a(A) belongs to d(A);
(iv) If PROOF.
then A
E
d(A).
The equality
(i)
is trivial. On the
other hand (since A is triangular) Lemma 3.39 implies that a(A) a clopen subset of a(A) such that a0nd(A) =0 and
(ii) Let let
be a Cauchy domain containing d(A)
such that o0nO=0. It iseas
ily seen that, if E =
then Ee =e
dx,
for all n = 1,2
Since E is an idempotent and ran E
V{en}n1=H, it readily fol
lows that 00 0. Since a(A) is a compact Hausdorff space, every component of
is the intersection of all the clopen subsets containing it. Let a bea component of a(A) and let {a: s A) be the family of all clopen sub sets of a(A) containing a. It follows from the first part of the proof that aand(A) 64
0 for all a in A. A fortiori, and(A)
=
(by an obvious argument of compactness). (iii)
This is a trivial consequence of (ii).
(iv)
Assume that (A_A)*y=O for some A
0(A) and some unit vec-
H; then
tor y 0 =
for all n =
=
(x-A)M, where
so that y
1,2,..., for all n = 1,2
V(e1,e21...
(n = 1,2,...) and therefore y j If A j d(A), then so that, y = 0, a contradiction. Hence, A E d(A). 0
PROPOSITION 3.41. Let T E L(H) and p Q = exists
(in
pO_F(T);
then
lim(A
the norm topology). Q is the orthogonal projection onto ker
ran(p—T)
n
}.
Assume first that p-T is onto. By Corollary 1.14(v), A-T
PROOF.
is onto for all A in some neighborhood =
l_Rr(AsT)
(A—T)
of p. It follows that is defined by (3.1), A
(where
converges in the norm to 1-
as A
(A—T)
p.
Put ran(p—T)m. Because (p-T) is semi-Fredholm, M is closed (i.e., is a subspace) and (p-T)M=M. Hence (p_TM)M=M. clearly (p—T)ker(A—T) = ker(A—T) and so ker(A—T) c M. It For A
follows that P
ker(A-T)
=P
P
ker(A-TM) M and the first paragraph shows that this has the desired limit. 0
COROLLARY 3.42. Let T c
LUll and
ps_F(T). The following condi
tions are equivalent (i)
p is a regular point of Ps_F(T).
continuous at Ap. PROOF.
(i) =>
(ii)
If nul(A-T) is a finite constant in some neigh ran
borhood of p then Corollary 1.l4(vi) shows that ker(A-T) c
is continuous at A = p.
By Proposition 3.41, If
is a finite constant in a neighborhood of p then Cor
ollary l.l4(vi) shows that ker(p_T)* CLAIM:
(3.9)
Since p-T is semi-Fredholm, (3.9) is equivalent to ker(p—T) C
Indeed, if ker(p-T) C
nn:l
(3.10)
ran(p-T)m, then 65
n = Thus, if y fore ran(p_T)*
Thereso that y = (u_T)*2y2
then y= (1j_T)*y1 for some y1
ker(U_T)*2
9
for some y2 E H and ran(U_T)*
1,2
H.
ker(ij_T)*3
By induction, there exists n =
1,2
y = (ii—
ran(p_T)*'1.
Hence,
This proves that (3.10) implies (3.9). The converse implication follows by taking adjoints.
Since (3.9) and (3.10) are equivalent, we conclude (as in the first part of the proof) that
(ii)
(i)
is continuous at A =
is continuous at A = then (3.10) holds. Hence (3.9) also holds and (i) follows from Corollary 1.14 (vi). =>
Let H
L(H) with
If
H for
where
=
all j
= l,2,...,k,
and let A
matrix of the form (3.3) with respect to this decomposition.
Observe that A is compact if and only if A.. is compact for all i and
H (i.e.,
j. On the other hand, since for all j,
H
via
H3
is
infinite dimensional)
= 1,
a fixed unitary map (j
2,...,k).
can be identified with the algebra of all k x k matri-
Then L(ti)
ces with entries in L(H1, K(H) can be identified with the ideal of all k x k matrices with entries in K(H) and (by taking the corresponding
quotients) A(H) can also be identified with the algebra of all k x k
matrices with entries in MN) (k=l,2,...). This identifications will play a very important role. The proof of the following elementary algebraic lemma is left to the reader.
the ring of a ring R with identity 1. Then
LEMMA 3.43. Let M12(R) be tries in
(i) If
R).
If
q
If
p
If i8
in
, then qd =
has
invertible,
PROOF OF THEOREM
a left inverse
in
,
p
66
(in
ker(p—T).
then ap
= 1 (in
then dq = 1.
3.38.
Ci)
By Corollary 1.14(vi), either ker
(A—T) c ran(A-T) or ker(A_T)* c ran(A_T)* for every A = ran(A—T)
1
then pa = 1.
is
(ii)
R).
has a right inverse
all 2 x 2 matrices with en-
ps_F(T).
If p
and so [ker(A_T)*]l
By Corollary 3.42 ltPker(
= V{ker(A—T):
A
A), whence it
[ker(A_T)*]l. Hence,
readily follows that ker(A-T) Hr(T)
+
0
T)
ps_F(T)
i
}
V{ker(A_T)*: A
ps_F(Tfl =
H
and immediate. (iii)
Hence, T admits a Let
under T is
matrix representation of the form (3.8).
{An}ni be a denuinerable dense subset of where
Corollary 3.42 flr(T) =
By
It is complete-
ly apparent that Tr has an upper triangular matrix with respect to an ONB obtained by Grain-Schmidt orthonormalization of a union of ONB's of the subspaces
in this representation (where
and d(Tr) =
d(Tr) has the meaning of Corollary 3.40). By Corollary 3.40, nent of ci(Tr) intersects d(Tr)
p(Tr)i every compo-
=
(Observe that ATr is on
c
ps_F(T) ) and c d(Tr) = be arbitrarily chosen (under the condition
to for each A
E
course), it readily follows that
{An}n:l.
Since {An}n:ican ,
of
and, a fortiori, that
and min.ind(A_Tr)=O for all A Ps_F(Tr) It is also clear that does not have any isolated point, i. e., either Hr(T) = {0} and a(Tr) 0, or a(Tr) is a nonempty perfect set
re(T)p(Tr)
and o(Tr)
Since psF(Tr) = re(Tr) intersects
it follows that every component of
In order to prove (iv), we only have to apply the arguments of (Indeed, observe that the equivalence of (3.9) and
(iii) to
(3.10) and Corollary l.l4(vi) imply that the content of Corollary 3.32 and (vi) follows from (iii), their proofs:
If A
re(Tr) c ae(T), a con
then A
E
(v) is
(iv) and
tradiction. Hence Ps_F(T) c pr(Tr)• A similar argument shows that ps_F(T)
c
(vii)
and then Assume that and H0(T)etlt(T) are infinite dimensional spaces.
Suppose that A
then A-c is invert-
ps_F(T) and ind(X—T) > -=;
ible and A—T is right invertible. By decomposing H as {ffr(T)Ho(T)]$ H
(T),
can be written as the 2 x 2 matrix
rJ.
= 0
A-c J=
q
By Lemma 3.43(i), p is right invertible. Since
67
*
ATr 0
is
it follows from the sante lemma that
Similarly, if ind(A-T)
right invertible.
we conclude from Lemma 3.43(u)
<
that
is left invertible. Hence A
nul(A-T)
e
is invertible, or
ps_F(To) and therefore either
0. Thus, in order to complete the proof,
0 or nul(A_T0)*
it only remains to show that
f
K0(T). The invar-
Assume that (A-T0)x=0 or equivalently (A-T)x i implies that (A—T)x
iance of
Hr(T)
Hr(T) such that
(A—T)x. Since
y-x
ker(A—T) c Kr(T)i x must be equal to 0. An analogous argument shows that nul(A_T0)*=O and so A-T0 is invertible, i.e. A p(T0). If either Hr(T) = {0} or = {O}, then the result follows by an even simpler argument. (viii)
p_F(T).
Let
is not an eigenvalue of T0, then
If
=ker(1I_Tr) and (by Lemma 3.12) we can
(vi) shows that we have
find a neighborhood
of
such that
This implies that
r ps_F(T),
for all A
On the other hand, the first part of the proof
Hence,
of (vii) indicates that (v),
is (ix) (x)
a contradiction.
ps_F(To) and, by (vii) and Corollary 1.14
an isolated point of
Therefore,
This follows immediately from (vi) and (viii).
Let A be a finite subset of p_F(T). It follows from (viii)
that A c
and, a fortiori (since A is closed), A is a clopen sub
set of o (T0), so that (by Riesz' decomposition theorem)
A 0
where
BJ'
A acts on a finite dimensional space,
=A and Aria (B)
Ø.
Therefore T admits the decomposition T o o o
where A=o(A) c
T12 T13 T14 A 0
o
Tr T12 B13
T23 T24
BT34
=
0
AB23,
B 0
T
0
i.e., a(T)nau(A)=a(A)nat(Tt)=a(A)n
a(B) = = 0. By Corollary 3.22 and its proof, there exist operators X, Y such that
68
lXl
lx 01
01
A
0
TO —
A
0
A0
1Y AB12 01 0
01
—
Hence
100 1XO
0
A
001 001
0
0
=
'PrO
*
0
A
0
0
0
100 1XO O1Y 010 001 001
TrT12B13
T— O1Y 010
T
B23
T*
*
*
=ASO
*
0
0
Ti
TA = A has the desired properties and A e PS_F(TA), where *
*
0
B
*
0
0
Ti
T =
r
From Theorem 3,38 and Lemma
3.36,
COROLLARY 3.44. Let T e L(H) (3.8); then
0
•
we obtain the following
with triangular
representation
TOT
T
3.4 Correction by compact perturbation of the singular behavior of erators The Weyl epeatrum of 'P aw(T)
It
L(HJ
=n{a(T+K):
is
defined by K e
K(H)}.
is easily seen that aw(T) is the largest subset of the spec-
trum that is invariant under compact perturbations and that (by Theocontains every complex A such that A—T is not a
rem 1.13(v) )
Fredhoim operator of order 0. Furthermore, it is well-known that aw(T) actually coincides with this set, i.e., aw('P)
= {A
(The inclusion
ct:
A—T
will
is not a Fredholm operator of index 0}. (3.1])
easily follow as a corollary of Theorem 3.48
below.) The Browder epectrum of 'P is the complement of a0(T) in the spectrum, i.e., aB(T) =
a(T)\a0(T). 69
Clearly, a(T) wIT)
c
aB(T)
aw(T)
reCT) and 0
ae(T)
B(T)
e(T) Cacl2.re(T).
PROPOSITION 3.45. Given T
L(ffl,.
there exi8ts K
K(ti)
such
that
P_F(T+K) = O,cl(T+K)=SB(T)andmin.ind(T+K_A)=min.ind(T_A)for all A Ps_F(T), moreover3 K can be chosen equal to a normal compact operator such that
=
A
P_F(T)}.
PROOF. Let
Tr T12 T13 Hr(T) T =
0
T0 T23 110(T)
0
0
Ti
be the triangular representation (3.8) of T and let {1tn}in<m (0
m
øo) be an enumeration of the eigenvalues of a0(T). Then T0 admits the representation
U2+T2
H2
*
T=
H o
.
I
0
where
H2...,
are defined so that
coincides with
the Riesz subspace of T0 corresponding to the clopen subset {ii1,ii2,...,
(1 n < m) and cult to check that a(T) c a(T0)\ci0(T0)= OB(To). of u(T0)
For each
It is not diffi-
aPS_F(T) be any point such that
let
It is easily seen that A
=
E aire(T)• Thus, if
is the operator obtained from T0 by replacing
n <
and
Tr T12 T13 T'=
T23
0
0
0
Ti
then it follows from Theorem 3.38 (Observe that (T')r=Tr and (T')z = a(T') =c3(T') =a5(T) and K=T'—T is a diagonal Ti!) that Ps_F(T ) normal operators with eigenvalues A1—p1 (with multiplicity equal to dim H1), A2—ii2 (with multiplicity equal to dim 70
,...,
(with
multiplicity equal to dim
n<m}=max{dist
1
3.46.
LEMMA
so that
Let T
p
L(H)3
and
E
such
exists a finite rank operator K
c >
0. Then there
that
IlK II
ps_E(T+K ). < xnin.ind(A—(T+K)) =min.ind(A—T)
(ii)
-,
Ps_F(T)\{p}, kl,2,... where T acts on a fi-
A
and T
If v
(iii)
nite dimen8ionai spacej
then
{v} and
(T+K) and T+K
E
where v
PROOF. By Theorem 3.38, p
op(Tr)uop((Ti)*)*• Suppose that u and denote by M the Riesz spectral subspace of T0 corre-
sponding to
If we put H
—
= H0(T)8M then
P
T12 T13 T14
0
T0jM T23 T24 M
0
P
T
0
p 0
0
34
H p
0
T
T
= 0
Since a(T)=(p}, we can find a (finite rank!) operator A E L(H) ={jj} and nul(T +A-p) =1. (Thus, if x < €/2, 0, then x is a cyclic vector for T+A.) Define x
such that IIAH
T +A * s't=
if
U
0
By Corollary 3.22, Sn —
i.e., 5" has an invariant sub— such that S"IR
space R complementary to
Let W
L(HOHi(T))
be an invertible operator such that =
Let f p)*, H
hell
c
ker(T+A—p),
= 1. Let
(T+A)eTt. 0 <
L(HOHt(TH,
and assume that
then
P, y
A
Hence
then = 0,
If x
and let e Hi(T) and z
<
A
= 0.
,E a(T+A) = {p}. It follows that z=0 and there-
Since ap(Ti)=Oi we see that y=0. If A=p, then z ker(T+A-p) and therefore either z=0, or 0 and (Ti_U)y = fore,
-e that
0.
But this is impossible because e i
We conclude
=0. 71
Observe that S2
=
T +A 0
H
eM
11(T)
T L
(T +A-A) * fee
so that (S2_A)* =
(T _A)*
0
£
p
is finite dimensional, it is not difficult to check that (where d(p) =dim H) and H c V {ker(S2_A)*: A
Since H =
ps_F(T)}, whence we conclude that HR,(52) Let S3=W
=HeHL(T) and
= S2.
S2W and set Tr T12 S13 Hr(T)
0TOIMS23M 0 0 S HeH(T) L
S
where S13= (T13 T14) and
(T23 T24); then K
rank operator such that
IIAII+IIWII.IlW
II
=
S-T
is a finite
lI.IIeefII <
and H0(S) =M and S0=T0jM. Hr(S) =Hr(T) and Hi(S) =HOH2(T) and Now (i), (ii) and (iii) can be easily verified by using Theorem 3.38. then we decompose P as
If
T14 H(T)
P12 T13
T23 T24H
0
—
where K
0
0
0
0
(TO)MT34M 0 TiHi(T)
is the (finite dimensional) Riesz subspace of T0 correspond-
Applying the above arguments to
ing to {p} and H
Tr T12 * T
o
p
instead of 5', we can find a finite rank operator K' such that IlK'
II
<
and {Tr T12}
S4=
+K'
and (S4)r=S4• satisfies Hr(S4)= In this case we define R13
S4
S=
0
R14 tlr(T)$I
(To)M T34 H
0
T
where
13]
T
and R14=
14
It is easily seen that K
24
23
0
has the desired properties. PROPOSITION 3.47. Let T K(H),
72
e such
that
L(H) and c > o; then
= S—T
there
exists K
(i)
cj(T)
a(T+K) =
(ii)
and (T+K) jH(A;T+K)
TIt{(A;T) for all A
=
min.ind(T+K—A) = min.ind(T—A) for all A
(iii)
ps_F(T).
c/2}. PROOF. Let A = {A pS_F(T)\[aO(T)up(T)]: Clearly, A is a compact set and A = Aflflp_F(T)\clO(T)) is finite. Appl1
ing Lemma 3.46 a finite number of times (once for each point in A), we can find a finite rank operator K1 such that UK]jl fies Ci),
<
and T+K1 satis
(iii) and
(ii')
=
Now we can repeat the argument of the proof of Proposition 3.45
in order to "push" the points of PSF(T+Kl)\cO(T) to by means of a compact perturbation K2, IlK2 II < which only affects the action of (T+K1)0.
It is easily seen that K=K1+K2 satisfies all our requirements. U
THEOREM 3.48. Let T
L(H) and £ > 0; then there exists K
KIll)
such that
£+max{dist[A,apSF(T)]:
<
IIK1I
and min.ind(T+K—A) =0 for all A
A
ps_FIT).
In
a(T+K) = {A
A—T is not a Fredhoim operator of index 0) = cw(T).
PROOF. By Propositions 3.45 and 3.47, we can find a compact opera tor K1 such that 11K111
<
c/2+max{dist[A,apS_F(T)]:
A
and the operator A=T+K1 satisfies the following properties:
=
ps_F(A) and min.ind(A—A) =min.ind(A—T) for all A Thus, in order to complete the proof it suffices to show that there exists K2
K(H),
PS_F(A) = PS_F(T). our requirements.) all A
Let A = {A min.ind(A-A)
such that min.ind(A+K2—A) =
IlK2 II <
0
for
(Indeed, K=K1+K2 will obviously satisfy all
A: and A1{A pS_F(A): dist[A,ap5_F(A)] 0). By Proposition 3.16 we can find y,z H such that
Pker(X_A)*Z
0, for all A
A1.
Let C =ayez, a > 0. We have min.ind(A+Ca_A) =min.ind(A—A)—l, for all A Since min.ind(A—A) =0 for all A
A1, a> 0.
A\A1, it readily follows that
73
p(A). Therefore, there is a positive constant
c
such
that
for all A for all A
II(A—A)xlI
II(A_A)*XM
for
all x
H,
lixil
Thus, if 0 A) = 0
and
l.
=
< €/6 and min.ind(A+C-
then
<
<
for all A Now we can apply the same arguement to A+C, etc. After finitely
many steps we obtain a finite rank operator C1 such that 11C111
min.ind(A+C1—A)
= 0 for all A
and a(A+C1)
c
<
interiorcy(A),6 (3.12)
Let m (0 =) be an enumeration of the components of 0 for all A a(A+Cl)npS_F(A) such that min.ind(A—A) A repetition of our previous argument shows Ui
that there exists a positive constant
such that the
<
relations (3.12) remain true if A+C1 is replaced by A+C1+B for any B in L(H) such that
IBU
then the arguments used for the construc-
If A1
tion of C
can be repeated here in order to find a finite rank opera-
n/2. such that min.ind (A+C1+C2 1-A1) = 0. By induction, we can construct finite rank operators C2 1' C2 2' n'S"' (n=l,2,...) such that K(H}, 1c211 < 11C2,511 < tor
min.ind(A+C1+C2—A) = 0 for all A and
c
ps_F(A+Cl+C2),
a(A+C1+C2) c
interior
o(A),6.
Since A
Ps_F(A+Cl+C2)}
<
we can apply the arguments of Proposition 3.45 in order to find a com-
pact normal operator
11C311
< c/6,
such that
ps_F(A).
min.ind(A+C1+C2+C3—A)=0 for all A Now take K2=C1+C2+C3. An operator T
L{H1 such that
min.ind(A—T) = 0 for all A will be
c
ps_F(T)
(3.13)
a smooth operator.
3.5 ApostoL—Foia?—Voiculescu's theorem on normal restrictions of pact perturbations of operators 74
Given T in L(N) we cannot expect, a priori, to find an infinite
dimensional invariant subspace M such that TIM is normal. An important approximation argument (due to C. Apostol, C. Foias and D. Voiculescu) asserts that some arbitrarily small compact perturbation of T always has that property. We shall need some extra notation to make it more clear.
3.5.1 Schatten p-classes
Recall that if K is compact, then (K*K)½ can be written as where A1= In:i An enøen with respect to some ONB {e 0 and An ÷ 0 (n ... An A3
(K*K)½
A2 >
•
The Schatten p-class CF'(K) of compact operators is defined by
{K
=
< oo},
K(I1):
E
0
<
p
=,
<
and C(K) = K(H). The reader is referred to [77], [107], [183] for the properties of these ideals of operators. In particular, we have: (a) (b)
If K
1'
= (c)
then K* c
(C(ti)) is a Banach space under the norm IKI for 1 < p
If 0 < p
<
q
(jKI
<
IK*
=
=, then
<
p
= IK*I p
= IIKII, resp.);
c CUfl (The inclusions
c
1 and K c then IKIq IlK then it is possible to define the trace of K
are proper); moreover, if p (d)
If K
C1(l1),
11
by
tr(K) =
of K. The result is independent of
(with respect to some ONB the particular ONB.); (e)
C1(tI)
= the
ideal of trace class operators
ly isomorphic to the dual K(fl)t
of
KU!).
This
is isometrical—
isomorphism is defined
where
by C1(H)
=
(f) isoniorphism is
tr(KX),
X
K(tI);
= K(g.l)tt is isometrically isomorphic to LUll. This defined by LIt!) A÷÷ •A C1 (K)
(g)
If A
LIt!)
is
a non-negative herinitian operator, then tr(A)
=
(0
tr(A)
is well-defined and the result is independent of the particular ONB of II.
75
(= the ideal of Hubert—Schmidt operators) is a
(C2(H),I.12)
(h)
Hubert space. If A, B e C2(H), then AB*
C1(HJ and the inner product
of C2(H) is given by = tr(AB*).
3.5.2 Normal restrictions
With the above notation in mind, we have the following THEOREM (T) and c
and K
E
>
3.49. 0,
Given T E L(H)J a nonempty compact subset
of
there exists an infinite dimen8ional subepace H
c H
C1(tl) such
that (so that jjKfl < sJ., < c c is a diagonal normal operator of uniform infinite multi
=
plicity and a(T) = ae(Tc) = PROOF. Let {Afl}nl be a sequence of complex numbers such that
{Xn}= card{m:
for each n=l,2
Am = A) =
It is obvious that
fore, we can find a unit vector e1
(3.14)
There-
c ote(T) c
such that II(A1-T)e111
H
< €/4.
Con
sider the decomposition e1
A1+t11
{e1}1
V{e1} + {e1}1 is a rank-one operator,
then
<
c/4
and
21 +
\/{e1}
is also a rank—one operator. Clearly, we can find a
finite dimensional subspace N1 V{e1} such that < c/4. Let R1 be the orthogonal projection of H onto Mt and define C1 by =
C
t11
R 0
It is easily seen that 1C111
0
<
0
A]
T—C1 =
0
0
+
1
1
+1T1 IM1I
e1
A1 0
0
A1
B1 N1 =
0
T1 D1
0
D1
T1Mt
0
B1A1
<
c/2
and
e1 N1
where N1 =
Thus, we can apply the same argument Clearly, di (T1) = to T1 in order to obtain a unit vector e2 N1, a trace class operator C2 such that 1C211 < 76
e/4 and a finite dimensional subspace N2
e2} such
that E2
V{e1,e2} D2 w2eV{e1,e2}.
A2) 0
T- (C1+C2) =
0
A2N2
B2
0
By induction, we can find orthonormal vectors and trace class operators C1,C2,... C
(H),
IK11
tial
V{en}ni
=
and T
Eff
T
T—K= where H
en
=
and
ICnIl <
<
such that K
bc
AH1 Xn en®en (strong limit of the par-
=
sums). The condition (3.14) guarantees that T
has the desired properEl
COROLLARY
3.50.
Given T
L(HLI nonempty compact subsets
c
CT) and Ore(T)t c > 0 and diagonal normal operators Ni and Nr c c uniform infinite multiplicity such that
rr c
and Fr c
c
o(Nr)
(rr),
*
*
0
A
*
0
0
N
Ni
such that IT—Lu
there
of
exists an operator L
<
PROOF. Applying the inductive argument of the proof of Theorem 3.
49 alternatively to T and to T*, we can find K
C1(H),
Ku1 <
£12 such that
T-K=
Mi *
*
0
A
*
0
0
Mr
where Mi and Mr are diagonal normal operators of uniform
tiplicity
and a(Mr)=rr, resp..
such that
Since
infinite mul-
c
c
and Ni and Nr are diagonal normal
c
operators
a(Nr)
c
(Fr)ct
of uniform infinite multi-
plicity, we can easily "spread" the spectral measures of Mi and Mr (con centrated in
and
tors
and ap(Mr)t resp.) in order to obtain normal opera
such
that
Nit
<
3c/2 and
< 3c/2.
The operator L obtained by replacing Mi and Mr by the normal operators
and
respectively, satisfies all our requirements.Q
77
3.5.3 Density of sets of operators with bad properties We shall say that a certain property (P)
(of operators acting on
a Hubert space H) is a bad property if: If A has the property (P), then ct+BA has the property (P)
(I)
and all
for all a
0,
If A has the property (P) and T
(II)
A, then T has the proper-
ty (P), and If A has the property (P) and a(A)nci(B) = 0, then A$B has the property (P).
(III)
Examples of "bad properties" are frequent in the literature; name
ly,
T is not cyclic,
(1)
(2) The spectrum of T is disconnected (or
has infinitely many components, or c components, where c is the power of the continuum),
has nonempty interior,
(3)
mutant of T is not abelian,
(4) The corn-
(5) T is not similar to a normal operator,
(6) T is not algebraic, etc, are examples of properties satisfying (I), (II) and (III).
THEOREM
3.51. If
is a bad property and there exists some oper
(P)
ator A with the property (P), then the set
L(H):
{T
T satisfies (P)}
is dense in L(H). PROOF. Let p
such that
A—pI
that IT-Lu
<
<
c/2
p(T) 0, we can find A €/2. By Corollary 3.50, we can find L L(H) such Given c
>
and
AR L= OB with respect to some decomposition H =
where M and M1 are
nite dimensional subspaces and A / a(B). (Observe that p
re(T)
and take
30e(T)
c
rr =
=
0
n/(l-fflAjI),
inf i-
0 <
6
<
then it follows from Corollary 3.22 that A+6A R
M Now (I),
B
0
=
(A+SA)OB.
(II) and (III) imply that M has the property (P). Since
< e, and
lIT-Nil
ily follows that T
{S
e can be chosen arbitrarily small, it read
L(HJ:
S
satisfies (P)).
0
An inductive repetition of the same argument yields the following 78
COROLLARY 3.52. If
is a denumerabie set of bad proper-
(Pa) is a bad property. Furthermore, if there
ties, then (F) =
exist operators A1, A23..., (for each n =
1,2,...),
An satisfies (Ps)
then {T
L(tf):
T satisfies (P)} is dense in
tIN).
3.6 Notes and remarks
The operators TAB were first systematically studied by H. Rosen— blum in [174]. Rosenblum's theorem has been proved there, except that the equality a = a (A) - a (B) is replaced by the weaker statement c a (A) — a
a
(B). The fact that this inclusion is actually an equal
ity is due to D. C. Kleinecke (see [157,Introduction]). Corollary 3.22, for the
special case when a(A)nc,(B) =
0, is
usually called Rosenb]nm's
corollary (see, e.g., [171,Chapter 0]). In [69), C. Davis and P. Rosen thai re-analyzed Rosenblum's results and proved that _clr(B) and extended Rosenblum's corollary to aL(B) and Corollary 3.22. Their arguments include the use of the Berberian—
Quigley construction (Lemma 3.3; see [45), [l72,p.25]) in order to prove Theorem 3.4 as given here. Finally, L. A. Fialkow modified Rosen bium's original argument (by using analytic one—sided inverses instead of resolvents) in order to show that
and
=
[87]. More precisely, Fialkow's results read as follows
then the following are equivalent
THEOREM 3.53. Let A, B fox'
is surjective;
(i)
(ii)
(iii)
ar(A)nai(B) = 0; contains the minimal ideal FIN); ran
is right invertible in L(L(HH. maps every norm ideal (in the sense of R. Schatten [183]) J c K(11) into itself, and each of the above conditions i8 also equivalent to each of the following ones is surjective for some norm ideal J; (v) is surjective for every norm ideal J; (vi) is right invertible in L(J) for some norm ideal J; (vii) (iv)
Clearly,
(viii)
is right invertible for every norm ideal 3.
Moreover, in this case ker spaces of L(II) and ker
and ran
and ran TAB are complementary subare complementary subspacea
of 3 for each norm ideal 3.
79
THEOREM 3.54. The following are equivalent for lAB: is bounded below;
(i)
(ii)
(iii) (iv)
o(A)nc(B) = is bounded below (where F7(H) is the set of all
operators); is left invertible in L(L(tfl}; is bounded below for some norm ideal .7;
(v)
is
(vi)
bounded below for every norm ideal 3;
TAB13 is left invertible for some norm ideal 3;
(vii)
is
(viii)
left invertible for every norm ideal J.
Moreover, in this case
and
ker TAB
ker
are complementary subspaces of L(U) (of .7 for every norm ideal 3, respectively).
Furthermore, Fialkow also considered the operator
induced by
=AX-XB. His results for
in the Calkin algebra, defined by
these operators can be sununed up as follows (see [59], [86], [87], [88], [89], [90]). THEOREM
3.55.
The following are equivalent for
jS surjectiVe; has dense range;
(i)
(ii)
(iii)
=
is right invertible in
(iv) inverse for
is
this case, a right
given by dA,
q,(Y)
is a Cauchy domain such that clre(A)
where
In
L(A(ti)J.
C
Q
and
=
0.
(A) is an analytic right inverse of (A—A) defined on a neighborhood of a:\c2
and
is an analytic left inverse of
Moreover, ker
borhood of £1.
and ran
defined on a neigh-
are complementary sub-
spaces of AU1). THEOREM
(i) (ii)
(iii) verse
for
3.56. The following are equivalent for is bounded below;
a&e(A)flore(B) =
has a left inverse in L(A(HH. In this case, a left in is given by dA,
is a Cauchy domain 8uch that cYte(A) c and are(B) riQ=ø, (A) is an analytic left inverse of (A—n) defined on a neighborhood of
where
80
is
and
an analytic right inverse of
Moreover, her
borhood of
and ran
defined on a neigh are complementary sub-
spaces of A(H). THEOREM 3.57.
only
if
(i)
=
a(A) no
o)Ia
is a Fredholm operator in L(L(HJ) if (B) =
0;
=
[0(A)
—
and
(A)—o (B)].
TAB is a semi-Fredhoim operator if and only if cYte(A)nar(B) (A)nO(B) 0 or (B) = o(naL (B) = 0. (ii)
=
Moreover, Fialkow also proved the following asymptotic version of Theorem 3.19(i) [90,Theorem 1.1]. THEOREM 3.58. The following are equivalent ran
(i)
(ii)
1)
is dense in L(H); =
o
0. and
There exists no nonzero X C1(H) such that BX = XA, Given Y L(H) and 0, there exists X L(H) such that
2)
(iii) — Y
K(U) and
—
<
c•
Combining this result with the proof of Corollary 3.22 and Lemma 3.36, we obtain the following asymptotic version of Corollary 3.22.
COROLLARY 3.59 ([90]). Let H1 and H2 be two Hilbert spaces, let A L(HJ), B
L(H2) and C
L(H21H1) and assume that ran
is
dense.
Then the operators
AC and
0 B
(acting
on H
H1eH2) are asymptotically similar.
The construction of one—sided resolvents (Theorem 3.8 and Corollary
3.9) is due to C. Apostol and K. Clancey [15] (see also [14]). strongly depends on a previous deep result of C. Apostol
The proof
(Proposition 3.16 [l0]).This result extend to the case of an arbitrary compact set a the one obtained in [26] for the case when a is simply connected. The results of Section 3.1.6 and 3.1.7 have been taken from J. G. Stampfli's article [189]. Indeed, Stampfli proved that if C is an irreducible C*_subalgebra with identity of L(H)
LtH)), then
= 2 min{IIT—AII:
A
c1}
for
(in particular, if C =
each inner derivation
(and the analogous result for the operators A, B C). Exten sions of Stampfli's results to derivations on certain C*_algebras and in
C
81
W*_algebras were given by C. Apostol and L. Szidó [31]. For the case of an inner derivation in the Calkin algebra, it is known that THEOREM 3.60 (C.—K. Fong E99]). If T (NT— L(ti)
A
8uch
It}. Moreover,
s
that
E
L(H)., then
= 2 mm
there exists an orthogonal projection X
=
The interested reader will find a very detailed analysis of the
spectra, left spectra and right spectra of large families of operators "somehow related" with the operators
operators; the operator x B2,...
(including tensor products of where {A11A2,...
and {B1,
are separately commuting families of operators in L(I1),.etc),
in [55], [79], [91],[92], [100], [187] and, very especially, in several papers by R. E. Harte [127], [128]. Very recently, the spectrum of the operator
has been completely determined by R. Curto
+
[68].
In [175], Gian-Carlo Rota proved that the backward shift of inf i-
nite multiplicity isa "universal model" for all operators T in L(H) such that sp(T)
(see also [130,Lecture X]). Two (very close) universal models for an operator T with arbitrary spectrum and the spec<
1
trum of the model equal to a suitable neighborhood of a(T) were independently obtained by D. A. Herrero [142] and D. Voiculescu [l94,Propo sition 2.1]. The model given here (Theorem 3.33) is the one due to Voi culescu. J. A. Ball obtained analogous models for commuting n-tuples of operators [41] and F. Gilfeather used Rota's original construction and an iterative argument to give a different (and independent) proof of Herrero's corollary (Corollary 3.35); moreover, Gilfeather [105] actually showed that the invertible operator W such that T' = WTW1 satisfies the conditions of that corollary can be constructed in the C*_ algebra C*(T) generated by T and 1.
Theorem 3.38 (except for those details concerning the structures of Tr and
given by (iii) and (iv) ),
Proposition
3.47 and Theorem 3.
48 are due to C. Apostol [10]. A simplified version of Apostol's trian gular representation has been obtained by R. Bouldin in [52]. The version given here (Theorem 3.38) combines the arguments of [10], [52] and some improvements of D. A. Herrero (items (iii) and (iv)), based on Corollary 3.40 which, in turn, is a particular case of [144]
(Theo-
rem 1 or Theorem 2). However, the proof of Corollary 3.40(i) given here is not the same as the one given in [144]. This new proof has been based on Lemma 3.39 (due to R. G. Douglas and C. M. Pearcy [75]) which will play an important role in the characterization of quasitriangularity (see Chapter VI below). 82
In [190], J. G. Stampf ii proved that, given T e K e
K(K) such
that a(T+K)
L(H) there
exists
(without any estimate on IIKII); more-
over (as observed by N. Salinas; see Remark on p.174 of [190]), the fact that a(T+K) = aB(T) for some K in L(H) is implicitly contained in that paper. A more general theorem, containing both results, has been
obtained by C. Apostol, C. N. Pearcy and N. Salinas in [28] (see also Section 4.3 of Chapter IV below.)
Theorem 3.49 is Theorem 2.2 of [23]. This clever approximation ar gument is one of the most important ingredients for the theorem on spectral characterization of quasitriangularity of C. Apostol, C. Foias and D. Voiculescu (obtained in that paper; see Chapter VI below). The result can be regarded as an asymmetric version of the analogous re
sult of C. N. Pearcy and N. Salinas for semi-normal operators [162]. These results (and many other related ones scattered thorugh the liter ature; see, e.g., [3], [139], [163], [179] and, very especially, [56] and [57]) are based on an argument due to F. Wolf [199]. (See also the partially related articles [71],[95] and [96].) Corollary 3.50 appears
in [27]. Theorem 3.51 is a mild generalization of Herrer&s theorem for non—cyclic operators. Indeed, the same result is actually true for the Banach spaces
(1
p <
=) and
Cc,
(see [146,Theorem 4]). The un-
derlying idea is contained in a paper of N. Salinas [178].
83
4 Two results borrowed from the theory of
The similarity orbit of T U(T) =
L(ti)
{UTU*:
contains U
(where (1(H) is the unitary group of L(H)) , portant role in approximation problems.
We shall see later that, since U(T) information about .S(T)
the unitary orbit of T
(1(H) }
which plays a very imC S(T), a large amount of
can be obtained from the structure of (1(T).
This structure is closely related with the C*_algebra C*(T) generated by T and the identity 1
LUll.
Most of those results related with closure of unitary orbits (and many others) connected with approximation problems can be derived from two deep results of the theory of C*_algebras. These results (the Brown-Douglas-Fillmore theorem and Voiculescu's theorem) will be stated here without proofs (suitable references will be given). Several consequences of these two results will be completely developed here.
4.1 Essentially normal operators
An operator T
L(H) will be called essentially normal if T is a normal element of the Calkin algebra or, equivalently, if the
mutator [T*,T]=T*T_PT* is compact. One way in which these operators arise is as compact perturbations of normal operators. The interest in essentially normal operators is due to two facts: a) Not all of them arise in that fashion; b) There is a rich family of operators of the form "normal+compact" whose structure is not obvious from their analytic expression. (This is, by far, the most important fact from our viewpoint.)
4.1.1 Brown-Douglas-Fillmore theorem
Essentially normal operators have been completely classified in the
84
famous article [60] of L. G. Brown, R. G. Douglas and P. A. Fill-
more. Their main results will be quoted here in the most convenient form for our purposes. (The reader is referred to the above cited arti cle for proofs.)
It is completely apparent from
tially normal,then
re(T) and
e(T)
ps_F(T), ps_F(T) =PF(T) and
A
the
definition that if T is essen-
therefore
<
for afl
re(T) =c(T).
THEOREM 4.1 (BDF theorem). If T1 and T2 are essentially normal operators on H, then a necessary and sufficient condition that T1 be unitarily equivalent to some compact perturbation of T2 is that ce(Ti) =
ce(T2) and ind(A—T1) =ind(A—T2) for' all A / a(T1). There is a special case of particular interest. COROLLARY 4.2 (BDF corollary). (i)
If T is an essentially normal
operator on H such that ind(A—T) =0 for all A c PS_F(T), then T is in +K(H)
{N+K:
N i8
normal, K i8 compact}
In particular, every essentially normal operator T such
(ii)
that cie(T) does not disconnect the plane is in Noit(H)
(iii)
No'ziIl) +K(H) is a closed
+K(tI).
subset of L(H).
4.1.2 Berger-Shaw trace
In order to make the above results useful for approximation purposes, a large family of essentially normal operators will be exhibited. We shall need some extra notation. Given T
L(H)
,
let
A(T)
(=
the weak closure of the polynomials
in 1 and T),
(theweakclosureoftherationalfunctionsofTwith poles outside a(T)), A' (T) = fA s L(H1 : AT=TA} (= the commutant of T) and A"(T) L(tl): BA=AB for all A in A'(T)} (= the double cornmutant of T) be the four (weakly closed) sublagebras of L(H) naturally associated with T. It is apparent that A(T) c The multiplicity subalgebra A of L(H)
is
=inf{card(r): (1 <
ij(A) =n
=). T a
c A"(T) c A'(T).
of a (weakly closed, identity containing) defined by
VIAX: A
H
A, x
r],
F
H}
(4.1)
is n—rnultioyclic (n—rationally multicyclic) if
(T)) =n, resp.).
An operator T
Luff)
is
called hyponormal if [T*,T] is a positive
semi—definite hermitian operator; T is called subnormal if there ex—
85
ists a normal operator N acting on a Hubert space R containing H such that H is invariant under N and T=NIH. It is not difficult to see that (Normal) =>
(Subnormal) => (Hyponorinal).
None of these implications can be reversed;
the operators
e. g.,
M+(r) of (3.2) are subnormal, but not normal. It readily follows from (g) of Section 3.5.1 that
(4.2) (0
tr[T*,T]
is
well-defined for all hyponormal T. The main re-
sult of E50] asserts that if Tis a hyponormal n-rationally multicyclic
operator, then [T*,T]
C1
(K); moreover,
t[T*,TJIi=tr[T*,T]
(n/,r) meas2(a(T)),
(4.3)
where meas2 denotes the planar Lebesgue measure.
auxiliary
In order to prove this inequality, we shall need an
re-
sult.
LEMMA
4.3.
then n
If
PROOF. It is trivial that 1j(A(T))
Assume that H
n+l.
1j(A(T))
P(Aa(T))
n.
=
j=l,2,...,n} for a suitable finite family c H and let {Akj}k_.l (j=l,2,...,n) be n pairwise disjoint dense subsets of p(T). If the positive constants n —l are suitably chosen, then Ilckj(xkjT) =
A
E
{Ckj)k1j12
< = and x.II J
therefore
n
-l
ckj(Akj_T)
is a well—defined element of H. 1
Now it is easily seen that, for n 2
suitably
chosen polynomials p11
we have
-,
so that
the same
A
e
0
(i +
A(T), j=0,l,2,...,n}; moreover,
argument shows that e
M for
all j=l,2,...,n, and all k=l,2
Hence (by Runge's approximation theorem),
M for
all j =
1,2,...,n and for all rational functions r with poles outside c(T).
It
readily
follows that
tains the vectors x1,x2,...,x card{xo,xi,x2,...,xn}n+l. THEOREM
86
4.4
(Berger-Shaw
is invariant under Aa(T). Since M con,
we
conclude that M=H. Hence, o
trace inequality). If T
LUll ts
a
hypo
normal n—rationally multicyclic operator, then [T*,T] e C1(H} and I[T*,T]Ji satisfies the inequality (4.3). In particular, T is eBsentially normal. PROOF. It will be convenient to split the proof in several steps. (I)
Assume that A is hyponormalandp-multicylic, i.e., k k k VfA x1,A x2...,A x}ko
H =
for suitably chosen p vectors x1, x2,...,
in H.
be the ONB of H obtained by means of the Gram-Schmidt
Let
process of orthonorxnalization applied to the sequence x1,x2, . . . ,x ,Ax1, 2
2
2
k
k
k
x11A x2,...,A x,...,A x1,A x2....,A
It is easi
ly seen that A admits the matrix representation a11
a12
a13
a14
.
a21 a31 a41
a22
a23
a24
.
a32
a33
a34
a42
a43
a44
a4
.
f
:
Hence, given m > p+2, we have (akr is defined equal to 0 for all
r
0)
0
_
k+p Lkm_p+1"Lj=m+l
2
Lim+1
Since [A*,A]=A*A_AA*
ki
'
Lk=m—p+lLi=m+1
0, it readily follows from (g) of Sec-
tion 3.5.1 and (4.2) that I[A*,A]li=tr[A*,A] =lim(m -* (II)
If
is a simply connected analytic Cauchy domain, q,:D +
(where D={X:
< l})
cmAm and
with symbol
(defined by
is a conformal mapping with Taylor series L(H2(aD)) is the analytic Toeplitz operator f e H (SD)), then T4, is a cyclic 87
1 is a cyclic vector is the canonical ONB of H
subnormal operator (e0(A)
[73). e(A)
If
and
=
(defined by
n=O,l,2,...), then =
1
tr[T 2
2
m
m=O
n=O
n
m
m=O
I
mIcl2= (l/lr)JD
n=l
=
m=n-'-l m
I
14'(A)12 dA= (l/,r)meas2W)
(where dA denotes the planar Lebesgue measure). Hence, (III)
Given
>
0, we can find finitely many simply connected
analytic Cauchy domains
c)q such that
= Uk:
C
1)
2)
if r
lxi
11Th);
5;
3)
and
4)
<
If
is
+
a conformal mapping (r=l,2,...,q), k
—rmmlm q
r=l
n.
1
max
is a hyponormal operator with
then
—
and cl(Ak)
Moreover, if
1 and
(i) eo,r
lIT
II}=hIrIl
denotes the function
(i=l,2,...,nk; r=l,2,.. L E Aa(T), j=l,2,...,n}, it is not difficult to
identically 1 in the (i,r)—th copy of H (aD)
..,q) and H= check that
= V
il,2,...,nk, 1 so that (IV)
kYjOGr2l j < n and j
i
(mod n)},
is nk-rationally multicyclic.
By Lemma 4.3, Ak is (nk+l)-multicyclic. Thus, applying (I) we obtain
to I
Ii
=
k([T*,T]11
[T,T.]11
(II) and 1) — 4) of (III), it follows that
From this <
=
88
Bk
(E/lr).
Hence,
[T*,T]11 <
Since k can be taken arbitrarily large and
can be chosen arbi-
trarily small, we conclude that (n/v)meas2[a(T) ].
I
U
4.1.3 Examples of essentially normal operators The operator M4(I') of (3.2) is subnormal and rationally cyclic
1 being a cyclic vector for Thus, by Theorem 4.4, M+(r) is an essentially normal operator. On the other (the function e0(A)
hand, since
[M+(r)*,M+(r)]_zz* M+(r)*Z_ZM(r)* Z*M+(r) _M(r)z*
H2(r) 2
Z*Z+[M_(r)*,M(r)]H (r)1
it readily follows from the same proposition that A fortiori, [M_(r) C1 (H2(r)) j = so that C1 (H2(r) M(r) is also essentially normal and Z is a Hilbert—Schmidt operator. (Indeed, Z is a finite rank operator.) .
,
From corollary 4.2, we obtain the following COROLLARY 4.5. Let =
be an analytic Cauchy domain with boundary r
and let M be a normal operator such that cy(N)
0e(M)
U. Then
M.M+(r) and NeM(r) are unitarily equivalent to compact perturbations of 1.1.
A different kind of examples can be constructed as be a nonempty bounded open set such that
follows:
Let
and let
be the "multiplication by A" operator acting on
The subspace A2(Q) spanned by the rational functions with poles outside ci is invar iant under
we shall denote the restriction of
By N+(U) and
to A2(cl) and its compression to N U2) Nifi)
=
+
0
respectively, i.e., G
N_W) A 2
(4.4)
As in the case when is a reproducing (Section 3.2), kernel space and it can be realized as a space of analytic functions defined of
furthermore,
89
=—l
=0 and for all p is a normal operator, N+(c2)
is a rationally cyclic
subnormal operator (Hence, N+(fl) is an essentially normal
=IIr(N+(ffl)II =max
operator, by Theorem 4.4) and {jr(A)I: outside
for
A
each rational function r with poles
Now it is clear that
is also an essentially normal opera-
tor such that
=1 for all and for each rational function
nul(1j_N÷(ca*)*)
r with poles
outside
ci.
Indeed, it can be actually shown that (see [50], [51]) (not ci Now it follows from the BDF theorem that
COROLLARY 4.6. If N is a normal operator such that cj(N)
then
=
(= ae(N))
is unitarily equivalent to a compact perturba
tion of N.
COROLLARY 4.7. If T is an essentially normal operator,
ind(A—T) =n} (1 that a(N)
=ae(N)
<
co)
and N is an arbitrary normal operator such
then T is unitarily equivalent to a compaat
pertubation of
(where
must be i.nterpreted as the 0 operator
on
the
case when c2Ø.)
Indeed,
the conditions
imply, in particular, that II[P
= —
=max{Ir(A)
I:
A
l/dist
[p,fl] and this suffices to guarantee that L is well-defined. Since
(l/rr)meas2(c2),
it is not difficult to check
that L is essentially normal.
4.1.4 An application to approximation problems It is easily seen that if N is a normal operator, then so is ever 90
y operator M in the closure U(N)
of
U(N).
In
this case, U(N)
admits
a very simple characterization. The following result is straightforward.
L(K) is normal, then
LEMMA 4.8. If N
L(tf):
ci(N)\a(N)}.
nul(X—M) =nul(A—N) for all A We
and
N is normal, a(M) =ci(N),ae(M)
have the following
PROPOSITION 4.9. If M is a normal operator such that c(M) is a perfect set, K is compact and
a(M), then N sTm M+K.
PROOF. It is easily seen that ae(M) =ae(M+K)
Let N be a
diagonal normal operator such that a(N) =a(M), i.e.,
respect to some ONB {e)i and
is dense in
with Assume, more
over, that all the eigenvalues of N have infinite multiplicity. By
Lemma 4.8, U(M)= U(N); furthermore, the BDF theorem implies that M is unitarily equivalent to a compact perturbation of N. In other words,
N can be chosen s,o that N-N is compact, and therefore M+K=N+C, where C= (M-N)+K K(tfl. Clearly, c(N+C) =o(M+K) =c(M) =c(N). Since N c U(M) c S(M), it follows that 14 stm N. Hence, it is sufficient to show that N N+C. Let be the orthogonal projection of H onto
then
is compact), and therefore 0 (n + oo). CPu
+ 0 -
(n +oo) (recall that C
[(l_Pn)N(l_Pn)+Pn(N+C)Pn]Il=
By using this observation and the upper semicontinuity of the spectrum (Corollary 1.2(u)), given c > 0 we can find m=m(e) such that IIC_PmCPmII < c/2 and max{dist[A,a(N)]: A O[Pm(N+C)Pm1} < e/2. Thus, (since a(N)
is
a perfect set) we can find an operator Fm
such that IIFm_Pm(N+C)PmIran pm" <
c/2
L(ran and a(Fm) is a set of m
distinct points 111m,112m,••• ,Iimm of the point spectrum of N.
Let
F
F(H) be the operator defined by: =0. Then
Firan
-
(N+F)II = IIC-Fli NWF and
ran P11 < e, N+F
Firan Pm=Fm
whence the result
follows.
0
COROLLARY 4.10. If T boundary a(tt)
and
containinig
L(HJ,
is an analytic Cauchy domain with
ci(T) and Mis a normal operator such that
ç(, then
N sTm
M
M
(co)
91
4.8,
PROOF. By Lemma
Corollary 4.5 and Proposition 4.9, (co)
M slm
and (by a similar
argument) M
M$M_(r)
On the other hand, by Corollary 3.37, M+(r)
whence we readily obtain M
M+(r
M.M+(r)
0
4.2 Matrix models for operators
L(H) and
Let T E
(1
let
ni
< oo)
ind(X-T) =n). Define
E
(so that N_(T)
N+(T) =n:i. where
0r(Dn)), and n=l
is defined by (4.4).
THEOREM 4.11. Given T e L(K), there exist K
K(N) and a decompo-
such that
aition
*
*
0
B
*
0
0 N_(T)
N+(T)
(4.6)
with respect to that decomposition, where B is a smooth operator and a(B) =Ge(B)
=ae(T).
PROOF. By Theorem 3.48, we can find K1
K(H) such that T+K1 is
smooth. Since c
which is contained in
C
it follows from Theorem 3.49 and Corol-
lary 3.50 that
*
*
0
B1 *
0
0
N
(with respect to a suitable decomposition of the space), where N+ and are normal operators such that c(N+)
0e
+(Tfl and a(N) =
respectively.
and that the
Observe that lim(n
stronger form of the Berger-Shaw trace inequality for the operators and
says that J
and
92
(l/1r)meas2(fl
It
ni
), 1
<
is not difficult to conclude that [N+(T)*,N÷(T)J, [N_(T)*,N_ and [N' (T)*,Nt CT)], where
and
N'(T)
are compact operators.
Hence, by the BDF theorem (Theorem 4.1; see also Corollary 4.6), N+ (N, resp.) is unitarily equivalent to a compact perturbation of resp.), i.e., there exists a compact operator K3 such that
(T+K1+K2)+K3
N+(T)
0
*
*
*
o
N'(T)
*
*
*
0
0
B1
*
*
o
0
0
N4(T)
0
o
o
0
0
N_CT)
N÷(T)
*
*
0
B2
*
=
0
0 N_(T)
where *
*
0
B1
*
0
0
N'(T)
B2=
Finally, by Theorem 3.49, there exists a compact operator K4
0$
acting on the subspace corresponding to B2) such that N+(T)
*
*
0
B
*
0
0 N_(T)
(T+K1+K2+K3)+K4
is a smooth operator. In particular,
where ci(B) =
E
pF(B):
ind(A—B)
Let K=K1+K2+K3+K4; then (see observations after Corollary 3.41) *
0 0
Since ce ÷(T)uae(N(T)) c must also be contained in
* *
0 N_CT)
we readily conclude that cle(B)
Moreover, since ae(N÷(T))uce(N_(T)) = Thus, in or c ae(B), it follows that ae(B) 0e(T)
der to complete the proof it only remains to show that pF(B)na(B)
0.
Assume it is not true. Then A-B is a non-invertible Fredholm operator for some complex A; furthermore, by stability properties of the index (Corollary 1.14(i)), we can directly assume that A does not be-
long to the nowhere dense set ae(N÷(T))uae(N_(T)). By smoothness, md 93
Now, if A p(T) (A—B) and it is straightforward that
then A
ind(A—T)ind(A_N+(T))+ind(X_B)=ind(A_T)+m, a contradiction.
A similar argument proves that A cannot belong to p(T) =p(N_(T)). Since A /
it follows that
O=ind(A-T) =ind(A-B) =m, a new contradiction.
We conclude that PF(B)na(B) =0. Therefore, the compact operator ic satisfies all our requirements. REMARK 4.12. In the above theorem (as in Corollary 4.7), we had to use the stronger inequality (not ci
in
order to guarantee that N+(T) and N_(T) have compact self-coxnmuta-
tore. Given c, 0 < c < 1, it is not difficult to construct denuinerably
many pairwise disjoint open connected subsets
0
the unit square ({a+i8: ..,
such that
of
for
l}) with
n=1,2
<
au
n1,2,..
In this case,
> 1—€ for all n=l,2
>
The following theorem provides a representation for T modulo compact operators, "synunetric" to the representation (4.6). Its proof fol
lows by the same arguments as in Theorem 4.11 and will be omitted. THEOREM 4.13. Given T
L(H), there exist C
K(tf)
and a decompo-
such that
8itiOfl
N_(T) *
*
0
B
*
0
0 N+(T)
(4.7)
with respect to that decomposition, where B is a smooth operator and a(B) =
ae(B)
REMARK
=
4.14. If T is essentially normal, then it directly follows
from the BDF theorem that T—F
N+(T)eBSN_(T) (for a suitable compact operator F), where B is a normal operator such that a (B) = cie(B) = aeCT).
4.3 Spectra of compact perturbations of operators 94
Now we are in a position to improve the qualitative parts of Propo sition 3.45 and Theorem 3.48.
THEOREM
4.15.
Let T e
If
L(tI).
is an enumeration of the
bounded components of ps_F(T) (where either M{l,2,...,m} or M is the set of all natural numbers), {dfl}nM 1.8 a family of non-negative integers, {Ak}kK (where either Kf1,2,...,r} or K is the set of all natu with no accural numbers) is an at most denumerable subset of mulation point in ps_F(T) and
is a family of operator8 acting
on spaces of positive finite dimension such that a(Jk) = then there exists a compact operator C such that (i)
a(T+C)
(ii)
{Ak}
K),
(k
>
and (T+C)OItIO(Ak;T+C)
=
k
K
(where (T+C)0 is the central operator in the Apostol'8 triangular representation (3.8) of T+C); and (iii)
min.ind(T+C—A)
COROLLARY
4.16.
for all A
Let T e LifE) and let a be the union of
then there exists K
some bounded components of that a(T+K)
K(t1J
and
such
0.
In particular, there exists
is con-
K(lI) such that
nected.
PROOF OF THEOREM
4.15.
By Theoreni 3.40 there exists C1
K(H) such
that *
T+C1
0 B1 * 0
M
0
where M is a diagonal normal operator such that M
M(OO)
and a(M) =0e
(M) =atre(T)s and at(Bl)=ai(T). By Theorem 3.48, we can find C2
Kill)
(which only modifies the
action of B1) such that *
0 B2 * , and
T+C1+C2
0 0
M
B2 is a smooth compact perturbation of B1. Clearly, T+C1+C2 is also smooth.
Let
in
ind(B2-A), A
Ak'S do not accumulate in (H) such that Ak - UkI + 0 (k +
we can find points 11k
=).
Let
=
Ak E
co). Sincethe (k
K) in } and J_ =
95
is any operator such that Ak e PL(B2)}, where for all k e K. It is easily seen that there exist compact pertur bations M+ and M of II such that M iiej_, respectively. < 1/k
Now we can proceed as in the proof of Theorem 4.11 in order to oh tairi a compact operator C3 such that 0
0
T+C1+C2+C3
*
N_:0
0
0
0
0
0
0
*
*
*
*
*
*
*
* * *
=
0
0
L0_
0000 0,
d
0
>
0),
*
0
B3
*
0
0
0
10
o,
in
where N+ <
0
*
0),
>
N_ =
•{ N+(Eln*)*
(dn):
J_IN_SN' and
in
N and
B3=
*
N'
*
0
B2 *
0
0
Finally, we can use Theorem 3.48 (as in the proof of Theorem 4.1))
in order to find a compact operator C4 (which only modifies the action of B3) such that
T+C1+C2+C3+C4
*
*
0
B
*
0
0
Let C=C1+C2+C3+C4 (clearly, C that a(J_)
c
k
K(ffl).
(B is smooth).
It
can be easily checked C
re(T)s
kaLre(T),
c lAk} and (since B is smooth) min.ind(T+C—A) =d
for all A
so that
E
dn > 01] c a(T+C) c On the other hand, if Ak
(B 0
dn >
then Ak
0)]. =
*
N
and Corollary 3.22 implies that T
Jk$Tk, where Ak E
ps_F(Tk). A similar argument implies that the same result holds for Ak whence we obtain (ii).
Since (i) and (iii) follow immediately from (ii) and our previous observations, we are done.
4.4 Voiculescu's theorem
96
El
An abstract C*_algebra is a Banach algebra C with an operation *:C +
C
satisfying the conditions:
(a*)*=a, a
(i)
(iii) (iv)
2
(v)
C;
(a+b)*=a*+b*, a,b e C; (Aa)*=Aa*, X C; a (ab)*=b*a*, a,b e C; and
(ii)
tall
= Ila*aIl, a
(An operation *:C
C
C.
satisfying (i) —
(iv)
is called an involution.)
with
The most obvious example of a C*_algebra is L(H),
*
equal
to
the usual'adjoint operation'. Given a (finite or denumerable) family {Tn} of operators in L(I1), we shall denote by C*({Tn}) the C*_algebra
generated by the Ta's and the identity operator, i.e., the minimal
norm-closed algebra containing 1, T1, T2,..., T,... and Every separable C*_subalgebra of L(H)
(containing
the ident-
{T} consists of a single operator, then
ity) has this form. If
we shall simply write C*(T) instead of C*({T}). A *_represefltatiofl p of a C*_algebra C on a Hubert space H
is
an
operator p from C into L(H) which preserves products (i.e., p is a homomorphism of rings: =
p(a)*
If
in
p(ab) =p(a)p(b)) and involutions (i.e., p(a*)
C has an identity e and p(e) =1
,
then p is cal
led unital; p is faithful if it has a trivial kernel. It is well—known that a *..representatjon p is faithful if and only if it is an isometric mapping from C onto p(C)
c L(fI)
(see, e.g., [37], [72], [176]).
The following classical result can be found in any of these references.
THEOREM 4.17 (Gelfand—Naimark—Segal theorem). Every abstract C
for
with identity admits a faithful unital i—representation p in
a suitable Hubert space
i.e., C is isometrically
rnorphic to a C*_algebra of operators. Furthermore, if
C
is separable,
can be chosen separable.
then
Since we are only interested in separable C*_algebras, every Hil— etc) considered here will be separable.
bert space (H, Two
*_representations
p1:C
L(H1) and
L(H2J are unitarily
p2:C
equivalent (or, simply, equivalent) if there is a unitary operator U such that p2(a) =Up1(a)U* for all a
from H1 onto
C; p1 and p2 are a sequence of unitary
approximately equivalent (p1 operators U :H -, H such that n
2
1
1
(a) —U p (a)U* n n 2
K(tf 1
7
(n=1,2,...)
and (4.8)
co) l1p1(a)
97
C. It is easily seen that
for each a
THEOREM 4.18. (i)
is an equivalence relation.
Let C be a separable Ca-algebra with identity
and let p:C +
be a unital i-representation of C. Let y be a unit al *_representation of iiop(C) (c on a Hubert 8paceil1. Then a
(ii)
bert x E
..)
Let p1 and p2 be two unital
on Fiji—
apacesti1and t12, respectively, and let
ff.}, j=l.2.
a
The the following are equivalent:
(1)
P]
(2)
There is a sequence of unitary operators Un:K2 +
such
that
a
for each a
lim(n + (8)
H1 +
p1(a)
and
in the weak operator topology, for each a (4)
(5)
of ker
(n=1,2,
H1
C.
There exist sequences of unitary operators (n=l,2,...) such that
of C on
ker
E
H1 and Vn:
p2(a), C.
ker p1ker p2, ker irop1=ker ,Tep2 and the *_represefltatjofls induced by p1 and p2. respectively, are equivalent. and the i-representations ker p1=ker p2, ker and induced byp1andp2,resp., are equivalent.
is irreRecall that a *_representation p of a C*_algebra C on reducing all opera
ducible if there is no non—trivial subspace M of tors in p(C).
COROLLARY 4.19. (i)
of a separable
Every unital
C*_algebra with identity is approximately equivalent to a direct sum of irreducible (ii)
Let p be a unital *_representation of a separable C*_algebra
In order that all unital i-representations of it isneceB C approximately equivalent to p be actually equivalent to
C on a Hubert space
sary and sufficient that p(C) be finite-dimensional.
COROLLARY
4.20.
Let p1, p2 be infinite dimensional unital
aentations of a separable Ca-algebra C with identity; then (i)
(ii)
98
If ker p1=ker p2 and neither p1(C) nor p2(C) contains any compact operator, then
non-
If ker p2
ker ,i°p1, then
a
4.5 Closures of unitary orbits
L(tl) and
Let T e
let C*(T) be the C*_algebra generated by T and
the identity. Denote by p0 the identity *_representation of C*(T) on H
(p0(A) =A for each A in C*(T)) and let S be any operator in U(T). It is easily seen that C' (T) =
{L
is
for
is a Cauchy sequence in L(H)
L(H):
sequence {LJn} in U(tf)
each
such that {UTU*};1 is Cauchy}
a C*_algebra containing C*(T) and therefore there is a (necessarily
unique) unital *_representation p1 of C*(T) on H such that p1(T) S. (This representation is defined as follows: If IS ÷ 0 (n + co) for a suitable sequence {Ufl}nl in u(H)1 then p1(A) = (norm)
lim(n
=)
Ac
C*(T).)
Since T e u(S), it is not difficult to check that p1 is also a unital *_representa tion of C*(T) on H and (clearly, this p1 is faithful), then p0 faithful; indeed, p1
p1(T) belongs to the closure of the unitary orbit of T=p0(T). 4.5.1 Operator—valued spectrum and unitary orbits The above remarks and the results of Section 4.4 have the following direct consequences.
L(H); then there exist sequences {Un}n:l and
PROPOSITION 4.21. Let T E (i)
U(T)
S
if and only if
(Vn}n:l of unitary operators such that +
S
and
(weakly)
+ T
(weakly).
If p is a unital *_representation of on there exist S1. S2 E U(T) such that S1 T$A, the identity h-representation of C*(T), then
(ii)
A,
then
p0 is
(iii)
=
p0
B=p(T) for some unital *_representation p of C*(T) if and
only if either T has a reducing subepace M such that TIM
+0
n=l,2,...) and If S E U(T)
B
or there
(for suitable
exists a sequence {Tn}fll in L(H) such that operators
and
H
(n + oo).
0, there is a unitary operator U such that S_UTU* e K(H) and IIS_UTU*II < c• There exists S e U(T) such that S is an infinite direct sum (v) (iv)
and
>
of irreducible operators.
99
PROPOSITION 4.22. The following are equivalent for T in LU!): (UT) is closed.
(i)
is finite dimensional.
(ii)
(iii)
where A
T
n
0
for some pair (m,n),
B
<
dense in 0, there exists K KU!),
COROLLARY 4.23. The set of all reducible operators i8
Furthermore, given T in LU!) and c < e, such that T+K i8 reducible.
L(H). IIKII
>
REMARK 4.24. Given an n-tuple T=
consid
er its unitary orbit LU!) We
(n)
u
:
shall say that T is reducible if T1, T21...,
have a joint
reducing subspace. Clearly, in this framework, the results of Section 4.4 imply that the analogues of Propositions 4.21 and 4.22 and Corolar y 4.23 hold for n-tuples. In order to determine U(T) following definition:
is
in
concrete
terms we shall need the
The operator-valued spectrum 1(T) of T
L(I1)
the set of all those operators A acting on some subspace of dimen-
sion n, 1
=, or
n <
LU!), and
A
quence {Tfl}nl in L(H} with Tn If S
U(T),
T is the (norm) limit of a se(for
suitable operators Tb).
we shall say that T and S are approximately unitari
ly equivalent operators (T
is an
S). It is easily seen that
equivalence relation and that T
a
S implies that T
i•
sim
S and S
sim
T.
With the above definition in mind, we have the following characteriza-
tion of U(T):
LU!). Then T
THEOREM 4.25. Suppose T, S
s
if
and only if
1(T) =1(S).
4.5.2 Concrete examples of closures of unitary orbits From Proposition 4.21(iv), we obtain PROPOSITION 4.26. If N e LU!) is normal, L is a diagonal normal operator in U(N)
and c > 0, then there exists U
N_ULU* 100
KU!)
and
IIN_ULU*II <
U(H) such
that
Hence, every normal operator is equal to a diagonal operator up to a compact perturbation of arbitrarily small norm It is not difficult to check that if T s Luff) then H can be writwhere and reduce T, TNOr is normal the normal part of T) and TpJ)TIHPJ,(T) = TIHN CT) (= the abnormal part of T) has no nontrivial reducing subspace M such ten as
that TIM is normal; more precisely, course, H
and
Nor(T)
H
Abnor(T)
(Of
can be
infinite dimensional, finite di—
mensional or even
PROPOSITION 4.27. If T, S E L(H) and T is essentially normal, then T s if and only if and nul(A-S) =Oe(T) = nul(A—T) for all A e[a(S)uo(T)]\ae(T). In particular, if K e K(H), then C KIthnorOOdl wher9 0d is the 0 operator acting on a sub8pace of dimension d, 0 d and d+dim PROPOSITION 4.28. If T is
U(T)
LUll:
(S
S
hyponormal and
then
TEN, N is normal and a(N)
c
aijT)}.
4.5.3 On normal and quasinilpotent restrictions L(H) and let {An} be a (finite or denumerable) sequence of distinct points of According to Theorem 3.40, there exists such that a compact operator Let T
where N
L(H
)
is a
N
*
0
A1 H11'
H1
a diagonal normal operator such that N (finite or denumerable) sequence of distinct points
of are(T) =(Jre(Al)• Applying the same theorem to A!, we can find a corn
pact operator K such that
Ni*
*
H1
T-1-K=0
A
*
H2,
0
0
N
H3
where Nr Let
aCNr)
ae(Ni) and rr={1Im}
ae(Nr) and con-
sider the (separable!) C*_algebra am'" .})
generated
by I,
orthogonal projections
and the images in the Calkin algebra of the (onto
—
Ni);
n =
1,2,...)
P.m (onto
101
m=l,2,...). Let p:C + L(H
)
be a faithful unital *_representation. Since
=0, it is not difficult to conclude that
0
B
0
0
N
By Proposition 4.21, we have and rr be closed subT such sets of aie(T) and are(T)i respectively. Then there eXiBt8 S
PROPOSITION 4.29. Let T e L(H) and let
that
*
N1 * S =
T$
0
B
*
0
0
N
N1 and Nr are diagonal normal operators of uniform infinite multiplicity such that a(N1) =Oe(N1) =r1 and O(Nr) =cle(Nr) =rrJ
respec-
tively.
COROLLARY 4.30. Let T E
L(H). If
r
is
and N is a normal operator such that c(N)
a closed sub8et of ae(T)
then there exist8
0e(N)
an operator A such that pS_F(T)\r,
min.ind(A_A)k=min.ind(X_T)k for all A
ps_F(A)
(4.9)
and 'P
i•
s im
NSA.
PROOF. Clearly r can be written as rr C Ore('P)
where
c OLe('P) and
are closed sets. Let S be the operator constructed in Pro-
position 4.28; then, by Lemma 3.36, we have 'P
a
s
Clearly,
=r, so that
0e
N (Lemma
4.8).
Let A be an operator unitarily equivalent to N$TSB. It is completely
apparent that psF(A) c p$F(T)\r. On the other hand, since p is faith ful, it readily follows that and (T)\r, P;:F(B)
=ps_F(B)up
F(B),
=0, whence we
diately conclude that A satisfies (4.9). Since (N 102
a
N$A, we are done.
0
Applying Corollary 3.50 to the direct summand of S unitarily equivalent to
(in Proposition 4.28), we obtain
COROLLARY 4.31. Let T E such that
there
and let r9, and
exists L€ L(fI)
be closed subsets c
such that Ni *
L= =ps_p(T)\[a(Nr)ua(Ni):l,
*
0
0
0
A
rr
respectively. Given c > 0 and normal operators c c (1(N)
and
Ni and N
L(tl)
N
min.ind(A_L)k=min.ind(A_T)k for all
Ps_F(L)J k=1,2,..., and
< 2€.
(Apply Lemma 4.8 to the result of the first modification.)
and assume that is an isolated point of is small enough to guarantee that {X: 0(1e(T)
Let T 0
L(t()
then dA
defines an idempotent commuting with
(in the Calkin algebra). This
idempotent can be lifted to an idempotent R E L(H) such that cle(RTRI ran R ) =
{ii} (RTR=RT=TR
(T)\{p}. Let W
modulo K(tl)) and
G(H) be such that
tion and let
jker R)
=e
is an orthogonal projec-
then L =
and Q is a quasinilpotent operator act-
B such that 1L
ing on the infinite dimensional reducing subspace Let
-s
Clearly,
of L.
L(HJ be a faithful unital *_representatjon of is a (not faithful, unless B acts on a finite unital *_representation of C*(L). By Proposition
dimensional
where 4.21(u), L It readily follows that T sTm
(00);
furthermore, since
is
faithful, we can use Corollary 1.8 to show
PROPOSITION 4.32. Let T ted point
L(H) and assume that Oe(T) has an isola
Then there exists a quasinilpotent 0 such that
T sTm Moreover, if
103
in some open neighborhood of
f
in
=
some open neighborhood of
(k=l,2,...), then Gk=O if and only if
=0.
4.6 Irreducible operators
According to Corollary 4.23, every operator has an arbitrarily small compact perturbation which is reducible. If reducible is replaced by irreducible, the result is still true. In fact, in this case the answer is much stronger and the proof is definitely simpler. precisely, we have LEtIMA
4.33.
The set of all irreducible operators is dense in t(H).
in
given T
c,
<
L(H}, p > 1 and 0
such that T+K
<
e
<
1, there
exists
in
is irreducible.
PROOF. Let T=A+iB (Cartesian decomposition). According to the Weyl-von Neuinann-Kuroda theorem (see, e.g., [153, p. 525,Theorem 2.3]),
c/3 such that A+K1 is a diagonal herthere exists K1 be mitian operator with respect to some ONB {en}ni of K. Let n=l,
the sequence of corresponding eigenvalues (i.e., 2,...). It is easily seen that now we can find K2 C C1(I1),
1K211 0. It is easily seen that R cannot i.e., N does not reduce R. commute with be the matrix of B with respect to the ONE Let B= (b.k) J
k—l 3,—
{en}fll and let 0 <
ri
< c/6.
Define 2
by 4
3
.
p2/2 p3/3 p4/4
.
.
ri4/4
.
.
.
.
.
r13/3 ri4/4
104
.
e2 e4
Then 1K11
< il3. Since the set
<
is at most denumerable, there exists some
such that
for all j,k=l,2 for some
Let
0
<
(4.10)
satisfying (4.10). Then B+K3
<
cannot be reduced by any nontrivial reducing subspace of A+K1+K2. Thus,
T+(K1+K2+iK3) is irreducible
0 THEOREM 4.34.
The set of all irreducible operators is a
(i)
dense 8ubeet of LUll.
(ii)
The set of all reducible operators is an F
dense subset of
L(tf).
PROOF. According to Corollary 4.23 and Lemma 4.33, it only remains to show that the set R(I1) of all irreducible operators is an. F0.
Let P be the set of all those
hermitian operators LI for which 0
H 1. Recall that P is exactly the weak closure of the set of all Since P is a weakly closed subset projections. Let P0=P\{X: of theclosed unit ball of L(H), it is weakly compact, and hence the metrizable. Since the set of all multiples weak topology for P is of the identity is weakly closed, it follows that P0 is weakly locally compact. Since the weak topology has a countable base, the same is
true for P, and therefore P is weakly c—compact. Let P1, P2,..., be weakly compact subsets of P0 such that u1 (n=l,2,...); the Let P={A e L(HJ: AH=HA for some H spectral theorem implies that u1 (k=l,2,...) and 0 (k + cc,). For Suppose that s such Since is weakly compact each k, find an Hk and metrizable, there is no loss of generality in assuming that the
sequence {Hk}kl is weakly convergent to H e CLAIM:
AHRA.
Indeed,
if x,y
H, then
- I
-
-
+ 0 (k
Hence,
+ All (weakly, as k ÷
similarly check that Hence, the norm
By taking adjoints, we can
+ HA and therefore A s
is closed (n=l,2,...) and therefore R(H) is an F
topology).
(in
0 105
4.7 Notes and remarks
The BDF theoremand the Berger-Shaw trace inequality have been pub lished in the "Proccedings of a Conference on Operator Theory, Halifax,
Nova Scotia, 1973", Lecture Notes in Mathematics Volume 345, Springer Verlag, Berlin—Heidelberg—New York, 1973 (see [60], [50]). The interested reader can also consult [61], [62], [49] and W. B. Arveson's survey article [38].
Lemma 4.3 is a portion of Proposition 2 of D. A. Herrero's article
[146]. The present proof of the Berger-Shaw trace inequality (Theorem 4.4) combines four ingredients:
(I)
is an argument due to D. Voicu-
lescu [l96,Proposition 1]. This argument has been independently discovered by C. Berger (unpublished result; see also comments in [196]); (II)
is a classical result (see, e.g., [50,Theorem 1]);
(III)
is an
"ad hoc" modification of the second computational lemma of C. A. Berger
is an observation of D. A. Herrero. An
and B. I. Shaw [49]; and (IV)
extended version of the Berger—Shaw trace inequality can be found in the above mentioned article of D. Voiculescu. The reader is referred to [66], [167] and [168] for a detailed account of the theory of hyponormal operators.
Proposition 4.9 is [l33,Lemma 1]. The matrix models for operators are due to C. Apostol [6] and Theorem 4.15 is a consequence of those constructions and Apostol's triangular representation. Corollary 4.16 (in fact, a stronger form of it) was proved by C. Apostol, C. N. Pearcy and N. Salinas in [28].
The notion (4.8) of approximately equivalent *_representations and all the results of Sections 4.4 and 4.5.1 are contained in the article [195] by D. Voiculescu, except for the equivalence between (1) and (3)
of Theorem 4.18, which has been independently obtained by W. Arvesonin the above mentioned survey article (see [38,Theorem 511) and by D. N.
Hadwin [112,Corollary 4.2]
(indeed, Hadwin established it in a slight-
ly weaker form), and Theorem 4.25 is also due to Hadwin [l12,Corollary 3.6].
Hadwin's reducing operator-valued spectrum E(T) has been modelled on J. Ernest's operator—valued spectrum [80] and the reducing essential matricial spectra of operators of C. N. Pearcy and N. Salinas. (See [160], [164], [165], [166], [179]. The article [63] of J. Bunce and N. Salinas is also relevant in this context.) The different pieces of ](T) in [112] correspond to the possibilities offered byProposition 4.21 (iii); namely:
noted by
106
1)
A is in the if
operator spectrum of T, de-
E(T);
2)
If n is a positive integer, then
the n—dimensional reducing operator spectrum of T, denoted by En(T), is
and the n—dimensional essential reducing operator spectrum of T, denoted by
is Ee(T)nL(cE'1);
3)
The infinite—dimensional
reducing operator spectrum ofT, denoted by
is E(T)nL(f1), and
the infinite-dimensional essential reducing operator spectrum of T, de noted by Ee(T)i is In [112] Hadwin also proved (Theorem 3.9) that U(T)
is always
arcwise connected, thus extending the corresponding result for normal operators, previously obtained by R. Gellar and L. Page in [104] (see also [85]).
Proposition 4.26 is essentially the I. D. Berg's extension to nor sal operators of the classical Weyl—von Neumann theorem on diagonaliza tion of hermitian operators via small compact perturbations [47].(This result has also been independently obtained by W. Sikonia in his doctoral dissertation; see [186].) Propositions 4.27 and 4.28 are essentially due to D. W. Hadwin [111] and to C. N. Pearcy and N. Salinas [162] (see also [160]). Proposition 4.29 was proved in [97] in connection with a problem of quasisimilarity of operators; see also [58], [160].
The problem of characterizing closures of unitary orbits has been developed by D. W. Hadwin in his doctoral dissertation [111] and in 11121. His ideas and techniques go beyond this particular problem. The interested reader can also consult Badwin's papers 1113], [1141 and [54] (joint work with A. Brown and C.-K. Fong), and the closely related articles [101] and [191].
Lemma 4.33 is a mild improvement of a result of P. R. Balmos. That result and Theorem 4.34 are contained in [120]. The proof of Lemma 4.33
given here is due to H. Radjavi and P. Rosenthal [170]. It is convenient to remark that Voiculescu's Corollary 4.23 affirmatively answers in the most complete form Halmos' Problem 8:
Is every operator the norm-limit of reducible ones? [123,p.919]. In [1161, D. W. Hadwin extended Voiculescu's theorem (and many other related results) to cover the case of non—separable C*_algebras and operators acting on non—separable spaces.
107
5 Limits of nilpotent and algebraic operators
As a first application of the tools introduced in the previous two chapters, it will be shown that the closures of the set of all nil potent operators and the set of all algebraic operators can be characterized very simply in terms of the different parts of the spectrum. Essentially the same kind of arguments can be used to obtain two important results about closures of similarity orbits.
5.1 Limits of nilpotent operators THEOREM 5.1. The closure of the set W(t{) of all nilpotent tars acting on II coincides with the Bet of all those A in L(tIl
satisfy-
ing the conditions: (i)
(ii)
(iii)
ci(A) is connected and contains the origin;
0(A) is connected and contains the origin; and ind(A—A) = 0 for all A e
Since a(Q)
{O} for each Q e W(H), the necessity of (i)
=
and (ii) follows at once from Corollary 1.6(1) and (iii), respectively.
Since A-Q is invertible (hence ind(X—Q) = 0) for all A 0, the necessity of (iii) follows from Theoreni l.13(iii). The sufficiency of condi tions (1),
(ii) and (iii) will be proven in two steps.
REMARK 5.2. Theorem 5.1 implies, in particular, that every quasinilpotent operator is a limit of nilpotent ones.
5.1.1 Normal limits of nilpotents
We have already shown (Proposition 2.28(u)) that N(H)
contains
a normal operator M such that a(M) = D(0,1), where = (A s ct: and radius r > 0. By Lemma 4.8, A—id < r} is the disk of center U(M)
108
= (N
s
L(U):
N
is normal and c(N) =
D(0,l)}
c
NUfl.
for
each Q in NIH) and each r > 0, it readily fol contains every normal operator whose spectrum is a
Since rQ s
W(tf)
lows that N(H)
disk centered at the origin.
5.3. Let
LEMMA
N be a normal operator such that and let Q
cumulation point
N(HL
has an ac-
then N qim Ne(A0+Q)
PROOF. Let {An}n:l be a sequence of distinct points of Oe(N) such A0
that
(n ÷
oo). By Lemma 4.8,
admits
Assume that Q s then Q matrix representation of the form °
Q12
Q13
0
a canonical k x k operator
ker Q1eker
.
.
ker Q
o
ker Q
= fo)
1)
and ker
eker
2
Q
ker Qkeker Qkl
0
(Clearly, ker
eker
= H.)
Replacing, if necessary, Q by we can directly assume that ker
(given by (2.1); 0
<<
(
is infinite dimensional
forallj=1,2,...,k. Given n
be the operator obtained by replacing the diIt is easily seen
1, let
agonal entries A0 in A0+Q by A +i,A that =
j
= l,2,...,k}
0
(n
By corollary 3.22 and the above observation,
N
N1
stm
Ne(A0+Q).
Recall that the Hausdorff distance in the family 8P(X)
Li
of
all
nonempty subsets of a metric space (X,d) is the pseudometric defined by
dH(A,B) =
>
0: A c
Bc
(5.2)
Moreover, the restriction of dH to the family 8(X) of all closed bounded
nonempty subsets of
X isametric space:
4.8 and the "spreading the spectral measure argument" (that have already applied, e.g., in Corollary 3.50) applied to-normal Lemma
109
erators with spectra equal to their essential spectra, yields the following simple (but very
result:
LEMMA 5.4. If M and N
0(N)
cy(N) =
are
and
normal operator8 BUch that 0(M) = <
c,
then
dist[N,U(M)]
dist[M,U(N)] <
<
c.
COROLLARY 5.5. If M and N are normal operators such that cl(N) a perfect set, a(M) then N
M.
It
PROOF. =
is
0(N) and each component of o(M) intersects 0(N),
o(M). Let
is completely apparent that oe(N) = a(N) 0
>
and let a(M) c
by open disks of radii E/4
and oe(M)
be a covering of 0(M)
centered at points of aCM). It is easily
seen that if the Xi's are properly ordered, we can find points
= l,2,...,m
j
in
such that, if A]=
}(r= m), then A1 c a(N), and 2
p-i
p
c Ah} <
:
e/2, h =
l,2,...,p—i.
be a finite set of normal operators such that
Let
and let
Oe(Mj) =
be nilpotent operators. By Lemma
5.3,
N stn
and, by using Propositions 2.28(u) and our previous observations, we see that the Q.'s can be chosen so that is arbitrarily small for all j = i,2,...,p1. It readily follows that p1
N
Ne(e.j=l
M.)
j and, by an obvious inductive argument, that N M
sim
o(M)
c
=
Oe(M) = {U.m1
so that dH[a(M)?a(ME)] < E. By Lemma
< c.
that
thatN
sTm
is a normal operator such that
5
c 0(M)
]
5.4,
(3/4)
we can find
such
Since t can be chosen arbitrarily small, we conclude
M.
PROPOSITION 5.6. If M is normal, then M
El
e N(t(1
if and only if
a(M) is a connected set containing the origin. PROOF. It was already observed that 0(M) must necessarily satisfy
the given conditions for each M in N(H). Assume that a(M) is connected and contains 0. Given c
110
>
0,
it fol
lows from Lemma 5.4 that there is a normal operator NE such that aCME) is the closure of a connected open neighborhood of the origin and > 0 be e ?'flH). Let < a. Clearly, it suffices to show that N a normal small enough to guarantee that D(0,n) c By Proposition 2.28(u), Lemma 4.8 operator such that = DC 0,
N(H)
and Corollary 5.5,
and
sim Since NIH) is invariant under similarities, it follows from Propo
s NIH).
sition 1.15 that
5.1.2
0
Spectral characterization of N(H)
Recall that an operator R is called algebraic if it satisfies an algebraic equation. If R is algebraic with minimal monic polynomial then it is easily seen that 0(R) = p(A) = Am} and, by Corollary 3.22,
H2
*
R = .
:
(5.3)
1j=l
0
A+Qm where H1= ker(R—X1) for r =
2,3,...,m,
1,
H
and Kr=
and
E
Nk
r
=
l,2,...,m.
denote the set of all algebraic operators acting on H.
Let
It is easy to see, by using the expressions (5.1),
(5.3) and Corollary
3.22 that the algebraic operator R is the limit of a sequence of operators such that R5 is similar to a normal operator and a(Rn) k. (= the degree of the minimal polynomial of consists of exactly R) points.
LIH), where N is a normal operator such that
LENNA 5.7. Let N,T
0(N) is a perfect Bet and a(T) c 0(N). Then N stm NIT.
PROOF. Let a that 0(N)
c
c
>
and let
0 c
c
be analytic Cauchy domains such and each component of
o(N). If M1, M are normal operators such that aCM1) =
and R is an algebraic operator such that 0(R) c from Lemmas 5.3 and 5.4, and Corollary 4.10 that
intersects and a(M) = ci,
then it follows
111
(00)•
N stm MOM1 stm
By Propositions 5.6 and 1.15, and Corollary 4.10, R can be chosen so that M+R11
R =
where
R21
R13
M1+R22
=
Is
Since
= 0,
c
it
follows from Corol
lary 3.22 that R
0 (00)
for
any X. Thus, if we choose X =
(where Z is defined by (3.2))
then
,
0
(00)
+SO0]
t4OTO[MOM1O Z
= (00)
Since
and dH[a(N) ,o(M) I
<
c/2,
it follows
from Lemma 5.4 that S(N) contains an operator unitarily equivalent to NOT+S', where llS'II
c can be chosen arbitrarily
<
small, we conclude that N stm NOT.
D
Now we are in a position to complete the proof of Theorem 5.1. Suppose that A Since 00(A)
there exists
L(H) satisfies 00
fl,
it
(ii) and (iii) and let c
(i),
>
0.
follows from Theorem 3.48 and (iii) that
K(f1), IIKlII < e/4,
such that A1= A—K1 satisfies
Clearly, A1 also satisfies (i)
=
Let p be a faithful unital *_representatjon of let
It readily follows that if A2= (Ak)
=
on (00)
then
and ci(A2)
c(A1). By Proposition 4.21(u) and (iv), there exists Finally, by Propositions 111(211< €/4, such that A3= =
K(tf),
4.29 and 4.21(u) and (iv), there exists 1(3 K, ,
, 2'
N 0
B2C2
C1
and
where N is normal and 0(N) = Oie(A1) Hence, if K = K1+K2+K3, then
112
K(tI),
such that
0
=
0(A1)
N
11K311 < c/4,
1(3
N A—K
N
C,
0
B1
0 W
0
2
—
—
N
IC1 0
0
—-
C
0
0
lB1 0
o
o
N
2
(5 4)
Since 0(B1)uo(B2) c 0(N), it follows from Proposition 5.6, Lemma 5.7 and Proposition 1.15 that there exist nilpotent operators Q. such that 11B1,N-Q111 L(H),
IIRII
<
€14 and IIN,B2-Q21!
€14. Hence,
<
there exists R in
€/4 such that
<
C10C2
Q2
Q=
A-(K+R)
o
k.
A straighforward computation shows that if
i.e.,
Q C W(U)
and therefore distlA,W(tI)] <
<
O,then Q €. Since
k+k can
be chosen arbitrarily small, it readily follows that A e N(H). The proof of Theorem 5.1 is complete now. 0
5.2 Closures of similarity orbits of normal operators with perfect spectra
5.8.
THEOREM
Let N
e
a perfect set; then S(N)
L(H) be a normal operator such that 0(N) j8 is the 8et of all tho8e operators A in L(I1)
satisfying the following conditions (i)
(ii)
a(A)
Oe(A)
0(N) and each component of ci(A) intersects 0(N); (=0(N)) and each component of %(A) intersects
and (iii)
ind(X—A) =
for
0
all A
os—F (A)
the necessity of the condiPROOF. Since o(N) = Oe(N) = tions (i), (ii) and (iii) follows from Corollary 1.6(i) and (iii) and Theorem 1.13(iii), exactly as in Theorem 5.1.
Assume that A e L(KJ satisfies (i), (ii) and (iii) and let c >
0.
Minor modifications of the proof of Theorem 5.1 (formula (5.4)) indicate that we can find a compact operator K, IlK II < €/3 such that
A—K
w 0
N and
,
(5.5)
0
c
j
= 1,2,3,4.
M, c(N) c By Lemma 5.4 we can find an operator M such that N c c(M) is the closure of a Cauchy domain and IN—Mu c(M) = ae(M)
113
L(ff),
< e/3. A fortiori, we can find R
c/3, such that B = A is the operator obtained from (5.5) by replacing the Ni's by nor mal operators such that < e/3, j = 1,2,3,4. <
(K+R)
Since a(M) is the closure of a Cauchy domain, it is very easy to prove that M = where the are normal operators such that a(A1)na(A2) = o(A3)nci(A4) = 0, a Cauchy domain and each component of contains a component of for all j = 1,2,3,4. By Corollary 5.5 and Lemma 5.7, N stm = (A1eA2)e(A30A4) and
sTm such that =
= 1,2,3,4.
j
it
<
e/3,
j
Thus, we can find operators
= 1,2,3,4.
Since
=
follows from Corollary 3.22 that c1ec2
M=C=
A' 0
0
Hence, N sTni
C
C3OC
and
j = l,2,3,4}
<
<
IIKII+I!RiI-'-c/3 < C.
Since £ can be chosen arbitrarily small, we conclude that N slrn A. El
5.3 Limits of algebraic operators THEOREM 5.9. The closure of the Bet ALg(H) of all algebraic opera tors acting on H coincides with the Bet of all those operators A in L(H) satisfying the following condition ind(A—A) =
0
for all
ps_F(A)
(5.6)
.
PROOF. The necessity of (5.6) follows from Theorem l.l3(iii). On
the other hand, if A satisfies (5.6) and e > 0, then we can proceed as in the
proof
of Theorem 5.1 in order to show that A-K has the form
(5.4) for a suitable compact operator K,
<
e/3, where N is a nor-
mal operator such that o(N) = ae(N) = 0(A) = o(A—K) D j By Lemma 5.4 we can find a normal operator M such that o(M) the
closure
of a Cauchy domain, c(N) c a(M)
c
and IN—MI!
= 1,2. is <
e/3.
Thus, if B is a suitable operator in L(H) unitarily equivalent to the result of replacing the N's by M's in the 4 x 4 operator matrix of (5.4), then IA—Bil < IIKIF4-IIN—MII
<
2c/3. Since M is normal and 0(M) has
only finitely many components, it readily follows from Theorem 5.1 that
Since a(M) is a perfect set and o(B.) c o(M), j = 1,2, it N AtgfH) is invar follows from Lemma 5.7 that N sTm MOB., j = 1,2. Since iant under similarities, it follows from Proposition 1.15 that .
Ug(tl),
j = 1,2.
Thus, we can find algebraic operators R1, R2 such that IIMOB2—R211
114
and IIB1SM—R111
<
< c/3.
It is completely apparent that if R1, R2 are algebraic operators, then so is R2 0
whence we deduce that dist[A,AL9(t{)] <
,1131.M—R111}
<
Since
can be chosen arbitrarily small, we conclude that A belongs to Atg(H). 0
Observe that the condition (5.6) is invariant under compact perturbations. By using this observation, our remarks at the beginning of Section 5.1.2 and Theorem 5.8, we obtain the following COROLLARY 5.10. ktg(HJ L(tl):
R
=
AILg(f11+K(H) =
=
is similar to a normal operator with finite spectrum} ind(A—A) = 0 for A
{R
e
tA e
all
L(tl):
From Theorem 5.1 and Remark 5.2, we obtain COROLLARY 5.11. Suppose that A e L(Ifl, ind(A—A) =0 for all A Ps_F(A) and 00(A) is a finite set and Oe(A) has finitely many compo-
4ents
cm. Let
be an arbitrary point of
Then there exists a sequence
of
o(R)
=
={A.}.m1UO (A),
j=l,2,...,rn.
algebraic operotora such that AIU(a(A);A) for all n1,2,...
0 (n÷=). COROLLARY 5.12. Let A be a finite nonempty subset of U and let S(A) T
=
e
c
L(tfl:
A};
then
S(A)
is the
set of all operators
satisfying the conditions (i)
(ii)
Each component of a(T)
or Oe(T)
= 0 for all A S(A)= {A e S(A):
ind(A—T)
Furthermore,
intersects
A; and
ps_F(T).
A
is
algebraic}.
5.4 Normal operators in closures of similarity orbits PROPOSITION 5.13. Let T such that cl(N)= a(T) and dim H (A;N) =
and let N be a normal operator
dint
each
isolated point
X of a(T). Then T sTm N.
115
It is completely apparent that if T and N are related as in the above proposition, then o0(N) = 00(T) and 0e(N) = OB(T) =
It
will be convenient to provide a separate proof for the case when
a(T) is totally disconnected. LEMMA 5.14. If T L(H), c(T) is totally disconnected and N is a normal operator such that c(N) = 0(T) and dimU(AN) = dim I1(A;T) for A of ci(T), then T sTm N.
each isolated point
PROOF. Let T and N be as indicated and let c
>
0. Then we can
find a finite covering
of c(T) by pairwise disjoint open sets and diameter < €14, for all j = l,2,...,m. 0
such that c2.no(T)
c D(A.,c/2)
then
Thus, if
By Corollary 3.22, T =
NIH(a(T.);N). It
N) for all j
= l,2,...,m).
(j
where
= l,2,...,m.
=
< €/2.
is completely apparent that
•m
j=l j
dim
By Corollary 3.35, there exists
such that Hence, T
Let
is easily seen that dim
On the other hand, it
€12.
<
and
= max{IIN.—T'.ll:
j
= l,2,...,m}
j = 1,2,... ,m}
max{IIN.—X.jl+IIX.—T'.II: whence we readily conclude that T
<
0
N.
PROOF OF PROPOSITION 5.13. Let a denote the set of all non-isolated points of
0, we can use Corollary 3.22 to ob-
Given
where A =
tain that T —
TIU(a,2;T)
and B =
It is easily seen that a(B) is at most denumerable and therefore totally disconnected. By Lemma 5.14, B stm NB= NIH(c(B);N). Let NIK(a(A);N) and let M be a normal operator such that o(X) = can be chosen arbiand IINA_MH < c. Since IIN_MONBII = IINA_Mil < £ and trarily small, it will be enough to show that A sTm M. By Corollary 4.30, Lemma 5.3 and Corollary 5.5 and its proof, if thenwe have
MA is a normal operator such that a(MA) = A
sim
MWC A
sim
where C is an operator such that a(C)
A fortiori, a(C)
c
c
which is a perfect set. Given ri C
C
c
>
0, let
c c(M)
be analytic Cauchy domains such that c(M) intersects a(M). Let By Lemma 5.4 we can find and, by Coroland <
and each component of
be a normal operator such that 0(M1) =
M normal such that a(M) = ae(M) = lary 5.12 there exist algebraic operators R1, R2 such that a(R1)uo(R2) 116
a(R1)na(R2) =
0'
and 11R2—M sM1e
<
By Lemma 5.3 and Corollary 5.5,
<
MeR1OCeR2.
A
so that
M slrn M
and a (C) are pairwise disjoint, it follows
Since a (R1), a
from Corollary 3.22 that
where
—
R1 X1 X2
L(C';X11X21X3) = C'
0
C' X3
0
0
R2
C and X1, X2, X3 are arbitrarily chosen operators.
It follows from (3.2), Theorem 3.23 and our choice of R1 and that X1, X2, and C' can be chosen so that IIL(C';x11X2,X3)
Hence, S(A)
—
contains an operator A' such that
Since
M (Lemma 4.8) and n be chosen arbitrarily small, we conclude that A slm <
5.5 Sums
of
can
two nilpotents
THEOREM 5.15. N(ff)+[N(H)ThNO'L(HH
= LUll;
1
N(H)+W(H)
+K (U). PROOF.
Let T e L(H) and p
3ae(T). By Proposition 4.29 there exK(H) and a decomposition H = are infinite dimensional spaces) such that ist K
T =
A
A
R
R
p M3
where
a(B) c ae(T). Let
xl
>
0
and let Mk be a normal operator such that a(Mk) = (k =
1,2,...).
Define
(with respect to the decomposition H = co6rdinate, occupies the
where the direct sum
mand
0
0
0
0
0
0
0
Mk M3
M2,k—1,2,...,
andlnl
Lk=
*
*
N1
0
B
*
N2
0
0
be the operator obtained from R by replacing the p's in the (1,1) and (3,3) entries by two copies of k = 1,2,..., let L be the operator obtained from by replacing R by Lk in the coordinate (k = = L+K.
1,2,...) and define S
It is completely apparent that T = mal and cl(p—M) = {A
e
a+e}
A—pi
Since p-Ne is noris a closed disk containing
the origin, it follows from Proposition 5.6 that other hand, it is clear that
=
then
If a(Se)
W(H)
N(tI).
c
cc:
= {X
On the
a+c}.
lxi
(by Theorem 5.1) and we are done.
Thus, in order to complete the proof of the first statement, we only oo) have to show that if nk + = (k fast enough, then a(S) = denote the orthogonal projection of H onto the n-th copy Let -..
and let P0 be the orthogonal projection onto H0.
of
Clearly,
if
= Kthen 11K511
(n + 'a).
0
By Corollary 1.2(u) there exist constants > > > c > ÷ 0 (k + 'a), such that > 0 and 1 be the first index such provided liCkli < 6k (k = 1,2,...). Let n1 n1 < n2 < inductively so that 1 and define that llKn II < k and II <
...
<
<
k
with this particular sequence. It is obvious that a(S) = cie(S )ua0(S) = a(M )ua0(S ). Assume that a0(S) 0 and let y ac(S); then lii =
> 0.
a+t+Tl for some
0
By Corollary 1.2(i) there exists
for
enough to guarantee that
all F in L(H) <
with
<
>
)
such that
n
-j)
L+(K—K
0
h be large
and observe that (if n0 is defined to
be equal to 0) S -K
IlFil
Let
= {(A,[,.h {R j—l
•L.}])+(K-K
(n.—n.
Since IIKn
II
<
and
clude that
is not contained in 0(M). it easily follows that if Since 0(R) c
118
= a(M), we con-
(Al
V =
IL.} +(K-K
then a(V) is not contained in
But, on the other hand, it is not
difficult to check that V
2HxnhII
IChil
)lran
LIJ+Ch, where Ch is compact and
Hence, by our definition of ohs 0(V) c
<
a contradiction.
Hence,
a(M),
=
=
whence it follows that T =
M) Finally, observe that if 1 = Q1+Q2+C, where Oe(Q1)UOe(Q2) c and C is compact, then l-Q2 = Q1+C and therefore l-Q2= =
Hence,
1 f
2D,
Oel0e
(1,½) =
0, a
so that contradiction.
N(H)-fN(H)÷K(H).
0
A simpler construction yields the following
PROOF. Let of <
cie(T)
c/2,
>
sp(T).
L(H), dist[T,N(H)]
PROPOSITION 5.16. For each T 0 and let
be a dense subset (1 By Proposition 4.29, there exists K K(HJ, Ku
(c
and a decomposition H = K01H1
are infinite dimensional
spaces) such that T = AIR+K, A c
L(H1),
L(H0), R
R=
A
N
*
*
0
B
*
00
T, c(R) = cle(T) and
N
denotes the i—
where N =
dentity on a subspace Rn
H1).
By Lemma 5.4 we can find normal operators = Oe(Mn)
Let
M =
0
I
M
n
;
t
and
<
clearly,
c(M) = Oe(M) = [ui
is
L(Rn) such that
UpO(T)
(tji:
connected and contains the origin. Thus, if S is the operator ob-
tained from R by replacing the two N's by M's, then a(S) = a(M)
ate(S)
=
and therefore (by Theorem 5.1) S
It
is easily seen that E =
a(A)\o(S)
ventually empty) subset
a(T)\c(M) is a finite (e-
of a0(A). Hence, All1(Z;A)
A =
=
*
H(Z;A)
A0
where H(E;A) is finite dimensional and a(A0) c a(S). It readily fol— lows
thatAIH(Z;A)
=
L+Q, where L is a normal diagonal operator such 119
= max{IA.I: 1 Since a(A0) c a(S),
that IILII
p}
j
sp(T) and 0 is a nilpotent.
<
it follows that
c a(S). Since S
the above inclusion implies that T1=
E
E
(Use Theorem 5.1). On the other hand, by construction
Will)
IIT—T111
<sp(T)+E.
= max{llLfl,IIR—S!I
Since
can be chosen arbitrarily small, we conclude that dist[T,N(ff)] <
sp(P).
It readily follows from Theorem 5.1 that the upper estimate given by Proposition 5.16 is very poor, in general; namely, if H is hermitian, = [0,1] and > 0, then there exist nilpotent operators Q1 and
such that IIH—Q111
<
c and IIH—(l—Q2)lI
sp(l-Q2) = 1, but dist{l—Q2,M(ll)]
Hence, o(l—Q2) =
<
and
<
Better estimates will be given in Chapter XI (second monograph).
5.6 The Apostol—Salinas approach:
5.17. Let T
LEMMA
An estimate for the distance to
NIH). Then for every a
ery pocitive integer k there II(Te0)
PROOF. We
>
0,
> sp(T) and ev-
Q c Nk(HOH) cuch that. Tk
—Qil
Let U and V be two isometries in LIH) such that VV*+UU*=l.
define for 1
j
k the
subspace
M. of Hell given by M. =
x x
i.e., N. is the image under an isometry of the graph of the transpose —1 Hence N1 is closed, i.e., it is a subspace.) of .
h. This is due It is easy to see that MjnMh= {0} if 1 < j,h k, j are to the fact that the second components of the elements in M. and be a sequence in N such N1 and let orthogonal. Let N = h, we can write uniquely = 0. Since M.nMh= 10}, that lim(m +
1
j
where < k, m= 1,2
TJxmIWa1VJ1UXm,j, for some Xmj€ H, =)
Since lim(m
=
0
,
lim(m 1
lim(m
120
0,
1
j
lim(m
= 0, 1
k, and hence the algebraic direct sum M is closed, i.e., j it is a subspace. Now define Qe by Q!MkSM1 = 0,
M1 =
Thus, the representing matrix of
MJ.
on
is of the form
000.. .00 000... * 0
Therefore it is clear that Q
=
0. Let
be the projection onto
ker v*k. Then —
i
TJ
=
i-.1vi-lUx]
= =
Tkxkeo.
Since IIXkII = czB
we conclude that
-
Q]IMII
IITklI/aB k—l
.
Hence
II(TW0) —Qfl —
Thus in order to complete the proof it suffices to show that
Notice
that
Rence
=
+n =
If} and therefore
z
for all xeff}.
0
=
0 THEOREM 5.18. Let T e
every
>
0,
>
such
Q
sp(T),
that
k UT!)
IT—Qil PROOF. Let 0 <
Then for
L(tI) and suppose that 0
y > 1 and every positive integer k there exists
< min{aB,
(5.7)
Since 0
e
we
can proceed as in the proof of Theorem 5.1 (formula (5.4)) in order to find K
K(H),
IX!
<
such that
121
0
0 IC1 0
0
R20
O
010
T-K=
H
c2
Hi
1
a(T).
where 0(R1) U c(R2) c max{I!R1k11,JIR2kjj}
<
is
small enouqh, then
<
Nk(Hj)l 1
<
—
0
Furthermore, if C
By Lemma 5.17 we can find II(R1,0)
2
j
= 1,2,
such that
<
j = 1,2. Thus, if c1ec2 H
—
2,
H1
Q2k
then
=
and
0
j
IT—Qil
=
The above theorem is especially useful to estimate the distance from T to Nk(H) for the case when T is a quasinilpotent. Namely, if 0(T) = (01 and
in
a sequence (Q } an
does not decrease very fast, then there exists N(H) where Qn has order k (k • , as n
nkn nJ/k
such that the rate of decrease of the sequence IT
as
is the same
Ii
the rate of decrease of the sequence
L(H) be a quasinilpotent operator and let N(H) such that Qk2k = o 1. Then there exiBts a sequence for some constant c = c(T,ó) > 0, COROLLARY 5.19. Let T
>
c(IITIjk+IITkIIl/k).
IIT_Qkll PROOF.
It is a
If k
direct consequence of Theorem 5.18:
Take a =
(5.7).
and B = 2
D
and we choose
-I-i)
=
and
ci =
B[(k—1)
in (5.7), then the left side of this formula takes the form yC(k) (IITII.IlTkII )l/(k+l) where C(k) = (k+l) k >
2
and C(k)
-,
1
-L)/(k+1)
(k—l)
(k •
COROLLARY 5.20. Let T
dist[T,N2k(H)]
122
Observe
that 1
<
C(k)
3
for all
=). Hence, L(H) be a quasinilpotent operator; then for
all
k
2.
In the converse direction, we have the following PROPOSITION 5.21. Let T e L(ti)kand suppose that there exists a Se quence
in N(ti)
= 0,
such that
1/k ao)IIT_QII
lim inf(n Then T
1,2,...
n =
=
and
0.
18 quasinilpotent. k
PROOF. Since
n
0, it follows as in the proof of Proposition
=
1.10(i) that
(tfrfl+ 1+1/k +
(lt"iI
()
=).
(n
5.22(i) Let T, a, and k be as in Lemma 5.17 and let R E with is an infinite dimensional space. Identify where Nm(Ko)i and let K2— tl0eker R. If ker R c REMARKS
0
R12 R13
.
.
R23
.
.
0
.
.
0
Rim ker R1Oker R°= R 9ker R .
R
3,m—1
R
3m
ker R3eker R2
R= 0
0 R4 ,
0
m
ker Rm 1eker ker Rmeker Rm
and Q e Nk(KGK1) is constructed as in that lemma, then Q
Rj2 R
.
R
.
R
R
o
ker
R o
into
ker
R
is a nilpotent of or
der at most k+m—1 acting on KWH0 and !!(TWR) —LII <
(ii) Assume that
1T1
aIITIFIB+ 0
<
d
<
, Re
Nm(U:PJ, a >0 and 8 >
sp(T), and let k be a positive integer. Then we can apply the above construction to TWO and
(where 1
s
and 0 acts on an inf i123
(k-l)d,
nite dimensional space). If s =
then the operator L admits a re
= 0 and
ducing subspace N of dimension [(k—l)p+l]d such that lIxll= l}
max{lIx-
is small enough to guarantee the existence of a unitary operator U onto Td such that
ping
dim M
= [(k-l)p+l]d
ct/2. If M =
<
and there exists L'
Nk+m_l(MIS
that
L'lI
<
IT
2c*/2
Furthermore, if nul R =
n
then M
UN,
LIN such
L'
k
2, then M can be chosen so that
dim M
where [r] denotes the integral part of the real number r. The details of the construction can be found in [l49,Lemina 5.10 1
and r150] (Lemma 2.3 and remark following it).
5.7 Salinas' pseudonilpotents
is
We shall say that T E Luff) every c
>
0
a pseudonilpotent operator if for
there exists a decomposition H
H into the di-
=
rect sum of a finite orthogonal family of subspaces if1,
...
,
such that if T11 T12
.
.
.
T21 T22
.
.
.
Hi
T
H
n
then the norm of T11 T21
112
0
T
ni
is
less than
124
T
n2
.
•
•
T
n,n—1
T
nn
H
n
It is clear from the definition that a pseunilpotent operator is
an operator that can be "obviously" approximated by nilpotents. On the other hand, if Q is a nilpotent of order n, then it is completely apparent (see (5.1)) that Q admits a decomposition of the above type with glower = 0. Thus, if we denote by 'P(I1) the set of all pseudonil— potents, then c
M(H)
c
'P(H)
(5.8)
N(H)
We shall see immediately that both inclusions are proper. interchanging the subspaces H1,
By
H2,
...
,
in the defini-
H
tion of pseudonilpotency itiseasytoseethat an operator T is in '1'(H) if and only if for every
>
0
there exists a decomposition H =
that the norm of Tupper is smaller than c. (Tupper is the triangu
for i > j
lar ooerator matrix with (i,j)-entries
It
= (def)
readily follows that 'Y(H) = THEOREM 5.23. Let A = H+K, whcre
th'n
A e
and
'P(H)
only ff A
is
PROOF. Suppose that A = H+K
Let
<
H
c-
H
is
0 for i
and
L(H):
(T
j.)
T*
liermitian and K is compact;
a compact quacinilpotent. is not compact and let
0
<
be a decomposition of the space with respect
=
to which hA lower 11< c; then II IA lower 11< c. lower lower Since H is hermitian, IIH Let H diag =H ® . II = IIH < upper lower 11 H with respect to the above decomposi-
IIM1eII <
tion. It is easily seen that IIHdiagII =
=
Iii lower
and, a fortiori,
upper—ii diag 11< 3c.
Since c can be chosen arbitrarily small, we
conclude that hAll = 0,
i.e., A is compact, a contradiction. On
the other side, if A
K(H)n'F(H), then
A
M(H)
and therefore
it must be a compact quasinilpotent (Theorem 5.1). Conversely, a compact and
if A is
is an in creasing sequence of finite rank orthogonal Projections such that
1 (strongly, n
to), then IIA_PnAU
0
(n -*
of the spectrum (Corollary 1.2(u)), given c > 0 positive integer n0 such that if n > n0, then O(PnAPn)
tinuity a
From the upper semicon-
Let
m >
there
trix
n0 so that
IIA•••P
All
Since ran
there exists c
is finite dimensional
exists a basis el,e2,...,ek of PH on which the representing maof the operator
is in upper triangular form. Observe
that the diagonal entries of this matrix are in absolute value less than Letting = V (e.}, j = 1,2,...,k, and definina Hk+1= ran we
lAlowerli
<
deduce that with respect to the decomposition H = e. Since
is arbitrary we
conclude that A e 'V(H).
0
125
It follows from Theorems 5.1 and 5.23 that if H is hermitian and
[0,1],
N(t()\'1'(K).
On the other hand, if B L(K) is the unilateral weighted shift, with respect to some ONB defined br-Be = cznen+l, n=l,2,..., a + 0 (n + then B is a compact
aCE) =
then H
0 for all k =
quasinilpotent, but B
1,2,...,
so that B e 'P(W)\N(I1)
Hence, we have COROLLARY 5.24. N(fl)
N(K).
'v(H)
The following result provides a simple way to construct operators with positive spectral radius.
in
LEMMA 5.25. Let T
L(Hj
and suppose that there exists a sequence
in N(K) such that
and
urn inf(n -*
Then T
£. For
0.
=
'I'(K)
PROOF. Given £ <
00)
each j,
i
onto ker 0n eker Q
> 0,
j
let n be sufficiently large so that k
let
P. be the orthogonal projection of H 3 Also let x K; then
-l .
,
=
Therefore
< c,
as desired.
D
COROLLARY 5.26. Let c = 1,2,...) be defined as in (2.1) and let S be the unilateral shift of multiplicity one. If either aB 00) = 00 or then the operator for infinitely many 0 tO T =
is
a pseudonilpotent with spectral radius sp(T) = PROOF. It is clear that sp(T)
11Th
00
IT
00
1.
1. On the other hand, by
Corollary 2.24 (or minor modifications of its proof), there exist oper ators such that IIT_QkIl < 4/k (k = + oo) since also Qkk=0, Leitnua 5.25 implies and +0 Thus
1,2,...). 0
T€'V(tfl.
PROPOSITION 5.27.(i) Let T e L(KJ be such that urn inf(n +
126
oo)
kilT
kl/k II
0.
Then T is pseudonilpotent and quasinilpotent.
{Ck}kl
(ii) However, given a sequence 1 = creasing to
tent unilateral weighted shift B = B1 in 'VUI) lim(k +
of po8itive reals ina compact quasinilpo-
arbitrarily slowly, there
oe)
such that =
PROOF. The first statement is an immediate consequence of Corol-
lary 5.19 and Lemma 5.25. = a2 =
Define B as in the proof of Corollary 5.24 with weights
= 1,
= 1 .,
where
%
1
1/2,
=
=
+1=
,
...
j—l
2
= l/j,
+1 /j = = j
is a strictly increasing sequence of natural num-
bers. Given the sequence {ck)k1, it is not difficult to define the se quence {n.}.1 inductively so that lim(j • follows
JJ
that for j
>
1, if n._1 < k
n., then
n.1/n.
j
:i
ckIIBII
>
lB
>
j—1
j—l
Since B is a compact quasinilpotent, B lim(k REMARK
A
It
nj_i
oo)
'+'(UI
cklIBll
(Theorem 5.23) but
=
5.28. Let B = B1 be as in Proposition 5.27 and let
'Y(H1 and lim(k +
=
but
ak is compact for no
value of k.
Very little is known about the structure of the operators. In [149,Lemma 5.1], D. A. Herrero proved that if H is hermitian and c(H) = [-1,1], then
dist[H,Wk(H)]
5/log k
for all k large enough. On the other hand, Theorem 5.23 and Lemma 5.25 imply that dist[H,NkIK)]
0(1//k).
But the gap between 0(1/log k) and 0(1//k) is still very large. (See also the results of Section 2.3.3.)
The following proposition follows immediately from the definition of pseudonilpotenty.
PROPOSITION 5.29. (i)
If A e L(H) has the form of Lemma 3.36 and
for all j=].,2,...,k, then A€ 'V(fl).
e
(ii)
If A e 'V(H), then U(A)
c
127
COROLLARY 5.30. If N is normal and 0(N) is equal to the closed unit disk, then N is a pseudonilpotent.
PROOF. Let N be the normal operator defined in the proof of Proposition 2.28(u). Since the matrix of (Lk_Qk)upper has no more than four non—zero entries in each column or row, a straightforward computation shows that if M= (M..). with respect to the decomposition k (j where is the subspace defined by
=1,2,... ,k),
]JekerU,:1
AQk
if. =
then IlMupperli
AmOk
whence we conclude that M
<
is a pseudonilpotent. Now the result follows from Lemma 4.8 and Proposition 5.29 (ii) 0
Theorem 5.23 and Corollary 5.30 suggest the following PROBLEM 5.31. Which normal operators are pseudonilpotents? PROBLEM 5.32. Let A (1)
If W
(ii)
Is
G(K),
is
WAW'
E
for all polynomials
Ap(A)
p?
PROBLEM 5.33. Is every quasinilpotent a pseudonilpotent?
5.8 Limits of nilpotent and algebraic elements in the Calkin algebra Most of the results of Sections 5.1 through 5.7 induce analogous results in the Calkin algebra; namely, we have THEOREM 5.34. The closure of the set N(A(K1)
of
all nilpotent ele
ments of the Calkin algebra coincides with the set of all those A c A(H)
satisfying (ii')
(iii')
the conditions:
Oe(A) is connected and contains the origin; and ind(A—A) =
Moreover, W(A(if)) = 7T
1[N(A(HJ) ]
0
for all A
c
= ir[N(if)] = is closed in L(11).
andN(K) +K(if)
=
[N(HJ+
PROOF. The necessity of the conditions (ii') and (iii') follows
exactly as in Theorem 5.1. On the other hand, if A satisfies Cii') and (iii'), then it follows from Theorem 3.48 that A+K satisfies the condi tions Ci),
A+K
N(f1)
(ii) and Ciii)
Theorem 5.1 for some K in K(H). Therefore
and, a fortiori, A = A+K
whence the results follow. 128
of
e
c 71[N(K)]
c
0
Similarly, from Sections 5.2 through 5.6, we obtain the following corol].aries.(All the proofs follow exactly the above scheme and will
be omitted.) Of course, similarity orbits in A(H) are defined with respect to the invertible group
A E L(H)
G(A(H1J=
is a Fredhoim
erator}.
Luff) be a normal operator such that Oe(N) is the set of 4(H) satisfytho8e ing the following conditions: THEOREM 5.35. Let N is a perfect Bet; then
all
(ii')
0(N); and 0
THEOREM 5.36. (i)
for all A
ps_F(A).
The closure of the set AZg(A(H)) of all alge-
coincides
braic elements of 4(11)
with the set of all those
in 4(H)
satisfying the condition (5.6). (ii)
AZg(A(H))
4(H):
=
R is similar to a normal operator
with finite essential spectrum) =
(iii)
4(11):
=
If A is a finite nonempty subset of
0e(A)
c
A), then A(H;A)
and A(H;A) = {A
is the set of all those
c
ouch that
each component of 0e(T) intersects A and T satisfies (5.6). Furthermore, A(H;A)
A(H;A):
=
A is aigebraic).
PROPOSITION 5.37. If T 0e(N) =
ThEOREM 5.38. 1
4(11)
and N is a normal operator
that
aw(T), then T sTm (i)
N(A(H))+{N(A(K))riir[Nok(H)]}
=
4(H):
however,
N(A(H))+W(A(H)). (ii)
4(11),
For each
dist[T,W(A(H))]
It will be shown later (Theorem 7.2) that if can be lifted to an operator Q THEOREN
> sp(T),
5.39.
y >
If
land
Wk(A(N)), then
Wk(H). By using this result, we have
4(11) and 0 e re(T)j then for every a > 0, every positive integer k there existsQ €N2k(A(H))
such that
IT!)
Finally, observe that pseudonilpotency admits the following expression in 4(11):
4(11)
is a pseudonilpotent if for every c >
there exists a decomposition T
=
where
0
is a finite
129
family of pairwise orthogonal hermitian idempotents, such that H
With this definition in mind and using the well-known fact that can be lifted to an hermitian opera
every hermitian element B of Alto
such
tor H0 in Lift)
that
we have the following
=
THEOREM 5.40. (i) N(A(H)) '1'(A(HJ) N(A(H)). Indeed, N(A(H)) contains quasinilpotents as well as elements such that sp(T) =
=
1; if
is a non-zero hermitian, then ii
H
(ii) Ifklim(n
such that
= 0,
(iii)
N(A(H))Vv(A(tl)). c NIA(H))
= 0 for a sequence
then ÷
If lim(k quasinilpotent.
0, then
=
is pseudonilpotent and
(iv) However, given a sequence E = (Ck}k_l of positive reals increasing to arbitrarily slowly, there exists a quasinilpotent in 'VIAIK)) such that lim(k
=
-*
PROPOSITION 5.41. If A
'FlACK)); then U(A)
c 'FlACK)).
5.9 On the spectra of infinite direct sums of operators
5.42.
THEOREM
Let {Tn}n:i
n =
e
ly bounded family of operators and let T =
rl
1,2,...)
n1
e
LIII),
11n'
(i)
=
a = (A /
is not uniformly bounded},
a(Tn): (iii)
moreover, a c
[U:1 a(Tn)]•
reCT) and
each component of a intersects
If there exists a function 0(r)
(iv)
defined
on (0,5)
such that (5.9)
<
for
a uniformwhere H =
then
a(T)
Un:l
be
, then c(T) all A e a(Pn) and all n = 1,2 The condition (5.9) cannot be replaced by the weaker condi(v)
tion =
possible
growth:
PROOF.
130
1,2,..., even
o(Tn) and all n =
for all A
Ci)
0(r) =
Let
x =
(5.10)
hr.
x
H,
A
C
if 0(r) has
the minimal
and assume that (A-T)x = 0.
0 or A
(A_Tn)xn so that either Then 0 = (A—T)x = each n = 1,2,..., whence the result follows.
(iii)
(ii) —
The inclusion u1 a(T) c a(T) is obvious. If A e 0,
then there exists a sequence {x
x
k—l'
E
lix
H
Ii = 1, such that
). Thus,
(k
0
+0
= 1 and
then
for
e
0),
(k +
Hence, a c ai(T).
oo)
The same argument, applied to T* indicates that a c ai(T*)* = Since (by (i)) a
ire
a
(T).
Conversely, if on A),
then
C
(for some constant C depending only
it is easily seen that A
p(T). Indeed,
C.
lI(x—T)'l!
Hence, a(T)
Let r be a clopen subset of a Let
a(T)]= 0.
and assume that
be a Cauchy domain containing 1' such that
C
a constant C such that A fortiori,
C
for all A Since
and all n = 1,2
for all A
is an analytic function defined (at least) in some neighborhood of £1, it follows from the maximum modulus theorem that
C
and all n
for all A
1,2
It readily follows (as above)
that A e p(T) and
C for all A i.e., r = is compact, it follows that each component of o
Since a
must in
tersect tUn:i (iv) This follows immediately from (ii).
(v) Let H be an hermitian operator such that 0(H) = [-1,1]. By Theorem 5.8, H
N, where N is a normal operator such that cl(N) =
0(0,1). Let T. Since P =
Tfl
H be such that + 0 (n + = 1/dist[A,a(H)] for all A
lows that the sequence
•(r) = hr and, moreover, then However, if A
satisfies the
condition
co)
and let it fol
p(H),
(5.10) with
= 0(H) = [—1,1] for all n = 1,2
D(0,l),
oo){infll(A_T)xIi:
lim(n
= inf{Il(A—N)xfl:
liXTi = 1)
e
lxii =
11
=
0,
so that
n l} for all A
e
D(0,1)
\[—l,1]. Hence,
a(T) =
= D(0,1).
we are in a position to show that the result of Corollary 3.40 is the best possible. Indeed, we have Now
131
EXAMPLE 5.43. Given a nonempty finite or denumerable bounded set and a compact set r such that every component of r interd = sects d, there exists a triangular operator A such that d(A) = = d and a(A) = I'. Furthermore, if d is a perfect set, then A can be chosen so that
n =
1,2
PROOF. Let N beanormal operator such that 0(N) = Oe(N) = r. By of algebraic operators Corollary 5.12 there exists a sequence such that ÷ 0 (j -* = (j = 1,2,...) and
Define A = of (v)) a(A) =
•j:1 u.1
By Theorem 5.42 (see, in particular, the proof = d and 0(A) =
is
a = {X j d:
due, where
not bounded below} =
a(N)\d
=
r\d,
so that a(A) = r. is easily seen that A is triangular and that the only possible
It
choice for d(A)
is d(A) = d.
This solves the first part of the problem for the case when d is denumerable. If d is finite, then the first statement follows from a trivial modification of the above argument. If d
is perfect, then it
follows from the
theorem (see, e.g., (122]) and Proposition 3.35 (and its proof) that A admits a compact perturbation A—K, where K E K(H) is normal, such that A-K is triangular, =
d(A-K)
=
d, nul(A_K_An) =
1 for
all n =
1,2,...
,
and a(A-K) =
r.D
5.10 Notes and remarks A classical example due to S.
Kakutani (see 1119,Problem 87],[l72,
p. 282]) with weighted shift operators indicates that a limit of nilpo tent operators can be an operator with positive spectral radius, so that neither the spectrum nor the spectral radius are continuous, in general. In his survey article "Ten problems in Hilbert space", P. R. Halmos raised the following PROBLEM 7.
Is
every quasinilpotent operator the norm limit of nil-
potent ones?
In view of Kakutani's example, Halmos remarked that the above question is
in the sense that the condition is already known
to be not sufficient. The right question should be: "What is the closure of the set of nilpotent operators?" Can it be charaôterized in simple terms?
In [129], J. H. Hedlund found several interesting examples of 132
operators with positive spectral radii contained in W(H), including the operator J This example was the key result for the •kl first proof, due to D. A. Herrero [132], of Proposition 5.6. An alternative proof of this result was later given in [26]. (The proof included here is different from both of them.) N. Salinas [180] extended Herrero's result to certain classes of subnormal operators and C. Apos tol and D. Voiculescu affirmatively answered Halmos' problem in [32]. (See also [5], [11].) A different proof of thesame result was later given by C. Apostol and N. Salinas in [29]. The complete spectral characterization of N(tI)
(Theorem 5.1) was obtained by C. Apostol, C. and D. Voiculescu [26]. Partial results in that direction were independently obtained by D. A. Herrero [133], [134].
The spectral characterization of ALg(H)
is
due to D. Voiculescu
[194] and the results on closures of similarity orbits were obtained by D. A. Herrero in [140] and [141], respectively. Theorem 5.8 has been extended in [42] to cover the case when ce(N) = a(N) but o(N) need not be perfect.
(N normal),
The original proofs of Theorems 5.1, 5.8 and 5.9 and Proposition on Voiculescu's theorem, so they are rather differ
5.13 do not depend
ent than the ones given here;
e. g. ,
the proof of Theorem 5.8 given
in [140] strongly depends on D. A. Herrero's result on the existence of universal quasinilpotent operators 1137]. In [135], [136], D. A. Herrero characterized W(A)
and
(in terms of parts of the weighted spectra; see, e.g., [78]) for the case when I-i
is a non—separable Hilbert space.
Theorem 5.15 improves two previous results [l33,Theorem 7], [149, Proposition 6.2] of D. A. Herrero. Proposition 5.16 is due to C. Apostol and N. Salinas [29,Theorem 3.5] and the results of Section 5.6 are contained in the same article, except for some mild improvements in Corollaries 5.19 and 5.20, based on Voiculescu's theorem. The observations of Remarks 5.22 are due to D. A. Herrero; they are related with approximation problems in the class of quasidiagonal operators ([149, Theorem 5.11], [l50,Section 2]; see also Section 6.2.1 below). The notion of pseudonilpotent was introduced by N. Salinas in his (in fact, a slightly
article [180], which also contains Theorem 5.23
weaker form of the theorem; see comments at the beginning of Ll50,Sec— tion 6]), its corollary and Problem 5.32. The results going from Lemma 5.25 through Remark 5.28 are due to C. Apostol and N. Salinas [29]. Proposition 5.29 containB two unpublished ro. Corollary 5.30 is an unpublished result of
D. P. iferre-
I.
lierg, who also
raised Problem 5.31.
The results of Section 5.8 are tacitly contained in the corre— 133
sponding articles on approximation of operators. In particular, L. A. Fialkow proved in [83] that if N is normal, then (WNV:
W,V are Fredholni operators and
=
=
S(N)+K(Il}.
Finally, Theorem 5.42 is a particular case of [144] (Theorem 1 or Theorem 2; see also [84,Section 2]) and 1l47,Lemma 1], and Example 5. 43 is essentially due to N. Salinas (see [84,Theorem 3.2]).
134
6 Quasitriangularity
Our next step will be the analysis of the all important notion of quasitriangularity and its characterization in spectral terms. Several consequences of this characterization will be developed as well. An operator is quasitriangular if it can be approximated by trian gular ones. The most obvious examples of triangular operators are the algebraic operators and the operators M_(r)
(defined by (3.2)). The
main step of the characterization of quasitriangularity says that all auasitriangular operators can be constructed out of algebraic opera-
tors, the operators M(r) and approximation. The introductory section of this chapter deals with a aeneral argument of approximation of operators by operators with a very simple structure.
Most of the results of this chapter are "invariant under compact perturbations" and therefore they can be immediately "translated" to results about closures of subsets of the Calkin algebra.
6.1 Apostol-Morrel simple models
We shall say that an operator S is a simple model if it has the form
5=
*
*
0
A
*
0
0
S_
(6.1)
,
where (i)
(ii)
a(S), 0(A) and a(S_) are pairwise disjoint; A
is similar to a normal operator with finite spectrum; is (either absent or) unitarily ecTuivalent to
(iii)
in
'
1
is a finite family of analytic Cauchy domains with pair-
where
wise disjoint closures; (iv)
S
is (either absent or) unitarily equivalent to 135
(h.)
h1
1
where
<
is a finite family of analytic Cauchy domains with pair-
wise disjoint closures.
6.1.
THEOREM
The simple model_s are dense in L(H}. More precisely:
and
Given T
there exists a simple mode1 S
0
>
c
o(S÷), cy(S_) c
ps_F(T)
all A
ind(A—T) for
such
ind(A—S)
c,(A)
ps_F(T) and
Ps_F
IT—
that =
Sli <
Let S be the simple model given by (6.1) and let N be a normal N. It is easily seen that (use (i) and Corollary
erator such that A 3.22) S
and It follows from (3.4) that r+=cyz (S÷) °re By using the results of Section 4.1.3 and
are normal operators such that
Proposition 4.27, we see that if N ,N
a(N
+
c
)
+
and a(N
F
c
)
I'
respectively,
ON+
S
Let
,
S ON
and
S
then
and(r),respectively,
be compact subsets of
and let M ,
M
S
be normal operators such that cy(M
)
=E
and
) =
E
< c, then we can use our pre (6.1) and ITvious observations and Lemma 5.4 in order to show that there exists an
resp.. If S has the form
operator S' such that IT- S'fl
<
2€ and S' admits a 3
x 3 matrix decomand S_ replaced by S÷,M+ and S_0M,
position of the form (6.1) with
respectively. Thus, we have the following. COROLLARY 6.2. Given T c L(H) and e S÷OM+ *
*
0
A
*
0
0 S_OM
0,
>
exists
(6.2)
where (i)
S÷, A and S
(i.m.)
+ M
(Sf) and
(iii)
the form of (6.1); +
are normal operators such that cy(M )flO(S÷) = =
ac(S),
resp.;
ind(A—T) for all A (a(SOM) is the closure of an analytic Cauchy
ind(A—S')
(iv)
=
)
resp.)
domain
sects a(S+)
have
——
and M
resp.) inter-
such that each c'o'nponent of
(a(S_), resp.) and c(S+)
c
o(S+)
(o(S
o
(v) 136
,
and o(A) are
pairwise
disjoint sets;
c
c
(vi)
5'
(vii)
and
IIT—S'II < 2c.
Similarly,
we can replace A by a more general kind of algebraic
operator, or by an operator similar to a normal operator with "large" spectrum, etc, i.e., Theorem 6.1 mustbe regarded as an argument to
construct very general distinct families of "models" dense in UK). A or S_ can be replaced by an operator in a very Each of the terms large class; some other examples of this situation will be given later.
PROOF OF THEOREM 6.1. Let T Cauchy domain such that
L(H) and c
3.50 to obtain normal operators Ni and
a(Nr)0e(Nr)
0.
c aire(T)c/8,
c
re(T)
>
is an analytic
If 'P
we can use Corollary
Nr such that a(Ni)
and an operator
'P
T1=
Ni*
*
0
L
*
0
0
Nr
such that min.ind(A_T1)k=min.ind(X_T)k for all A e ps_F(Tl), k =
< c/4.
and IIT—T111
Clearly,
1,2,
and therefore
c0(T1) is a finite subset of Cl
*
0
L1'
where C1=T1IK(a0(T1);T1) and a0(L1) =0. By Theorem 3.48, there exists (with respect to the same decomposition), 11K111 < K1 E K(tfl,
El4 and Cl
*
0
L2
where L2L1+K{ is a smooth operator. It is not difficult to check that
Ps_F(T2) = and ind(L2—A)=ind(T2—A)ind(T—A) for all A =
=
P;_F(T2)
=
ps_F(T2).
be the components of p_F(T2)na(T2) and let be the components of p_F(T2)no(T2). It is easy to see that
Let
=
,
0 for
all i
h,
=
0 for
=
(i=l,2,...,m)
h,
all j
(j=l,2,...,n).
and
Define +
where
(k.)
(k.)
1
1
H A
i =
l,2,...,m,
M
j=l
and (h.)
(h.) ,
137
where
j = l,2,...,n,
A
*
*
0
T0
*
0
0
T
r
and let
be the Apostol's triangular representation (3.8) of T2. Then a(M+)
u
a(M) c
e(T) c azre(T2) and Propositions 4.29 and 4.26 imply that we can find K2 K(KJ, such that (with respect to the same de composition as for T2),
< c/4
11K211
and
*
*
*
Br *
*
*
*
*
M+ *
T+K
T=T+K= 22 3
*
*
0
T
0
0
0
*
o
=00
T
0
0
0
B&*
0
0
0
0
M+
o
*
*
=0
B
0
0
*
o
M
M
where *
Br *
B0=
T0*
0
0
*
B
and
0
are smooth operators. M According to (3.2), we have 0 Br
and
S
* +
(k.) 1
0
T3
0
0
* *
*
*
*
*
B0
*
*
O
0
0
o
0
0
(h.)
*
0
B0
*
0
0
i=l
R)
=
R
0
0
S_ (6.3)
*
1
follows from our construction that
T3IH(c0(T1);T3) =TIH(00(T1);T)
0
s_
(k.)
It
*
0
where
S÷ T12 T13 =
(h.)
j=l 0
(T
3
)
=
C1, B01H(cj0(T1);B0)
(T 01
)
(B 00
)
=
o(K),
C1 and
C1. Moreover, T3 is a compact pertubation of T1, so that and ind(A—T3) =ind(A—T1) for all A ps_F(T3). Assume that A
and ind(A-T3)
and A-S_ is invertible, so that
is finite; then
=
and (A—S_) are invertible in
the Calkin algebra. It follows that (A—n) must be invertible too, i.e., (A-R) is a Fredhoim operator; moreover,
ind(A-R)=ind(A-T3)
If In 138
ind(A-T3)
this case,
then
- ind(A-S÷) - ind(A-S_) =-= and
A-S_ is invertible.
so that A
On the
Oire(Ti)
other hand, A Since A-S
and therefore A
is invertible and A
are(R)•
ma 3.43) that A
by construction.
(kt))we conclude (Lern
e
Hence A
then
Similarly, if A
we conclude that either ind(A-T3)
is finite and A—R is a Fredholin operator of index 0, or and A
e
Hence a(R) is the disjoint union of
=ciO(Tl),[a(R)npS_F(R)]\ ind(A—T3) is finite) and
= {A =
'I'u{A
ind(A—T3)=±oo}, R
where c(R1) =
ci(R)\a0(R),
and ind(A—R1) =0
Oe
for all A By corollary 5.11, we can find an algebraic operator B1 such that
c(H1) =Oe(Hi) c
'V
and
€14.
1R1—H1l1 <
A fortiori,
C1 R12 H
0
is algebraic, 1
does not intersect o(S+)uO(S_) and
o(H) [ci
R12] liRi—Hill <
€74.
Furthermore, by using our remarks at the beginning of Section 5.1. 2, we can find an operator A similar to a normal operator with finite spectrum, such that a(A) c
= 0) and
hR—All
(so that a(A)n[o(S+)
u
< €74.
Set T13
S=
0
A T23.
0
0
S_
Then IT— IT— T1h1+h1K111+11K211+IIR— All < c. It is completely apparent that S is a simple model which satisfies all our requirements. 0
REMARK 6.3. It follows from (6.3) and the properties of the opera tor R that if I is an arbitrary perfect subset of 'V, then R can beuni formly approximated by operators similar to C1,N, where N is a normal operator such that c(N)=Z (use Theorem 5.8). This indicates that the central piece A of the model S can be replaced by many other operators (for the purposes of approximation; see also corollary 6.2 and remarks following it).
139
6.2 Quasitriangular operators
It is completely apparent that an operator A is triangular (see (3.6)) if and only if there exists an increasing sequence such that
finite rank
1
(strongly, as n
of no)
and APE-
= 0 for all n= 1,2 - Pn)APn This formulation suggests an asymptotic generalization. An opera-
tor A is quasitriangular if there exists an increasing seauence n ÷ no) and of finite rank projections such that such that
+ 0
of
The set PF(H)
(n + øo).
all finite rank orthogonal projections, ordered
by range inclusion, is a directed set. Thus, given T in L(H), P
11(1-
P)TPII is a net on that directed set and it makes sense to define q(T)
no lim inf(P
1)
(6.4)
(where P runs over PF(K); q(T) is the modulus of quasitriangularity). (or (QT) (K)) denote the class of all quasitriangular op-
Let (QT)
erators. It will be shown that q(T) is equal to the distance from T to the set (QT), so that an operator A is
if and only if
(QT) admits many other different characterizations,
q(A) = 0. In fact,
the most important one being the characterization in terms of the different parts of the spectrum. This can be summarized as follows: THEOREM 6.4. The following are equivalent for A (i)
(ii)
(Vi) angular,
(vii)
+ 1
(strongly) and
_Pn)TPrJI + 0, as n +
q(A) = 0.
(iii)
(v)
L(H).
There exists a (not necessarily increasing) sequence
in PF(K) such that (iv)
E
A is quasitriangular.
A
is the limit of a sequence of triangular operators.
A = T+K, where T is triangular and K
Given
is
E
is compact.
0, A can be written as A = T +K • where T
>
compact and
<
—
P;_F(A) =
COROLLARY 6.5.
(QT) is a closed subset of L(K), invariant under
similarities and under compact perturbations. Furthermore, if and
is tn
c.
p is a polynomial, then p(A)
A
(QT)
(QT).
PROOF. Every triangular operator is obviously quasitriangular. Now the equivalence between (i) and (iv) indicates that (QT) is triangular}
{T
L(U)
:
is a closed set.
The equivalence between (i) and (v) makes it completely apparent that (QT)+K(I1) 140
(QT). On the other hand, since every operator similar
T
to a triangular operator is also triangular (to see this, we can use,e. g., the existence of a suitable increasing sequence of finite dimensional invariant subspaces),
it readily follows that if A has the form
of (v) and W is invertible, then WAW1=WTW1+WKW1, where WTW1 istri angular and WKW1
Finally, gular and C
K(H).
observe that p(A) =
p(T+K) =p(T)+C, where p(T) is trian-
K(KJ.
REMARK 6.6. The proof of Corollary 6.5 only depends on the equiva lence between (i),
(iv) and (v) of Theorem 6.4.
6.2.1 Equivalence between the formal and the relaxed definitions of quasitriangularity It is trivial that the definition of guasitriangularity implies the weakened form (ii) of Theorem 6.4. On the other hand, if q(A) = 0
and {e}1 is an ONB of H, then there exists P1 C PF(H) such that ran P1 and ll(l—P1)AP1II < 1. Assume that we have used the conditionq(A) in PF{H) such P2 ... < = 0 inductively to obtain a sequence P1 ran and 1/j for j=l,2,...,n. Then we can that e. e
use the same condition in order to find 1'n+l c PF(H) such that < l/(n+l). Hence, the condition and 11(1-P en+l c ran +111 q(A) =
0
(QT).
implies that A
Assume that
(ii) =>
0
(n ÷ =)
in PF(H) such that
quence
Let
0
for
some seP
PF(H).
be given and let Rn be the orthogonal projection onto
<
P <
and since
+ 0
(n ÷
=) for
all X C
there exists n0=n0k) such that IIPnX_ xl! (C/4)IIxIi for all x n0. Let n n0 and let y ran P and all n Pn(ran IIYII = 1, Y = U,
P
ran
x, X
(l—c)
-l
,
P. Then lxii
lix
nXll+IIPnXII
=
< lIPn
<
PxIl
xiftliPli.ll
ran Furthermore, if y n0' iiRnY_PYIHIPnY_YIi
for n
p, ilyli
<
land
over all Ilyll
(l+IIPII)
/4)llxll <
= 1,
<
= 1, then Py=y and Ry=P Y. Hence,
C
for
all y
(ran P)
P),I!yii
[(ran
F) VP (ran = (ran Taking the supremum (for each fixed n
all n > no. If y
then
<
c. Combining this with the previous
statement we have that liRny_Pyli =
(E/4)Iixli+l so that lxii
and
we obtain
Define Rn so that ran Pn=Pn(ran
<
no) of
Thus uRn_Pu ÷ 0. be the projec—
and let
141
PFUI), L Then Ln P), it follows that
tion onto ran jection onto
Thus, since
0
and, since Rn is the pro-
P
0
=
(n
0, we obtain
and IILn_PnII +
0.
Hence, q(A) =
(iii)
(iv) =>
is a Cauchy sequence of quasi
Suppose that
triangular operators and that IA-
0
such that R
c/2 and that
(iv)
(I)
and (vi) => (v) =>
(i) =>(iii) , we
n0(e) so that
and Il—A Rh
P
Il(l—R)Ano RhI+!j(l—R) (A—Ano )RII
(vi) =>
Since
oo).
(n
see that given c > 0 and P e PF(H), we can find n0 =
It follows
<
IA—A
n0 II-4-II(l—R)A no Ru
<
The first two implications are
trivial. On the other hand, it is completely apparent that if K
K
(H),
then
lim sup(P =
urn
1) II(l—P)KII =
lirn
sup(P
1)
sup(P
1)
(6.5)
o.
II(l—P)KPII =
is the increasing sequence Thus, if A has the form (v) and in PHI-I) naturally associated with the triangular matrix representation of T, so that + 1 (strongly) and (l_Pn)TPn=O for all n=l,2,..., then urn
inf(n
Hence, A
6.2.2 Two
LEMMA
lower
6.7.
=)
lI(l_Pn)APnII=lim
inf(n
co)
I(1_Pn)KPnII
El
estimates for the distance to (QT)
For each operator A
q(A)
ly) such that
there is a sequence {Rn}n_u
LU-I)
and Rn+kARn
oo)
(n
be an ONB of U and let
PROOF. Let (II)
so that e1 e ran P1 and hI(1—P1)AP1II PF(U) so that ran P <
creasing and
lirn
1. Since q(A)
(ii
0. Choose P1 e
PF(U)
and
inf(n
oo)
is in-
Pn)
nTT'
+ °°). Furthermore,
it
read-
since ran
i€ follows that
for n
Define R1=P1, and therefore the equations
142
2.
0 for all k
Choose recursively
<
orthogonal finite rank projections,
k2.
>
1
Rn
Dran Pn_lVran It is completely aPParent that
ily follows that tI(l_Pn)APnII + q(A)
ran
0.
(QT).
of pairwise orthogonal finite rank projections with
P
=
hR +1AR
II
+ q(A)
2. Then the Rn'S
are
pairwise
Rn=l (strongly) and Rn+lARn= (n
=). Finally,
observe that
(n=l,2,...) imply that Rn+kARflO when 0
PROPOSITION 6.8. For each A L(K), dist[A,(QT)] = q(A). Furthermore, there exists T c (QT) such that IIA—Til = dist[A,(QT)]. PROOF. If q(A) = 0, then choose T=A. Assume that q(A) > 0 and choose
as in the above lemma.
Define
if
,
11o
=
S
t"n+1
R
fl'1
n = 1,2,...,
,
1
and let
Observe that Sn + 1
=) and
(n
q(A) for
all n. Now define
(strong sum), T = A—B.
B =
Since the Rn's are pairwise orthogonal, we can write
n
Rn; therefore IIB*BII =
l}
q(A)2, so that
q(A). Rk. It is immediate that
Let that
Hence T
0(n + co).
=
=
(QT) and hA-
= IIBU
q(A). Thus, in order to complete
the proof, it is enough to show that IIA-XlI
q(A) for all X
(QT).
Assume that IIA-XII < q(A); then
urn inf(P ÷ 1)
urn inf(P
hI(1—P)APII
< jjPi—xII+lim inf(P
so that q(X)
q(A)
1)II(l—P)XPII >
0, proving that X fails to be quasitrian
gular.
0
PROPOSITION 6.9. If A
L(H)
increasing sequence
-*
in PF(H) 8uch that
0, then there exists a compact operator I(2=Q and A-Ks is triangular.
and
>
0
o.) for some
(n
+ 1 (strongly, a €,8uch that IIK€hl
PROOF. Start with the sequence {Rn}ni of Lemma 6.7. Since g(A) = 0, we shall have Since < c/2n for all n = 1,2 is a finite rank operator, we can define K
=
KUI).
(The sum converges in the norm.) The product of any two sunmiands in the
definition of
is 0; therefore K2=0. The orthogonality of the Rn's
implies that
K*K
=
n:l
ran
and therefore
sup{IIR2A*R2+uhh.11R2+1AR2II:
n
1}
<
2
143
Hence,
<
To see that
is actually triangular, observe that (with so that = O•
0
The results of Section 6.2.1 and Proposition 6.9 imply that (i) =>
(vi)
and therefore, that any two of the conditions (i) through (vi) of
Theorem 6.4 are equivalent. This suffices, in particular (as observed in Remark 6.6), to complete the proof of Corollary 6.5.
L(tI), define
Given T
minimum modulus
the minimum modulus m(T)
(the essential
of T by
me(T))
m(T) =min{A T*T)½)} =
ae
(me(T)
(6.6)
min{X
Clearly, me(T+K)=me(T) for all K K{H). By using the spectral and T*T and the Weyl—von Neumann theorem, it is
decompositions of
not difficult to prove the following
L(H); then
PROPOSITION 6.10. Let T
If
(i)
sup{m(T+K):
me(T) =
(ii)
operator of positive index, then
T is not a
urn
inf(n
÷
oo)
K(H)} = max{m(T+K):
K
K(H)}.
me(T) for any ON sequence
ITXnII
H and there exists an ON sequence 00)
K
such
of
that me(T) = lim inf(n
IITYnII
(iii)
If E(.) is the spectral measure of H = (T*T)½, then the
smallest non-negative a such that rankE((a—c,a+c)) is infinite, for every c
>
0, i8 a = me(T). Given > 0 and p >
(iv)
1
such that nul(H+Kc_me(T)) = is (right) invertible if and only if me(T) (v)
left
>
<
if
that
is Fredholm case, me(T) me(T*).
(vi)
ran T is
closed and nul
(vii)
T
0
(me(T*)
(me
>
00
(viii)
If
T1
is
another operator, then
1TT111.
Ime(T) _me(Ti)l < 11TT111 me(A — T)
is continuous for
me(Ai_T)_me(X2_T)I
144
0
(ran T* is closed and nul T*
T <
0, resp.). if and only if me(T) > 0 and me(T*)
and only if me(T) >
(ix)
>
re8p.).
(T*)
0,
c,
there exists K
moreover, xl
A2!.
>
0.
In
We have the following COROLLARY 6.11. Let A C Luff) and assume that O;_F(A)
disttA,(QT)]
0;
then
A
A p;_F(A). By Proposition 6.10(i) there exists K(H) such that me(A_A) =m(A+KA_X). By Lemma 3.39, the distance from A+ KA to the set of all triangular operators cannot be smaller than m(A+
KA_A). On the other hand, by Corollary 6.5,
(QT) is invariant under
compact perturbations. Since the triangular operators are dense in (QT)
(equivalence between (i) and (iv) of Theorem 6.4) and the above observations hold for all A in p_F(A), we conclude that
distlA,(QT)]
A
3p_F(A)
A
and (by Proposi-
c
tion 6.l0(ix)) me(X_A) is a continuous function of A, the above supremum is actually attained for some A e By Proposition 6.l0(vi), this maximum is strictly positive. 0
6.2.3 Spectral characterization of quasitriangularity Corollary 6.11 implies, in particular, that Ps_F(A) =0 for all A (QT). Conversely, if Ps_F(A) =0 and c > 0, then according to Theorem 6.1 there exists a simple model
c
K1
R
B
0
S_K2
where R is algebraic and S_ has the form of (6.1) (iv), such that
< c.
then S_ is absent and
(If
R is obviously tri-
angular with respect to some ONB of K1.
On the other hand, if
is an analytic Cauchy domain, then either
by a direct argument (based on Runge's approximation theorem [lO2,p. 281) or by using Vitushkin's theorem [l02,p.207], we can easily see is uniformly A / Since dense in the space of all continuous functions defined on
that the linear span of the functions the mapping A +
is a norm-continuous mapping from
then
denumerable dense subset of
=
onto
where P_ denotes the orthogonal projection of l])
=AP_ER—A)
into
is an arbitrary
it is not difficult to deduce that if
it
follows that
readily e (P_ER—A) is triangular with respect to the Gram—Schmidt or-
thonormalization
of of the sequence
Since
Now it is easily seen that if
for all A —l
=
is a denumerable dense sub— 145
is a denumerable dense subset of ker(S-pk)
and
set of
(k=l,2,...), then S_ is triangular orthonormalization of the family
respect to the Gram—Schmidt
is triangular. Since a(R)na(S_) =0,
Hence,
is also triangular. By taking
S
1/n,..., we conclude that A is the limit of a sequence of triangular operators, i.e., A satisfies (iv). The proof of Theorem 6.4 is now complete.
U
6.3 Biauasitriangular operators Let (QT)* =
Lilt):
e
T*
(QT)}. An operator A
LUll is biquasi-
triangular if both A and A* are quasitriangular. Hence, the class (BQT) of all biquasitriangular operators is equal to the set (QT)n(QT)t
6.3.1 Block-diagonal and auasidiagonal operators An operator Be Lilt)
is
block-diagonal if there exists an increas-
(strongly) and PnB=BPn
ing sequence
in PF(H) such that
for all n1,2
Clearly, this is equivalent to saying that B =
-*
1
=B!ran(Pn_Pn...1) (P0 =0; n = (Pn_Pn (It is obvious that the Ba's act on finite dimensional spaces.)
where
•fl=l
1,2,...).
as above
An operator A is quasidiagonal if there exists such that
A-AP n
n
-*
0
1
(n
Let (QD) and (BD) denote the classes of all quasidiagonal and all block-diagonal operators, respectively. The main properties of the qua sidiagonal operators will be summarized in the following three theo-rems.
(Their proofs are either very simple or minor modifications of
the proofs of the analogous statements in Theorem 6.4 and wi]1 be omitted.)
THEOREM 6.12. The following conditions are equivalent for A
(i)
L(lt):
A i8 quasidiagonai;
There exists a (not necessarily increasing) sequence 0 as n ÷ 1 (strongly) and PF(I1) such that (ii)
-in
(iii)
The modulus of quasidiagonality qd(A) =lim inf(P
where P runs over PF(H), (iv)
is
1) flPA—APII,
(6.7)
equal to 0;
A is the limit of a sequence of block-diagonal operators;
(v) AB+K, where B is block-diagonal and K is compact; (vi)
146
Given c
>
0, A can be written as
where
is
is compact and UKU <
block-diagonal,
THEOREM 6.13. For each A
LU-I),
maxfq(A),q(A*)}
q(A)
Moreover, there exists C THEOREM 6.14.
qd(A) =dist[A,(QD)].
(QD) such that
(QD) is a closed subset of LU-I),
invariant
under
unitary equivalence and under compact perturbations. Furthermore, if A c (QD). In particular, Not(H) + Kill)
E(QD), then C*(A)
c (QD).
But (as 'roven in [188,p.14,Example 1.31; see also [149],[150]) the classes (QD) and (BD) are not invariant under similarities and this makes it impossible to give a spectral characterization of (QD) (analogous to Theorem 6.4(vii)). As we shall see immediately, such a characterization is actually possible for the larger class of all bi— quasitriangular operators. (It is completely apparent, from the definic (BQT).)
tion, that (QD)
6.3.2 Characterizations of biguasitriangularity
THEOREM 6.15. The following conditions are equivalente for A (i) A is biquasitriangular; There exists a (not necessarily increasing) sequence
(ii)
in PF(K) such that
+ 1
+ 0 and
(strongly),
0, as n
—
Given
(iii) in PF(I1)
L(tl):
such
>
that
0,
there exists an increasing sequence
+
(strongly),
<
and
< c/4fl for all n=1,2 ,q(A*) }
(iv) (v)
= 0.
A
(vi)
A A
{R
L(I-I):
R is similar to a normal operator with fi-
nite spectrum). (viii) (ix) (x)
(xi)
(xii) (xiii)
(xiv)
A c {R
A
{R
A c {R
A
A
LIH): LU-I):
LU-fl:
(R e {R
LU-fl:
L(fi):
R R
N, N is normal). B, B
lJok(f1)
+K(N)).
a(R) is totally dieconnected}. interior c(R) =0). R
C, C e (QD)}.
=P:_F(A) =0. (Staircase representation) Given c >0 there exists
c, and a family {Rn)ni c PFIN) of pairwise orthogonal projection6, such that A—Ks admits a matrix representation of the form 147
A1
Ck
D1A2E2
R3
C2
R4
0
D2A3
R5 (6.8)
AB n n C
n DA n n+1
0
where R
=
ran
2n-1
R
2n
R 2n-t-1
n = 1,2
There exists a decomposition U = H10H2
(xv)
R
into two infi-
of II
nite dimensional subspaces such that = [D1
A
D
(6.9)
22 such that
where DJ is a quasidiagonal operator acting on j
= 1,2,
(xvi)
and K is compact.
Given c
>
0
1
0
al
KU-I),
there exists K
B*K where
=
B
such that
<
1
1610
22
is a block-diagonal operator acting on the infinite dimension
subspace H). such that Ce(Bj) = Given e > 0 there exists K (xvii) NOD1 0
A—Ks =
*
NOD2
0
0
0
0
j
*
*
*
*
NOD3 0
*
ae(Dj)
= c(A),
IIKH
<
c, such that
M1
M3'
NOD4M4
where N is a normal operator and the D.'s such that c(N) = ae(N) =
= 1,2.
K(H),
are
block-diagonal operators
j = 1,2, 3,4.
PROPOSITION 6.16. (BaT) is a closed subset of LU-I),
invariant
un-
der compact perturbations and under similarities. Furthermore, if A (BQT) and p is a polynomial, then p(A), p(A*) PROOF. 148
Since (BQT) =
e
(BQT).
and (QT) and (QT)* are closed, in
variant under compact perturbations and invariant under similarities (Corollary 6.5), (BQT) has the same properties. The second statement
follows from Corollary 6.5 as well.
U
PROOF OF THEOREH 6.15. The equivalence between any two of the con ditions (i), rem
(ii),
(iii) ,
(iv)
or (xiii) follows immediately from Theo
6.4. Let (BQT),
...
tors described in (v),
(vi),
denote the families of opera-
, ...
,
(xii),
respectively. By Corollary
5.10 and the equivalence between (i) and (xiii), we have
(BQT)..= (BOT). Since I'Jcn.(H)
{N
o(N)
is finite) c
we easily see that
c
(BQT)ViIi
=
c
(BQT)iX
=
=
(QD)
c (BQT)
(BQT).
Similarly, since (by Corollary 1.14(i) and Theorem 1.13)
=
L(HJ:
c (BQT)x c (BQT). c {B
(BQT) =
pF(B)
0) = (BQT),
we see that (BQT)x =
(BQT).
=
(BQT).
Hence, any two of the conditions (i) (xvi) => (vx)
=>
(xiii)
(xiii)
are equivalent.
The first implication is trivial. On the
other hand, it is easily seen that if A satisfies (xiii), then A—K has the form (6.10) for some K in K(H). Since (BQT)
is invariant under com
pact perturbations (Proposition 6.16), it suffices to show that every operator of the form (6.10) belongs to (BQT). Since (BD) c (QD) c (BQT), it readily follows from the equivál.ence between (i) and (xiii) that oe(Bj) and =0 for all A
Ps_F(Bj)s j=l,2. A fortiori,
= 0
for all A
and ind(A—K—X)
Hence A-Ks satisfies (xiii). Since Ke is compact, it is completely apparent that A also satisfies (xiii). Assume that A-K admits the representation (6.8) (xiv) => (i)
K(I()); then
(for some
PF(H), P
and
(since K
(iii) => K
Since
+ 1
(strongly, as n
(l—P2) = 0, so that A—K are triangular operators. Hence A-K (BQT). A fortiori is compact), A (BQT). = 0
+
(xiv)
and
Define P1=P0=0 and }.
=
(l_P2n_i)A(P2n.i_P2n3) and (P2
—P2
2)A(l—P2) are
(6.11)
finite rank 149
operators and '
<
it
is easily seen that the series (6.11) converges in the norm, K and lIKE11 < 2 <
K(H)
It
is straightforward to check that A—K admits the matrix repre
sentatiori (6.8.). (Roughly:
A— (l—P1)A(P1-P1) has 0's everywhere in the first column, except for the term A1; A— (l—P1)A(P1-P — (P2—P0)A (l-P2) has 0's everywhere in the first two rows except for the terms A1, B1, C1, etc.) (i) => (xvi) Since A (BaT), it follows that oire(A) = Proposition 4.29, there exists S A, A—S E K(H), 11A <
Ni *
*
B
*
0
Nr
0
By
c/2
and
A and Ni and N are unitarily equivalent diagonal normal operators of uniform infinite multiplicity such that O(Ni) =
where A'
o(N)=
Assume that Ni
=
Nr =
(diag{A1,X2,
..
.})
Let
.
be an ONE of H.
Now we proceed essentially as in the proof of Theorem 6.4(iii) > we find P1 PF(K) such that ran P1 contains e1 and a vecker(N —A1), and PF(H) I(1—P1)SP1II < c/5. Now we find P2 tor 4 such that ran P2 contains ran ]?1, e2 and g2 j ran
(i). First
P1 (j
= 1,2),
and 11P23(1—P2)ll
Assume that P
1
P
2—<
...
PF(H) such that ran
< c/5. < p 2n have been chosen. Now we choose contains ran
ker(N -A.) and
where
(H)
such that ran
j
ran
then we choose
<
2,...,n+l), and
e2+l and vectors =
1,
c PF
contains ran
2
and
and
lIP2
+2)11
<
El5
By induction, we have constructed a sequence {P}1 satisfying
(iii) with A replaced by S and c replaced by c/2. Define P1=P0=0 and let
Cc =
It follows from the previous step (see (6.11)) that C
c/2,
K
=
(S—A)+C
c K(H),
IIKII
<
c
and A--K
admits a stair-
case representation of the form (6.8). Furthermore, by construction, 150
An is an eigenvalue of Ak and an eigenvalue of Ck for infinitely many
values of k, for each n=1,2 . More precisely, c
O&re(Ok'°l
A
(6.12)
Ck).
Now observe that
B1 0
A1
0
A2 D1 10
0
0
.
.
B2 10
0
.
.
I
0
0
A3 D2101
.
0
.
R1
R2
B.
A—Ks =
(6.13)
0
A4D3 C3 0
Since the column of A n contains A n in the main diagonal and Oss in any other entry, it is easily seen that
0,
A1
A2
A3
D1 B2
R3
D2B3
0
0
R5
:
A-K
-
— —
—
——
...
C1
R2
• n=1
I
I
A
B
n
—
•n=1
0
C2
II
CU n
1 2
0 0
where
K
n=l
1
R
2n—l
and II 2 = 0nl
Clearly, Xi=0n1
R
2n
and X2=01
are block—diagonal operators
and K2, respectivley) and Oe
(acting. on
c
ae(A)
Combining this observations with (6.12) we conclude that
(i)
=>
(xvii)
Let K, X1=01 An and X2=0
be the opera-
tors constructed in the previous step. Recall that An
for
all k > n.
Thus, if k
n we have
-
151
A
=
A2k_,OA2k
*
n
" n,k
0
Enk 0
0 0
0
*
M
R2k_leMfl,k = 0
A2k R2k
F
n,k
n,k
Wnk
where Mflk is one-dimensional, Fflk= Enk$A2k R2k)eMfl,k and
The same argument, applied to (A.,klOA2k)* shows that A
2k—i
•A
G
—
2k —
n,k
*
0
An
n,k
is one-dimensional, Gnk
where
(k n). cp(Gn Since we have infinitely many possible choices of k for each n,
and
we can easily see that ]S[Ok:l(A4k_2OA4k)]
An =
nl
G
*
n
0
=
*
N
No0;lFm'
where N= (diag{A1,A2,. ..})
the operators Gn have the form Gn,k for
a suitable subsequence
and the operators
F
for a suitable subsequence {k
m,k
m } m=l
have the form
so that 0
,
n=l
G
n
F and 0 m=l m
cp(Gn)ncp(Fm) for all n
are block—diagonal operators. Moreover, A
and m large enough (depending on r), whence we easily conclude that
ce(Onl Gn)
Fm)
Define D1 =0 n=l D1
G
N
2
0m=l F m ;
N
0
0
0
D1
n,
—
D
10 NJ 0D20 0
0
—
0 NO
—
0
then *
NOD1 0
N0D2
D2
0
Similarly, we can obtain *
NOD3 0
NOD4
and, a fortiori, NOD1 0
A - K
=
0
X2
(xvii) =>
152
NOD2
0
0
0
0
(xiii)
c0(A)
*
* *
NOD3 0
* * *
NOD4
Observe that ceDj)
GLre(A)g j
= 1,2,3,4.
re0Dj)
=
Now the result follows exact-
(xiii).
ly as in (xv) =>
The proof of Theorem 6.15 is
now.
6.4 On the relative size of the sets (QT),
El
(QT)*,
(BQT),
[N(f4)+K(lf)
and NIH)
THEOREM 6.17. Each of the following inclusions i8 [U(H)+K(K)]
N(H)
(BQT)
c
(QT)
c
proper:
L(Ifl.
Moreover, each of these subsets (except LIlt)) is nowhere dense in the next one.
Furthermore, the same is true for the chain of inclusions (BQT)
[lJ(l-lJ+K(l1)]
c
LIH)
and (QT)n(QT)* is nowhere dense in LIlt).
PROOF. Consider the first chain of inclusions. Since each set in the chain is closed, in order to prove that a set is nowhere dense in the next one it suffices to show that its complement is dense in the containing set.
Observe that "T / (QT)" is a "bad property" (as defined in Section 3.5.2) and the unilateral shift is a concrete example of an (i)
non-quasitriangular operator. By Theorem 3.51 L(fl)\(QT) is dense in LIlt)
Let T
cii)
(BQT)
and
>
0. By Proposition 4.29, there exists
(BQT), IIT—T111 < €12 such that T1
for some x
0 T2
=
where o(T2) =c(T).(Take r,={X},
c
rr=Ø.)
Proceeding as in the proof of Theorem 3.51, we can find A+fl+6S*
T
where
0
=
*
T2
u
is a backward shift of multiplicity one, such that
and
=0, is dense in (QT).
T
(iii)
Let T
[W(H)+K(HJ )
and
>
0.
If
then
ae(T)
0 and it follows from Theorem 5.34 that there exists Q such that IT—
<
fore (BQT)\[N(H)+K(H) 1 (iv)
If T
NIH),
Clearly,
(BQT)\[N(K)+K(ll)
j
and there-
is dense in (BQT). 0
and
,
then we can use 153
the argument of the first part of the proof of Theorem 3.49 inorder to obtain a finite rank operator F (T+F such that hF C < 0 C c 0. It is easily seen that T+FC e IU(fi)i-K(K)]\N(tf), so that LN(tf)+
Ktff)]\N(K)
is dense in IN(K)+K(I1)].
By taking adjoints, we conclude that (BQT) is nowhere dense in (QT)* and
nowhere dense in L(fj]
.
Hence, both (QT) and (QT)*
are nowhere dense in L(f-() . A fortiori, so is their union.
0
REMARK 6.18. Theorem 6.17 indicates that NIH), [N(H)+KU-() J, (BQT), (QT), (QT)* and (QT)u(QT)* are "very small" subsets of L(H).How ever (as proved in Theorem 5.15), even NIH) is large enough to guaran tee that L(H)
= N(H)+N(K).
On the other hand, we also have
(L(H)\[(QT)u(QT)*]}+{L(H)\[(QT)u(QT)*]} =
L(H).
Indeed, by Proposition 4.21(u), given T such that
(A, B acting on
L(H) there
infinite dimensional spaces).
is the unilateral shift of multiplicity one and r > T1+T2, where
T1=
and
exists K
211Th!,
then
T2= (_2r_rS)$(2r+rS*)
It can be easily checked (use Theorem 1.13(iii) and (v)
)
that
T1+2r and T2-2r are Fredholm operators of
1, and T1-2r and T2+2r L(H)\[(QT)u(QT)*j. are Fredhoim operators of index -1. Hence, T1, T2
6.5 A Riesz decomposition theorem for operators with disconnected sential spectrum The classical Riesz decomposition theorem asserts that if 0(T) is the disjoint union of two nonempty clopen subsets 01 and 02? then the space can be written as the algebraic direct sum K = K14-K2 of two invariant subspaces ofT, K2, such that o(TIH.) j=l,2. (See Chapter I.) This result is false, in general, if we merely assume that the essential spectrum splits into two clopen subsets. Namely, if T = where
is
the annulus
position is impossible:
If H =
under T, Oe(TIK1) ={A:
Al
(A—TIH1) =0 for all x
in
= l}
U:
1 <
<
3}, then such a decorn
where K1 and H2 are invariant ={A: Al = 3}, then md
and ae(TIH2)
and therefore ind(A-TIH2) =-1 for all A in
Since 0(TIH2)nD(O,l)0, we conclude that ind(X—TIH2) =—l for all A in D(0,3). Therefore, ind(A—TIK1) =ind(A—T) —ind(A—TIH2) =+1 for all A
154
D(0,l) and this implies that D(0,l) c
a contradiction.
A more careful analysis indicates that two nonempty clopen subsets
splits into
does not have two
but
and
and H2, such that and oe(M+(3 +AIH2) for any A L(H) such that hAll < 1. How= ever, it readily follows from the BDF theorem that is unitarily coMplementary invariant subspaces,
equivalent to a compact perturbation of 3SWS*, where S denotes the uni
Al = 3} and
lateral shift of multiplicity one. Clearly, cYe(38) = oe(S*) ={A:
IX)
=l}.
Consider the bilateral shift B defined by Be =e n+l for n < 0 and n Ben =2 ne n+l for n > 0 with respect to an ONB {e n }—oo(fl<= ; then a(B) = D(0,l), Ge(B) ={O}uaD(O,l) and ind(A—B) =nul(X—B) =1 for all X€D(0,]) \[0). Either by general results about the structure of bilateraiweight ed shifts (see, e.g., [103], [131], [184]) or by a direct argument, it can be shown that B does not have any pair of complementary invariant subspaces. However, if Bm is the result of replacing the rn—th weight by a 0
(i.e., mB =B_2me m+1 øem
for some m , 0, then B
)
m
admits two corn
plornentary reducing subspaces H1= V{en}n
—
S*!) quasinhlpotent andB These two examples illustrate both the possibility of restoring the validity of Riesz decomposition theorem for some small compact per turbation of T (where T
L(14)
has a disconnected essential spectrum)
and the 'index obstructions" that make it an impossible attempt.
PROPOSITION 6.19. Let T
LIII) and let r be a clopen nonempty sub
set of aire(T) contained in a unique component = > and ind(A
inf{ind(A—T):
suchthat
of = n > 0
A
—T)=n for every X in a component
of
(6.14)
whose boundary
intersects
Given c
>
0, there exists K T - K
K(H),
IlKhl
<
such that
A$R,
= r, a(R) =ru{x e and nul(A—A)=nand nul(A_A)*=0 for all A
where R is a smooth operator,
ind(A—T)
>n}, ce(A)nQ =
0
C
PROOF.
Proceeding
as in the proof of Theorem 4.11, we can find Ce
K(H) such that
T-C = where V
0 B
(interior
a(B) =rua0,
a&re(m =rua1s 155
and
ind(X—T)
(A
>
n} (Obviously, ind(A—B) =
ind(A—T) —n, for all A It readily follows that B is the algebraic (not necessarily ortho gonal) dfrect sum
of
an operator G whose spectrum does not intersect and an operator J' such thatclL Further(J') =r and a(J') because ind(A—J')
(QT)
more, .1'
0
for all A
(Use Theo-
6
rem 6.4.) Hence,
* *U
tl
1
2,
T-C= OG*H2= 0 0 J
J' and
v* =
L(H0),
G
0
00 03 C30 C33
C
H
0
then *
T—
C30
—
J+C33
(QT) and C33 is compact, J+C33 also belongs to (QT) and
Since J
therefore (by Theorem 6.4) J+C33=J0+C0, where creasing sequence in PF(113) such that oa)
and C
1
(strongly, as n
is compact. Furthermore, by Proposition 3.45 and its proof,
=0. Since C3
C0 can be chosen so that —
0
—
(l_Pn)C3o
K1(n)
is
for an in -*
+
compact and
0
is compact, the operator 0 113
(n ÷
Let A4(n) be the restriction of T-K1(n) to its invariant subspace 114(n) =H0eran
and let 14(n) be the compression of J0+CQ to the
=
subspace 11
Clearly, there exists n0 such that if K1=K1(n), A4=A4(n) and 14=14(n0), then flKJJj < e/2 and a(M) g2nr,2. By Proposition 3.45 and Theorem 3.48 there exist compact operators 1(4
c/2,
and
1(5 such that
A=A4—K4 has the desired properties and a0(M-K5) so that R=M—K5 also has the desired properties. 11K511
<
Thus, if
K
<
Ø,
and
A* T-K Since a(R) is contained in 156
=
0
R
and
-
c
it follows from Coro}
lary 3.22 that T-K REMARKS 6.20.
AIR.
(1)
0
By taking adjoints, it is easily seen that Pro
position 6.19 remains true if (6.14)
< sup{ind(A—T) and "c(R) =Fu{A [ZnpS_F(T): and nul(X_A)*=0 for all A ind(A--T)
is replaced by n
A
:
ind(A—T)
<
(6.15)
0
and nul(A—A) =n
>
is replaced by "o(R) =ru{A
0 and nul(A—A) =0 and nul(A_A)*=_n for all A
<
C
(ii)
An especially important case of this proposition is the one
corresponding to the case when r
is
a singleton (or, more generally, a
case will be analyzed with detail in
finite subset of Chapter VIII.
6.6 Notes and remarks
The concept of quasitriangularity plays a central role in the proofs of the Aronszajn—Smith theorem [36] on the existence of invariant subspaces for compact operators, and in the proofs of several gen eralizations [39], [51], [118]. However, this important concept stayed hidden for many years until P. R. Halmos explicitly stated it (and some
of its related properties) in his article [121]. The notion of quasitriangularity immediately drew the attention of many operator theorists who
used
it to extend the previously known results on the ex-
istence of invariant subspaces for operators "somehow compact operators" (see, e.g., [9], [70], [158],
related
to
[161], [192], [193])
or worked on the structure of the quasitriangular operators and its irn
plications. The most important result in this line is the spectral characterization of quasitriangularity (i.e., the equivalence between (i) and (vii) in Theorem 6.4). The equivalence between any two of the properties (i) -
(vi)
in
Theorem 6.4 is due to P. R. Halmos [l2lJ (see also [l23,Problem 4]), who also proved that (QT)
is closed, invariant under compact perturba-
tions and closed under the formation of polynomials. (The simplified given here is due to G. R. Luecke proof of the step (ii) =>
(iii)
[154].) That (QT) is also invariant under similarities is an observation of W. B. Arveson [12l,p.29l]; the first proof of this fact (very different from the present one) appears in [75].
In [121,Pheorem 3], Halmos proved that the unilateral shift of multiplicity one is not quasitriangular. The argument of the proof was 157
used by R. G. Douglas and C. M. Pearcy in (i)
[75,Theorem 1] to show that
(vii).
=>
The final step,
(vii) =>
(i)
proved to be the most difficult
In [75], Douglas and Pearcy proved that every operator with a finite spectrum is guasitriangular. Halmos extended this result to operators T with the property that
has analytic capacity zero [124]. (The
reader is referred to [102] for the definition of analytic capacity. In particular, if o(T) has analytic capacity zero, then a(T) is totally disconnected and has planar Lebesgue measure zero.) C. Apostol proved in [4] that every decomposable operator (in the sense of C. see [67])is quasitriangular, thus extending the previous resu]ts,
and introduced the notion of "modulus of quasitriangularity" (q(T) ).
In
[19] (see also [21]), C. Apostol, C.
and L. Szidó obtained
several partial results about non-quasitriangular operators. Finally, the goal was attained by C. Apostol, C. Foias and D. Voiculescu in a series of papers published in the Revue Roumaine de Ilathématiques Pures et Appliqueés:
[21], [22], [23], [24],
[25]. (The main result
was previously announced in [20J.) In fact, a first proof of the 1mph cation (vii) => (i) was obtained in [23] and a second one, including a stronger statement, was obtained in [25].
Simultaneously (and independently), L. A. Fialkow obtained several relevant results on the subject, including a proof of the equality q(T) =
dist[T,
(QT)].
A simplified proof of the main theorem of C. Apostol, C. Foias and D. Voiculescu was given by R. G. Douglas and C. Il. Pearcy in [74].
The four proofs of this result (i.e., the two obtained in [23] and [25], Douglas and Pearcy's version and the one given here) strongly depend on constructive techniques of approximation. The present proof (largely based on Voiculescu's theorem, which was unknown at the time of publication of the other three proofs) provides a more natural approach.
The relevance of the achievement of C. Apostol, C. Foias and D.
Voiculescu goes far from the significance of their main result. It is a) The introducworth mentioning, at least, two other consequences: tion of a large number of constructive techniques that have become
standard tools since then (including Theorem 3.49 of Chapter IIfl. b) From a heuristic viewpoint, the result strongly suggested that the only real obstructions for many problems of approximation are thos2 de-
rived from the stability of the semi-Fredholm operators and the upper semicontinuity of separate parts of the spectrum. (This is, precisely, the aspect of the Approximation of Operators that has been continuous— iy emphasized throughout this monograph.) 158
Finally, it is convenient to recall that the notion of quasitrian gularity also admits several other alternative definitions; namely, R. G. Douglas and C. M .Pearcy [751 have also shown THEOREM 6.21. (Addendum to Theorem 6.4) The following conditions are equivalent for A e
L(HJ
A is qua8itriangular.
(1)
(viii)
>0 andasequence
Thereexists
of finite rank op-
erators converging strongly to 1 such that each
÷ 0 (n + c on (ker F)1 and (ix) There exists a sequence
is bounded below by
of positive semi-definite
compact operators that converges strongly to 1 and satisfies -*
0
(n
Furthermore, if (FC÷,)
denotes the directed set of all positive
semi-definite finite rank contractions acting on H (partially ordered by the relation: H,
A1 < A2 if and only if
=lim inf(A (A
for all x
as usual) and
FC+),
(ii)
(iii)
II(l_A)½TA½11
then we have [2l,Section 3]
THEOREM 6.22. If T E (i)
1)
L(H),
q(T) =q (T+K) ½q5(T)
for all K
q5(T-I-A)
q(T) =lim(X
then
q(T) for all A
K(H).
a:.
co)
In [l92],[l93], D. Voiculescu analyzed certain notions of "extend ed quasitriangularity" in Hilbert spaces (where the role of K(H) is played by a Schatten p—class
On the other hand, P. Meyer—
Nieberg defined quasitriangularity in arbitrary Banach spaces (the definition coincides with the usual one in case the underlying Banach space io a Hilbert space) and proved several invariant subspace theorems in this new setting [158]. C. Apostol [7], [8] and C. Apostol and D. Voiculescu [33] analyzed the properties of this newly defined quasi triangularity. Among many other results, the last paper extends the Apostol—Foias-Voiculescu ' s spectral characterization of quasitriangu-
larity to the case when the Banach space is either C0
or
1
p
<
The introductory result of this chapter (Apostol-?lorrel simple models) is one of the most important "tools" of approximation. Theorem 6.1 has been proved by C. Apostol and B. B. Morrel in [27] where ithas been applied to obtain (among other results) a spectral characterization of the closure of the set of all operators with spectrum contained 159
(This result will be proved and anal-
in a fixed nonempty subset of yzed in the second monograph.)
The existence of quasitriangular approximants (an operator T in (QT)
L(H) if
is a quasitriangular approximant for A
(QT)])
IIA—Til
= dist[A,
is due to C. Apostol, C. Foias and D. Voiculescu 121,Theorem
2.2]. The proof presented here (Lemma 6.7 and Propositions 6.8 and 6.9)
is a refinement obtained by R. A. Smucker in his doctoral dissertation [188]. (Proposition 6.9 refines Halmos' result in [123,p.905].) The es sential minimum modulus me(.) has been analyzed by several authors; na mely, in [95] the main theorem states that seven conditions are equiva lent, and each of these conditions is equivalent to me(T) =0. Proposition 6.10 (essentially equal to [53,Theorem 2]) sums up the most vant properties of the essential minimum modulus and Corollary 6.11 is just a quantitative version of [75,Theorem 1].
Theorems 6.12 and 6.13 are due to P. R. Halmos [123,Problem 4]. Halmos' article and the doctoral dissertation of R. A. Smucker are the approxibasic references for quasidiagonality from the viewpoint of (The interested reader can also examine [149], [150], [155] mation and [l97,Chapter XIV].) In particular, Theorem 6.14 is also due to R. A. Smucker [188,Theorem 5.5]. The equivalence
between any two of the conditions (i) -
(xiii)
of
Theorem 6.15 is either an immediate consequence of the Apostol-FoiasVoiculescu result on spectral characterization of quasitriangularity or a consequence of Voiculescu's theorem on algebraic limits of algebraic operators (Theorem 5.9). The equivalences between (i) and any of the conditions (xv) or (xvi)
are due to C. Apostol and C. Foias [17].
The staircase representation (xiv)
is a result of C. Foias, C. M.
Pearcy and D. Voiculescu [98] and the representation of (xvii) is one of the steps of Voiculescu's proof of Theorem 5.9 [l94,Proposition 3.2].
C. Apostol, C. Foias and D. Voiculescu [2l,Theorem 2.8] proved that (QT)
is
nowhere dense in L(H)
(and, a fortiori, that (QT)u(QT)*
is nowhere dense). In [155], G. R. Luecke proved that each of the sets of the chain
Wok(ff)+K(tlj (except (QT))
C
(QD)
c
(QT)
is nowhere dense in the containing set and D. A. Herrero
E149,Theorem 3.1] proved that the same is true for the chain (QD)
c
(BQT) c
(QT).
Hence, Theorem 6.17 admits the following companion result. THEOREM 6.23. Each of the
160
inclusions is proper
Nonifi) c No/t(fl)+K(H)
(QD)
c
(BQT) c
(QT)
c
each of theee subsets (except L(I1))
(QT)u(QT)* cL(H),
is
nowhere dense in
the next one.
Finally, the results of Section 6.5 are due to J. BarrIa and D. A. Herrero [42].
161
7 The structure of a polynomially compact operator
An operator T is called polynomially compact if p(T)
K(H) for
some monic polynomial p. Every algebraic operator is polynomially compact (p(T) =0) and so is the sum of an algebraic operator and a compact one. Indeed, this is the most general example of a polynomially compact operator (Theorem 7.2 below).
The structure of these operators did not play any special role in the previous chapters. The intrinsic reason was that in those chapters we considered the set of all algebraic operators (or all
nilpotent
op-
erators) and the structure of the whole set "hides" the structure of many interesting subsets. In order to analyze several more specific approximation problems, we shall need a finer analysis of these operators. This is the content of the present chapter, as a preparation for the results of Chapter VIII on closures of similarity orbits.
7.1 Reduction to
(essentially) nilpotent case
The following auxiliary result will help to reduce many problems about polynomially compact operators to the case when the essential spectrum is the singleton {0}, i.e., to essentially nilpotent operators.
(T is essentially nilpotent if T is a nilpotent element of the
Calkin algebra.)
and assume that a(T) is the disjoint union of finitely many clopen subsets Let = 0m (j = l,2,...,m); then K = Ki+K2+•••+Km and A A., A. S(T) (i) {A e L(H): j= j=l If A S(T) and c(A) is the disjoint union of finitely man LEMMA 7.1. Let T
L(I1)
(ii)
y clopen sub8ets ..,m, then A (iii)
ci,...,
such that
for
each j
= 1,2,...
(j = 1,2,. ...m). where A. e S(TIH.) If T is algebraic with minimal monic polynomial p, p(A) = i
j),
then 0(T) is the disjoint union of
m clopen (finite or denumerable) subsets 162
D
0m
such that
= l,2,...,m.
Furthermore, if A oe(T) = {A.} for each j o(A) = ci(T) and ae(A) = oe(T) = • (iv) = o(T) for a11 A in .S(T), then
p
If S(T)= {A LU-I): A (v) If T is polynornially (a8 in (iii)) and TIH. 0
and K
j=
,
compact,
has minimal monic polynomial
where L.
(for each j = l,2,...,m),
K. =
A1,
then
S(T)
=
LUI1),
then T = L+K,
where L
0
and
L(H),p(L)
Kit-f).
PROOF. (i) Since is a finite family of pairwise disjoint compact sets, it follows from Corollary 3.22 that T where
L(M.), T,
T.
= l,2,...,m.
j
Till.,
(Observe that o(A)
.,, where 131,3—
W. .:M. + N.
ci..) Assume that W= (W. .).
cj(T)
(i,j
13 1 —' J 3 m an invertible operator such that then A.=W(e. T.)W A.) j=l j j=l j=lj j m P.). Proceeding exactly as in the proof of Corollary 3.22, w = ;
we conclude that W.
.
13
= j
=0 for all i
l,2,...,m. Therefore = l,2,...,m. (ii)
M.
and W.. is invertible for all j JJ ._i N. and A.=W..TW.. T. = Tjl1. for each j
3
:1
33
3
JJ
3
:i
Proceeding as in (i), we can write T
T.
j=l,2,...,m. A., where ci(T1) =a. c Clearly, it suffices to show that A1 is unitarily equivalent to an op-erator in S(T.) (for each j and A
= l,2,...,m).
N1, i,jl,2,...,m; n=l,2,...)and
If Wn= j=1
A.-W (e.m j
T.)W _lii + 0
n
n
=),
(n
then we can combine the argu-
ment of the proof of Corollary 3.22 (as in (i)) with a limit argument in order to show that
n'1 + 0
(n ÷ =) for all i
that we can directly assume that M. =N.) and IA. —W.. 3
3
(so
=
1.
T.W..111 + 0 J3,fl
3J,fl 3
3
(n
and cy(T)\cy (T) can on ly accumulate at whence we immediate deduce that is thedis joint union of m clopen subsets such that cYjncie(T) = 0m — (A.) (j=l,2,...,m). If A then A S(T) S(T) and therefore (by Proposition 1.7) =0. Hence ae(A) c ae(T) It follows that ci(A) is a finite or denumerable set and therefore it is a totally disconnected (iii)
Clearly, cie(T) =ci(T) —
,
set. By Corollary 1.6, cie(A) =ci(T) and ci(A) =q(T). (iv) (v)
where T.
This is a corollary of (ii).
Let W be an invertible operator such that Till.. Then there exists operators
=
T.,
L. and Kt = K. such 3
that
(j=l,2,...,m). Then T=L+K, where L=W
isfies the polynomial equation p(L) =0 and K=W
—l
m
—l
3m
K)W
sat Kill)
0 163
7.2 The structure of a polynomially compact operator THEOREM 7.2. Let T s m p,
and assume that p(T)
L(tl)
= 11j=1
0 for some poiy-
a compact operator
J. Then there
(A —
K such that p(T—K) =0.
LEHMA 7.3. Let P, R
be two orthogonal projections with corn
LIt-I)
pact product PR; then for every jection F <
>
R such that IP(R—F)II
0
there exists a finite rank pro-
<
be an increasing sequence of finite rank pro-
PROOF. Let
KIN),
jections converging strongly to R. Since PR =).
Thus,
7.4. Let H be a po8itive operator, 0
LEMMA
0
IPR(R_Fn)fl
for a sufficiently large n, IP(R_Fn)II =
(n
<
H
<
C
0 (where C i8 a
constant); then there exist an invertible operator W and a sequence
of
pairwise orthogonal projections such that WH =
(i)
HW
=
and
= WP
(ii)
(iii) (iv)
ran P H = m
1/C,
liwli
for all n =
=
ran 1m P 11w
ran < 2C.
H c
c
n=l
1,2,...;
ranP n
=
(ranH)(m=l,2,...):
PROOF. Let H=f A dE(A) be the spectral decompositionof H and set
(n=1,2,...). The sequence
H'
is pairwise orthogonal ,
for all n, and it is also clear that
=
satisfy (iii). Define W=f[oc]f(A) dE(A)
the
0,
1, if A = f(A) =
l/(2 A)
,
if n =
Then WH =
n=l,2,...,
HW
=
=
110,C] Af(A) dE(X) =
sup(A
[0,CJ) dE(A). Hence,
[0,C]) f(A)
(A
sup(A
COROLLARY 7.5. Let A, B
there exists a projection P
where
(2
A
—n
C,2
—(n—l)
C],
1,2 for all
1/C, W is bounded below by inf
invertible) and t('=f[0 [O,C]) f(A)1= 2C.
0
Then KIN) LIft) such that AP and (l—P)B are both
L(H)
and assume that AB
.
compact.
PROOF. Let A=13H1 and B*=VH2 be the polar decompositions of A is compact and this reduces our problem and B*. Then H1H2 to show that there exists a projection P such that H1P and (1-P)H2 are
both compact. Hence we can directly assume that A=H1 and B=H2 are positive seui—definite hermitian operators. 164
Two projections t4 and N will be constructed so that AN, NB, and (1-N) (1-N) are compact. Then (l—M)B will also be compact, since (l—N)B = (l—M)NB + (1—N) (l—N)B.
By applying Lemma 7.4 to A and B, we obtain invertible operators (comW, V and two families of pairwise orthogonal projections
muting with A) and {Nk}k:l (commuting with B) such that and
A'
Thus, we can write 2_m_nMmNn)Nk =
MJWABVNk
Therefore M.Nk is compact for each j,k1,2 k
Nj)Nk
Thus the product
is also compact. By applying Lemma 7.3 to this product, we Nk such that
obtain a finite rank projection Fk
<
N1) (Nk_Fk)II 2—k—j
<
In particular, IMj(Nk_Fk)II
k =
1,2
k, k=l,2
Similarly, there exist finite rank projections G. <
k
II(Mj_Gj)NkJI
gers k,
j,
j,
j=l,2
N. such that
Thus for all positive inte-
we have 2—k—j
<
(7.1)
I(M1—G1) (Nk_Fk)lI
L(K) by
Define the projections N, N
M=
Gj)+Pker A'
N=
Fk)+Pk
B
Then
AM=
23G.,
Gk)+Pker
which is compact because
0
(j
=) and G3G1=O if i
j.
Similarly, NB is compact. Observe that (1—M)
!1j+Pker
Gj)+Pker
and
(1—N) =
(Nk_Fk).
Thus (1—M) (1—N)
(Nk_Fk) .
(7.2)
Now (Nk_Fk) =
is compact because MJNk c K(H). It follows from (7.1) that (1-N) (1-N) is the norm limit of the compact partial sums of (7.2). lience, (1—N) (1—N)
Kill).
0
PROOF OF THEOREM 7.2. By Lemma 7.1, we can restrict ourselves to 165
for some k 1. We proceed by induction over k. If the case when 2 and then T is compact. Suppose that Tk E K(I1} for some k the result is true for k-i.
By Corollary 7.5, there exists a projection P such that TklP, (l—P)T
then
K(H);
A T
ranP
X
K21 K22 ker P
=
where K21 and K22 are compact operators and A=Tran Observe that 0
T—
k—i
0
P=
K21 K22
0
Hence, Ak1E K(PH). By our inductive hypothesis, there exists a com-
L(ran P) such that (A_K11)kl=0.
pact operators K11 Define
K=
K11 0
K21
22
then
A-K11 X
k
(T—K)=
k-i
A-K11 X
0
0
0
*
0=00
0
A-K11 X
0
0 ci
0
COROLLARY 7.6. Let Q be an essentially nilpotent operator of es-
of compact opera
sential order k; then there exists a sequence tore such that 0
(i)
(n
oo);
(ii) Q—K
(iii)
—
QIH(100(Q)\D(0,1/n);Q) and
where Fn
is
nilpotent.
so that rank that
e
if Q isquasinilpotent, then the Ks's can be chosen is finite for all j such rank Q] < =— and rank
K(ff).
(n=l,2,...) .
PROOF. Let subset of ci(Q) and — —
FXM nfl 0
o
isaciopen
n
i4
and
where
Clearly,
so that
By Proposition 3.45 (and its proof) there exists a normal compact < i/n and isquasi such that
operator
nilpotent. By hypothesis,
it readily follows that
= 0. By Theorem 7.2, BnLn+Cn, where 166
and
=
Since
o(B) ={O}, it follows from the upper seruicontinuity of the spectrum (Corollary 1.6) and Proposition 3.45 that there exists a firite rank operator G
n
such that IC
n
-Gn
1/n
nfl ={O}.
and ci(L +G )
Define R =L nfl +G and n F
n
Then K e K(H), (i). On the other hand,
X
0
0
n
R
0
ON+(C-G) +
—
i.e.,
<2/n
{K}:1 satisfies
(ii) and (iii) are immediate consequences of k
k.
(L +G n is a finite rank n k quasinilpotent; therefore, Rn is a nilpotent. A fortiori, so is In the case when ci(Q) = 0 the R's can be chosen so as to satis fy the condition"rank R < rank QJ =- and rank R is finite for n n all j such that K(H)", by an "ad hoc" modification of theprevious the above constructions:
Observe that R
=
)
fl
,
argument.
o
Corollary 7.5 yields the following COROLLARY 7.7. Let
be a finite family of operators acting
on H and assume that the product A1A2.. •Am is compact. Then there exist compact operators K1, K2,..., Km such that (A1-K1) (A2-K2)
(Am_Km) =
0.
PROOF. By Corollary 7.5 and an obvious inductive argument, we can find orthogonal projections P1, P21..., P1 such that K1=A1P1, K. = (j=2,3,...,m-1) and Krn= (l_Pm_i)Am are compact operators. Then, (A1-K1) (A2-K2) (A3-K3) .
.. (Am_i_Km_i)
[A1—A1P1J[A2—(1—P1)A2P2][A3—(l—P2)A3P3]..
=
=
x
[A1(l—P1) ][A2(l—P2)+P1A2P2]{A3(i—P3)+P2A3(}.P3)]
(1-P A P ...[A )+P ]CP A] m-2 m-l rn-i m-i rn-i rn-i m
[A1(i—P1)A2(1—P2)][A3(i—P3)+P2A3P3].
=
+P2A1P1][P1A] = Ar •
... = [A1(i—P1)A2(l—P2)A3(l-P3)
][Ar(l_Pr) +Pr1AP][A+i(l_P+i)+PA+iP+i] =
.
= 0.
[A1(l-P1)A2(l-P2) 0
7.3 Restrictions of nilpotent operators 167
is the k x k Jordan nilpotent cell (k=l,2,....) de
Recall that fined by (2.1).
LEMMA 7.8. If Q is a nilpotent of order k > (a)
vant subspace of Q such that QIM
1
and M is an invar-
for some cardinal a, 1
then there exists an invariant subspace N forQ suchthat closed and II
a
<
M+N is
M+N.
PROOF. Let M.=Mnker QJ, j=1,2,...,k. By induction on j we shall c ker QJ such that prove that there exists an invariant subspace and
1)
2) N,+M.ker
Q]•
For j=l we can take N1=ker QeM1. Suppose N. has been construct—l < k. We have Define (N.)nM = Mnker Q = Q
ed for some j
=
It is completely apparent that W.+W1 is closed and that N. 1nM = {O}.
If x€ker
,
then QX=Y.+Z., where
follows
it
and z.
Since QIM Hence
for some
that x
Q (j=l,2,...,k), if we define N = Wk
we are done. LEMMA 7.9.
Let Q be a nilpotent of order k and let (5.1) be
(i)
the canonical representation of Q; then
maps
QP1 injectively into ker QrBker
ker
for 1
r
<
p and p =
2,3,. ..,k. (ii)
Assume that ran Q
k-l
Then there exist a subspace R k-l and a constant > R M = Q 0
contains a subspace M of dimension a. such that
0
.
(7.3)
.
r
and for 1
for all x
Qkl)i of dimension a such that
(ker
p
k—i. The subspace R =
Q and is actually equal to the algebraic direct sum
R÷QR+Q2R++QklR
(iii)
Furthermore, Q1 = QIR Qkl and Q2 If, moreover, M = ran
PROOF.
(i)
then
=
For each p=2,3,...,k-1, we have 0 0
.
.
.
0
ker Q1eker Q°
Q12Q23.. o *
kerQekerQ
0
o
168
ker Qkeker Qkl
(The first p—i columns of the matrix of
are 0's and the (i,p)—en-
try is equal to Q12Q23. .
By definition of the representation (5.1), it follows that QP'l into ker Q1eker Q0 and ker(Q12Q maps ker 'p
•
Q1 p
-l
c
ker
-l
-l
(0}. n[ker Q eker Q is injective. A fortiori, Therefore, Q12Q23. . 0r,r40r+,r+2 is also injective for 1 r < p. . . Q _, p Thus, if M c ran Qk-i =Q12Q23...Qk..lk[(ker Q k-l )] i is a space (hence, closed) of dimension a, then there exists a subspace )
Q
•
(ii)
c
(ker Qkl)1
such that Q12Q23.
÷
M
is an isomorphism; this
shows, in particular, that dim R0=a and Q12Q23...Qk_l,kIRO is bounded below.
Now it is not difficult to show, by induction, that 0+1 maps isomorphically onto 0r,r+l0r+l,r+2 and 0r+1,r+2 therefore there exists a positive constant 5 such that (7.3) is satis•
k-l,
and for 1 r p < whence it readily follows that Q3R0 must be closed (i.e., a subspace) for all j=0,l,2,...,k—l. Let for some p, 0 < p k—i and some x and let z = R fied for all x c
p+l Q
k-i ..+Q
p+2
°
Ily—zil
Xk_l E go; then
Xk_1l where
(1/11Qk—l—P11) IIQk_l_P(y_z)II = (1/11Qk—l—P11)
.
,
= (1/11Qk—l—P11)
(o/IIQk_l_PII)IIxll
.Qk_l,kxII
whence it follows that the subspaces and QPRa form a pos itive angle. Therefore the algebraic direct sum = QiR is closed .2 •k—l +Q R +. and direct, i.e., R = R . .+Q is a subspace of H conR 0 0 0 •
taming
M
Q
k—l
0
R0.
It is completely apparent that R is invariant under Q and that the action of QIR can be described by: Q:Q?R0 isomorphically for j=0,1,2,...,k—l, and Q:Qk 1R -* {0}, whence it immediately follows that *
Q
R
0
and assume that M
= ran
Q
k-i
(c
R
). Observe that
=
[Qi
k-i
and
= M.
Corollary
It follows that ran
*
Qk_l]
Q2kl={Ø},
i.e.,
7.6 implies, in particular, that if
o
{Q} 169
(i.e., QJ is a compact operator, not of finite rank;
and rank
k, then Q is the limit of a sequence of nilpotent operators such that rank < rank for all n for all
see Section 1.3)
(n,j=1,2,...)
j
e K(K) for all n=1,2 By using CoQk=Q rollary 7.7 we can obtain an analogous result for the case when and j
and Q-R
PROPOSITION 7.10. Let Q be a nilpotent of order k and assume that
Qr Qkl Qk2 = are non-aero finite rank operators, and rank for j s,s+l,...,r—l, but QS1 / K(K)(l s r k). Then there K(fl) for aZ-i n, exists a sequence {Rn}n1 in Nk(tI) such that
rank
=
QJ for j
rank
s
increasing for each j,
= r,r+l,...,k—l, j r—l, and HQ—
is strictly
{rank 0
(n ÷
argument,
PROOF. By Lemmas 7.9 and 7.8 and an obvious inductive we have 1
,•_k
where W is invertible, F tegerS,
rank
=
Q
j
j—r 1
k—l
0,
are non—negative in
the J
for
T
j=s,s+1,...,
is not compact.
r-l, but
If s=r, there is nothing to prove. Assume that 1
s
<
r
<
k and
let 0
T12 T13
.
.
M1
T23
.
.
M2
0
.
.
M3
0
0
0 Ti,_i,r Mr_i 0
T€
be the canonical matrix of
where
M =
Mr
L(M1 •with respect to the decomposition
j=i,2,...,r. is a compact operator with rank =-.
Observe that T12T23.
Thus, by Corollary 7.7 there exists compact operators K12, =0.
1(r—1,r such that
Given n
1, we can find compact operators C._,
r—l) such that n <
<
j
,j,n
(j=i,2,...,
and
rank(T12_C12,n)(T23_C23,n)....(Tr_i,r_Cr_i,r,n)
Let T.1 .—C.1 ,j,n j ,j j 170
(rn)
—l
<
by
T by replacing ,
j=2,3,...,r.
It is easily seen that T—T'
n
K(I1),
(rn)1,
<
=d
is not compact and rank(Tn)
is finite.
Now we can repeat the same argument with T"
Wr(M)
KIM), flT—Tjj
such that
<
in order to obtain
for
2/rn,
j=s,s+1,...,r-l, is not compact and j=r-l,r—2. After r—s steps, we obtain an operator
< 1/n,
KIM),
that T_Tn
is finite for e
such
is finite and larger than
rank
s—l n for j=s,s+l,...,r-l and is not compact. Passing, if necessary, to a subsequence we can assume that
j
is strictly increasing for each j, s
rank
< r—l.
Finally, define R =W(FeT)W1. It is completely apparent that the sequence
satisfies all our requirements.
D
7.4 Operators similar to Jordan operators
is
A Jordan operator J in LU-f)
an operator unitarily equivalent
m.
to
',
where 1
m
<
are distinct complex numbers, 0
°°,
and mj is finite for all j
THEOREM 7.11. If Q
= 1,2,... N (II).
,m.
then Q is similar to a Jordan operatOr
if and only if ran QJ is closed for all j=l,2
PROOF. If J is a Jordan nilpotent operator, then it is completely
is closed for all j1, 2,.... It is also clear that if Q=WJW1 for some invertible W, then JJ) is closed for all j=1,2 ran
apparent from the definition (7.4) that ran
Assume that Q Since ran
Qkl
Nk(K) and ran
is closed for all j=1,2,...,k—l.
is closed, it follows from Lemmas 7.9 and 7.8 that Q
has an invariant subspace Mk with invariant complement Rk such that (0
QIMk
= (l/k)dim
Mk
oo)
and
QI
Rk e Nk_l(Rk) . Now we
consider QIRk. By induction, we conclude that Q is similar to an opera tor of the form o j=l
It
is easily seen that
cz.=cx.(Q) =dim (ker Qnran
QJ) ,
and that this set of cardinals is a similarity invariant. Theorem 7.11
says that this is, in fact, the only similarity invariant of Q. More precisely, we have the following.
171
and ran and ran Q2J COROLLARY 7.12. (i) Tf Q1, Q2 = are closed for all j1,2,..,k—l, then Q1 Q2 if and only if for all j=1,2,...,k—1. (ii) If Q Nk(H) and ran QJ is closed for all j=1,2,...,k—l, then Q
Q
XQ for all A
e
By Lemma 7.1, we have
then T is similar to a Jordan oper
COROLLARY 7.13. If T ator if
p
is
and only if ran
q(T) is
closed for all polynomials qjp, where
the minimal (monic) polynomial of T. Operators similar to Jordan operators are actually dense in
(H). We shall need an auxiliary result. LEMMA 7.14. Let Q Nk(H) with canonical representation (5.1). Then Q3 has closed range for all j (j=1,2,...,k—l) if and only if has closed range for all j (j=1,2,...,k—1).
PROOF. Observe that
*
.
— o...oo
r%J —
0...00
.
.
.
0
.
.
.
0
where
has closed range
If
is bounded below by > 0 for all (for all i=l,2,...,k-l), then 0i m=1,2,...,k-j. Asi=l,2,...,k-l, and is bounded below by (ker Q))I, ker Q1eker i=j+1,j+2, sume that ...,k, = 1 for all n=l,2,..., and ÷ 0 (n -- oo). It is not difficult to see that 0
<
(n
and, a fortiori, that Jlx"II
=), i=l,2,...,k—j, 0
(n
no),
a contradiction.
QJ)]1 is closed, j=l,2,...,k—l. is bounded Conversely, if Q] has closed range, then QJ1 (ker Hence, ran
below by some positive constant bounded below by 172
Then
and therefore it is left invertible (j =
isa]so
1,2,
.
.
.
This clearly implies that ran
is closed for j=i,2,...,k-l. 0
THEOREM 7.15. Let Q c Nk(U). Then there exists a sequence
such that
in
+ 0 (n +
and (ran Q1)
=
ran
for all
j=l,2,...,k—1 and for all n1,2 PROOF. Assume
that
Q has the canonical representation (5.1) and
.=V.H. be the polar decomposition of Q._1 •, jj J ,J is injective, ker H. = {0}. Define Q n by (5.1) with J ,J J replaced by Q.1 =V.(H.+l/n), j=2,3,...,k, n=l,2 let
:i
,J
Since Q._1
.
'3
J
.
,J,n
3
Then
3
is closed f or all Nk(U) and (by Lemma 7.14) ran (n=l,2,...).On the other hand, it readily follows from the definition that ran for all j1,2,..., c ran c (ran Q3) k-i and all n=l,2 o
=
l,2,...,k—1
By using the arguments of Lemma 7.1, we obtain
COROLLARY 7.16. Let T be an algebraic operator with minimal monic
polynomial p. Then there exists a sequence {Tn)nl in Atg(H) such that 0 for all n1,2,..., IIT_TnH ÷ 0 (n ÷ oo) and [ran q(T)]=ran P(Tn) all and all n=1,2
7.5A
similarity
invariant for polynomially coripact operators
PROPOSITION 7.17. If p(T) K(H) for some (monic) polynomial p, then the following are equivalent:
(H (ii)
(iii)
T
J+K, where J is a Jordan operator and K is compact.
-. 5 for
some Jordan operator J.
ran q(T) is the algebraic sum of a subspace Hq and the
range Rq of a compact operator Rcf for each
(iv)
0 is an isolated point of
for each polynomial
qi
(v)
p(T) is similar to a Jordan operator for every unital 4-re-
presentation p of (vi)
is similar to a Jordan operator for some faithful unit
ci i-representation p of (vii)
is similar to a Jordan operator for every unital i-re
presentation p of 4(H). The equivalence between (iv) and any of the stater'tents (v),
or (vii) follows
easily
from
(vi)
corollary 7.13 and the well-known 173
fact that ran A is closed if and only if 0 is an isolated point of cl(A*A). The equivalence between (iv) and (iii) follows from the analysis of the possible structures of an operator range (or Julia manifold
(ii)
see [71], [96]) and (i) => => (iv) are trivial implications. In we shall order to complete the proof (By showing that (vi) =>
(i)),
need the following auxiliary result. LEMMA 7.18. If T
L(H), o(T)
=c1uc9, where 01 and 02 are two dis
joint compact nonempty subsets, H1 is the Rie8z spectral subapace corresponding to 01, H2 = A C H1
T
=
B H2
0
is the orthogonal projection of H onto H., j=1,2, then P1 and
and
belong to C*(T).
Moreover, there also exists a unique operator X:H2 + H1 such that
lx AC lX 01
0
and C*(T)
with identity is in-
PROOF. It is well-known that a verse closed. If the idempotent 1 Y H1 E
0 0 H2
is the projection of H onto H1=H(c1;T) along H(c2;T), then E is the limit of rational functions in T and therefore 1
l+Y*Y0
hi1 EE*=
B, E*=
0
belong to C*CT1, where
and P1= (norm) lim(n -* =)
0
is a sequence of polynomials converging
uniformly to 0 on the real seciment [-1/3,1/3] and converging uniformly
to 1 on the real segment oea = P2TP2, AWB =
Since
÷
and
=01 is disjoint from c(B)
lary 3.20 thab t
is clear that Pfl—P11 AWO = also belong to C* (T).
- T
021 it follows from Corol—
is invertible and
=j
dA,
where r is the (suitably oriented) boundaryof a Cauchy domain
taining a(A) such that c(B)nfl=Ø. 174
con-
Given A —l
X(A) =
(le(X—B))
o
—l
=
belongs to C*(T) and (by taking limits of suitable Riemann sums) we conclude that
lX W=
0
i
dA
1r X(A)
belong to C*(T). Since
and
A fortiori, W,
c*(T).
it immediately follows that C*(T)
o
Now we are in a position to complete the proof of Proposition 7.
17. Assume that there exists a faithful unital *_representatjon p of C*(T) on H
such
that
is similar to a Jordan operator with
minimal polynomial p, p(A)
Since p is faithful, itfol
lows that
p is also the minimal polynomial of Hence, by Theorem 7.2, T (or some compact perturbation of T) is an a1-
gebraic
operator with minimal polynomial p. Let T1 T2 *
(7.5) 0
T
be the matrix of T with respect to the decomposition N
=
where the R11s are inductively defined so that (n=l,2,...,m). Applying Lemma 7.18 (and an obvious inductive argument) to T, we can find a(unique!) invertible operator W in C*(T) such that and C*(T) A fortiori,
A2 •
A
*
= p(T) =
=
A.)V1
0
A
m
with respect to a decomposition
=
where
denotes the
(i=l,2,...,m), orthogonal projection of K onto is similar to a Jordan operator with minimal polyand
nomial 175
Clearly, this reduces our problem to the case when
is a sin-
gleton; moreover, replacing (if necessary) T by T—X, we can (and shal' directly assume that
1. Then
for some k
but
and 0
A12
Hp,1 A23
0
Hp,2 *
0
H
-
p,3 ,
(7.6)
0
0 0
j=l,2,...,k. A3eker Since A is similar to a Jordan operator, it follows from Lemma 7.
where H
14 that the operators A.
.+i:H J,J p,J
are bounded below and 0 is +H p'J. A*iAJ for all j=l,2,...,k-l; .
an isolated point of the spectrum of
cy(A*JA))] = therefore there exists > 0 such that k-l Thus, if denotes the spectral measure of
is finite for all j=l,2,..,k—l. liz follows that 0 isan isolated point of (j=l,2,...,k—l) and o(T*JTJ)] is finite. Hence, contains a closed sub k disjoint from a(T* T and therefore (by the StoneWeierstrass theorem) there exists a sequence of polynomials converging uniformly to 1 on [0,ct]and converging uniformly to 0 on a(T*JTJ)]. It readily follows that {rn(T*JTJ)} converges C*(T) (j=O, in the norm to the orthogonal projection l,2,...,k), and =P.=(norm) lim(n ÷ =){r(A*JAJ)_r n n J J 3 c C*(A) is the orthogonal projection of A for all onto )} j=l,2,...,k. T*)TJ, then rank
)
Since p is isometric, there ed.st compact perturbations
C*(T)+K(H) of L.-L._1, j=l,2,...,k, such that R1, R21..., Rk are pair wise orthogonal Rj=lP and allj 1,2,... ,k.
be the matrix of T with respect to the decompoj=l,2,...,k. Since AP. and p is isometric, it follows that T.. must be compact for all (i,j) such that 1 k. Thus, up to a compact perturbation, we j Let T= (T. .). 1,] k
sition
where
i
can directly assume that T has an upper triangular matrix of the form (7.6)
(with the A1J'S replaced by T..'s) with respect to the above de-•
composition.
Now observe that 176
p(1W WJ=2
p(R1+
)
= j=2
is invertible in C*(A), whence we immediately obtain that +
ran
<=.
.
If
j=l2,...,k—l) is a Fredhoim operator with 0
(c
T.
indT.
J 1
k=l, then T=0 is a lifting of T, and we are done. Assume
that k > 1; then, by using the above construction and an inductive ar-
gument, we can assume that (l_Rk)(T+Kk_l)Iker Rk (ker
is
U.)
equal to Tj2
0
0
0
*
.
•
eod k-i 0 0
with respect to a decomposition ker Rk=
(,k_i
d
k-i
(0
d. <
where
is a finite rank operator, is a subspace of finite codimension of U. and T' is bounded below for all :K'. + j,j+i j+l for all j),
.
3
j=l,2,...,k-l, and mite dimensional space
Let
j
denotes the zero operator acting on the fi— k-i dk 1 —
be the orthogonal projection of tiki onto
Since
is a Fredhoim operator with negative index, k is also a Fredhoim operator with md
Tk_l
ran
0 and
Rk_lPk_l,k=dk
anis2.
and ran
morphism of Hubert spaces between k_lWk
Thus, if u
(dim U
there exists a finite rank operator
<
then
such that if
then
T+Kk is unitarily equivalent to the orthogonal direct sum of an operator maLrix of the form (7.6), with the A. .'s replaced by the T! .'s,
acting on the finite dimensional space
and the operator 0d
k.
k
Since j,j+i j+l + ran 3,3+1 is an isomorphism of Hubert spaces for all j=l,2,...,k—l, it follows from Lemma 7.14 and Theorem .
.
7.11 that T+Kk is similar, to a Jordan operator, and this completes the
proof of Proposition 7.17.
El
REMARK 7.19. It is convenient to observe that in the above proof (i.e.,
(vi) =>
(i)
in Proposition 7.17) we did not use the 177
Indeed, this hypothesis is an easy consequence of p(T) is any of the hypotheses (i), (ii), (v), (vi) or (vii), but false, ingen eral, if we merely assume (iii) or (iv)
(see [30,p.868]).
7.6 Nice Jordan operators A Jordan operator J, given by (7.4) will be called nice if, for
(j=l,2,...,m),
is an infinite cardinal for at most oneval is an infiand very nice if, for each
each j ue of k,
nite cardinal for exactly one value k3 of k and
for k
Na-
mely, Q is a nice Jordan nilpotent if and only if (a.)
k
for
very nice Jordan nilpotent if and only if Q
a
some s.
LEMMA 7.20. Let C be a Ca-algebra with identity 1 and let t (C)
2). Then the following are equivalent
(k
is invertible for all
i
is invertible for some
('ii,
element v
and an
-
C
—l
such that
t.
There exists a faithful i—representation p:C
(iv)
j
= 1,2,. . ,k—l. j, 1 j k—i.
There exists a faithful unital i—homomorphism T:L(cr)
(iii)
1
j
k—i,
+ L(H)
and j,
such that the sequence H
+
ker p(t
exact
H
H
+
k—j
= ran p(t)and ker p(t j) = ran j
)
(v) For every faithful unital i—representation p:C
k—j p(t )).
+
the
sequences +
j — l,2,...,k—l, .
$
—
are exact.
for every faithful unital 'Lrepresentation p:C
p(t)
(vi)
L(H ). PROOF.
(i)
>
(ii)
and (iii)
(vi)
>
=>
(v)
=>
(iv)
are trivial
implications -
(v)
=>
(i)
and (iv) => j
(ii).
closed, so are ran p(t )*=ran
Since ran
j p(t*
)
and ran p(t
and ran are k—j )*=ran p(t* k—j
Observe that [ran
=
ker
=
ran p(tkJ)
[ker
p(tJ)*]1 =
ran
=
ker
and
178
=
[ker
=
[ran
.
.
p(tkJ)*]1
so that ran
ran
=
+ran p(t])* =
H
and ran
=
ran
=
Hence p(tkj+t*j) is invertible inL(H). Since p is faithful and unital, it readily follows that must be invertible in C. (ii) => Assume that is invertible (for some h, 1 h k—l) and let p:C*(t) + L(H ) be a faithful unital *_representatiop and Tk-h +T* h is invertible. Let If p(t) T, then T E
(iii)
T12
O
Hi
O *
H3
T= O 0
(where H.=ker
j=l,2,...,k—l) be the canonical represen-
tation of T. Now T(k,h)
k-h
Ii
=
¶r
T(k*,h*)
0
where T(h)
l,k—h+1 *
2,k—h+2 T(k,h) = 0
T(h)
h,k k—h,k (T'
'
k-h-l ,k-l
)*
0
T(k*,h*) *
)*
l,h+l
Since
and
is invertible,
it is not difficult to conclude that the operators
m,m+h are invertible. Since
:11
m+h
H
, m=l,2,...,k—h,
m is injective for all j=l,2,...,k-l, an
elementary inductive argument shows that
+
is invertible 179
for all j=l,2,...,k-l. In particular, this means that all the subspaces
(j=l,2,...,k)
have the sarte dimension.
with (via a fixThus, up to a suitable identification of onto we can directly assume that
ed unitary mapping form
is a positive invertible operator for each j=l,2,...,k—l. k—j +T* () are also invertible Clearly, Tmn+j (m=l,2,...,k—j) and T
T.
.,
operators (on their respective spaces) for all
TkJT*kj
then
and
(Tk_j+T*
=
If A.
)
.(Tk
* = (Tk_i+T*j)
)
(T*kJ+T3)
is invertible and A.B.Tk jT*kTj=0. Since A. and B. are positive,R. —l
=
(j=O,l,2,...,k) is an hermitian idempotent; it is easi-
ly seen that
is the orthogonal projection of II
j=l,2,...,k. Let
j = 1,2,... ,k.
denote the identity on
seen that for each i, 1
i
<
onto Ky
Now it is easily
k, C*(T) contains the operator
.
i+11i+1"
=
H1
0
0
K2
01 0
0
denotes the identifica-
(the empty entries are 0's), where tion between
and H1
(i.e.,
the operator
i
Sincethisistruefor alli,l
<
k, it is not difficult to con-
clude that all k x k operator matrices (with respect to the decomposition H
with "constant entries" belong to C*(T). Hence,
01 0
Hi H2
1
0
0
H c
0
•
01 0
180
Hk_l
C*(T).
If W=T
then 10
1 B13 1
0
0
.
.
.
.
0
I.
0
1
JHk
0
Let (1 B13 1
.
.
.
H. 0
I.
.
.
1
then
1B'24 0
B' 0 2,k—l 2k B' 0 3,k—1 3k B' B' 0
B'
.B'
1
B35
.B'
0
1
25
0
Itt2
4,k4 4k
.
01ff4
.
.
0
:
:
0
1
:1:
0
OJHk
Similarly, if
B'24
B'
25
1
.
.B'
B' 2k 1ff1
.
1
B'
4,k4
.
0
.
B'
4k
:
:
.
.
1113
1 1
then
181
1 B5 B36
0
0
1
B6
0
1
.
.
84k .
.
D=W21CW2
0
0
002 003 004
0
.
0
0
0
86k
=
0
1
0
00k—2
0
By an obvious inductive argument, after k-i steps we obtain Q =
V1TV, T=VQV1, where v=wwlw2...wk_l c Since p is faithful and unital, we deduce that there exist
C*(t)
c C such that p(v) V and p(q) Q. If is a matrix with constant entries in C*(T), then there
(unique) v, q
M=
exists a unique m Define i:L(&') cal ONB of 0
k
C*(t) such that p(m) =M. + C*(t) c C as follows:
and A= (A..).
k
.1
is the canoni-
If
with
is the matrix of A
re-
spect to this basis, then t(A) =m.
is
The properties of p make it clear that T *...representatjon. Since
a faithful unital
=q, we are done.
Observe that
IJ
is a unitary operator. Hence, so is
l,2,...,k-l. Let n=[k/2](= the integral part of k/2)and for ae C
assume that
-1; then
it follows from
the proof of Proposition 1.10 that II(a
k—n +a* n
k—n )
n
Ia
)II
k—n
k—n
n
II+!Ia* n
II
<
and therefore ak n+a*n is invertible in C. By Lemma 7.20, we conclude that if a NkIC) and < then a (Compare this result with those of Section 2.3.2.) Norevoer, if and Ila-bi! < Bk/(IIvII.IIv H), then Iv
and therofore b
r
7.21.
Thus, the subset 182
=
Ihf'(b—a)vII
<
Hence, we have
The set C: a (where t is some faithful unital *_homomorphism of into C) is open in Uk(C) (k1,2,...). COROLLARY
c
II
Let C be a C*_algebra with identity 1.
is open in Nk
is
Nk(A(lfl):
=
invertible)
(7.8)
(H)) and
(T e L(H):
is invertible) (7.9)
intersects Wk(U) in an open subset of
L(H) and Tk is compact, but then the following are equivalent J+K, where J is a nice Jordan nilpotent and K is compact.
PROPOSITION 7.22. If T
2),
k
T
(i)
T = WJV+K, where W, V are Fredholm operators Buck that J is a nice Jordan nilpotent and K i8 compact. T = WJV+K, where W, V are Predhoim operators such that (iii)
=
(ii)
I, J
=
isa very nice Jordan nilpotent and K is compact.
is invertible for all j1,2,...,k—l.
(iv)
is invertible for some j,
(v)
1
j k1.
There exists a faithful unital *_homomorphism such that and an invertible element There exists a faithful i-representation (vii)
(vi)
and
j,
j
1
+
< k—i, such that the sequence p
p
p
p
is exact.
Par every faithful unital *_representation p:A(H) +
(viii)
the sequences (7.10)
p
p
are exact for all j1,2,...,k—1. for every faithful unital *_representation
(ix)
+
We
shall need an auxiliary result.
7.23. let T =
j
H and 0
= 1,2.
<
dim
112
<
= and
Then, under the identiin A(H)
be a unitary mapping from 111 onto
PROOF. Let U:H1 -'.
and let S=UT1U* in L(H)
111
is unitarily equivalent to
fication of H with 1111
+
with
Suppose II =
be in L(I1);
;
it
Define
suffices to prove that
L(H) is
by Vx = xOO. Now UV*TVU* = S,
rank VU*=nulVU*)*=nul V*=dim
an isometry and co-
is finite; thus
is
unitary in
A(H) and it follows that PROOF OF PROPOSITION 7.22.
V:111
a (i)
>
(ii)
and (iii) =>
(ii)
are triv 183
ial implications,
(jjj)
(ii) =>
follows immediately from Lemma 7.23,
and the equivalence between any two of the conditions (iv) —
lows from Lemma 7.20 and its proof. (Take
(ix)
fol-
C=
and
A(H) .)
Since
If T has the form of (iii), then j
is
closed (Proposition 4.22), it easily follows from Proposition 4.21 for any faithful unital h-representation p of A(H) on
that p(J)
of dimension
a Hilbert space H Assume
(iii) => (ix)
By Proposition 7.17 and its proof, T=W(J÷K)
such that p0(T)
where W a. = ao,
By Lemma 7.20,
that C*(T) admits a faithful unital *_representation p0
then
G(H), K c K(H) and j ] (co) (oo) J •q where 1
L(H)
Let p:A(H)
with k—l
s < k and R
If
=0.
be a faithful *_representation. Since LI(J)
is
closed (Proposition 4.22), it is easily seen from Proposition 4.21 that 0,
a
0,
Lkl=O
H
H
and therefore the sequence
H
is not exact. Since T = WJW', it readily follows that the sequence (7. 10) cannot be exact for j = s, contradicting the equivalence between (ix) and (viii) We
k-i
conclude that
< =,
i.e.,
on a finite dimensional space. Since J see that (ix)
=>
is
(=)
where F acts a nice Jordan operator, we J
(i)
The proof of Proposition 7.22 is complete now.
U
COROLLARY 7.24. Assume that Q c L(H} is a quasinilpotent operator is invertible (far some k such that = 0 and 2).
Then Q is similar to a nice nilpotent operator if and only if rank
is
finite.
PROOF. The necessity of the condition "rank since ran Qkfliustbe closed if Q Assume that rank Q k
0k
nilpotent
je.
<wand
sional space H0
issimilartoan±ce
finite. Then rank Q
Q3 for all j
and rank
is a
k.is
and, R
is -
such
by
Lemmas.
Qk
7.9
is finite" is clear,
riilpotent.
j.is finite for all 3
k such that rank where and 7.8, Q
Hence, Q
nilpotent of order k acting on an infinite dimen that Rk—i +R* is invertible. It readily follows a
ran R* is a Fredholni operator and there fore ran R*J is closed for all j=i,2,...,k-1. A fortiori, ran R3 is also closed for all j=l,2 k-i. By Theorem 7.11, R is similar to a Jordan nilpotent J of order k. (Proposition 7.22), but Since R is invertible, so is 1 k—i from (5.1) that R*:(ker R*)'
184
this is impossthle unless J is nice. JWF and J$F is a nice Jordan nilpotent.
Hence, 0
COROLLARY 7.25. If Q sition of Q is given
Wk(H),
0k—l
IJ
0 and the canonical decompo-
(5.1) with ker Q3eker
=
IL,
j=1,2,...,k,
then the following are equivalent:
(polar decomposition), where H. (Q*
If 0j j+i =
(i)
and
+
is a (necessarily injective) par
tial isometry, then nul
<
= for
k—
+0* for all j=1,2,...,k—1. (jj)
If Q nul
(iii)
nul W.
(v)
(polar decomposition), then nul
for some j, 1 < j < k—i. < for all j1,2,...,k—1. k—i. = for some j, 1 <
nul W.
(iv)
all j=l,2,...,k—l.
<
j
PROOF. Given e
>
0, let Q
be the operator obtained from .0 by re-
by V.(H.+c) . By Theorem 7.15 and its proof, 0
placing
is simi-
lar to a Jordan niipotent of order k. =>
(i)
(ii) ,(iv) . If
Q satisfies Ci) ,
then
Q
is similar to a nicE
Jordan nilpotent. (Use Theorem 7.11 and its proof.) By Proposition 7. is a Fredhoim operator. On the other hand, it is easily
22,
seen from our construction and (5.1) that
k-
k— J+Q*J)]- =ran(Q
and [ran(Q
for all c
>
0
=ker(Q C and all j1,2,...,
k—i, whence the results follow.
(i)
(iii) =>
ran(Q
then M= ran W.
JJ
c..
N,
J
(iv) => ker(Q J±Q*J)
for
Conversely, if nui
(i)
k-
+0*
)
some j,
j=1,2,...,k—l,
has infinite codimension in H. Since (ran
it follows that nul W*=oo, a contradiction. J
Similarly, if nul for some j, I < j < k-1,then kker(Q J+Q*J)isinfinitedimensional and therefore
nui W. ==, a contradiction.
(iii)
Since (ii) =>
and (iv)
=>
(v) are trivial implications, we
are done.
o
the form of Corollary 7.25 and is finite for all j=l,2,..,k—i, then the following are equiva-
COROLLARY 7.26. Assume that 0 has
nul lent
(L)
(ii) ci) =
(for (iii)
ci)
(for (iv)
vs not tnvertvble.
0
If H. = f A dE. (spectral decomposition), then rank k—i. all c > 0) for some j,
I
If R. =
all c rank
>
f
A
0) for
j decomposition), then
(spectral
some j,
1
<
(for all c
j
rank
< k—i. >
0) for all j=l,2,...,k—i. 185
PROOF. Since nul V3 is finite for all j=i2...,k-l, it is not difficult to conclude from (5.1), Corollary 7.25 and Proposition 7.22 that the following four conditions are equivalent is invertible;
(a)
is a Fredholm operator for all j=l,2,...,k—i;
(b)
(c)
is a Fredholm operator for all j=l,2,...,k—l;
(d)
is a Fredhoim operator for some j,
1 <
k—i.
j
The contrapositive of this equivalence yields the result.
that Q c Nk(H). rank for j=s,s+1,...,r—i,
COROLLARY 7.27. Assume
rank
or s
is finite for j=r, and
either s1
is not invertible. Then there exists a sequence
2 and
in
0
such
Nk(II)
that
+ 0
and
(n
;
J..5+1
i-s independent of
(ti-i.)
j—l
n and rank
Q
j—l
for
j
=
r+l,r+2,.. .,k; (iv)
(v)
{rank
is strictly increasing for js+i,...,r—i;
is a Jordan operator of order and
s
is not in-
vertible.
PROOF. By Proposition 7.10 there exists a sequence {Rfl}nl in Nk (H)
such that Q—R
,k—i
,
(rank
Rn]
j
r-i, and where Ln5= 0 and Fn= is easily seen that
for j=r,r+i,. K(H) for all n, rank Rn3 =rank is strictly increasing for each j such that s 0 (n + °°). By Lemmas 7.9 and 7.8, acts on a finite dimensional space. (It
is independent of n for j=r+1,r+2,...,k.)
Since is compact, it follows from Lemma 7.23 that 0 and either s=1 or s > 2 and is not invertible. Thus, inorder to complete the proof, it suffices to show that if T c W5(H) 0 and is not invertible, then T can be approximated arbitrarily by operators similar to Jordan models with the same properties. Let c > 0 and let
0 T12 T13 0
.
T23 0
.
.T154T15
M1
.
M2
.
M3 (7.11)
0
M5_1
0
0
186
M5
be the canonical decomposition (given by (5.1)) of T. Since is infinite dimension is not difficult to see that M. = ker T3eker — ÷T* is not invertible, it follows al for all j=l,2,...,s. Since T from Corollary 7.26 that either (ran has infinite codimension is not closed, for some i' 1 or ran k-i. i
j,j+l
If
V.H. be the polar decomposition of T. (ran j j infinite codimension for some h, 1 h k—i, =ran Vh has Th rethen we choose T as the operator defined by (7.11) with placed by j=l,2,...,s—i. It is easily seen that IT—Til = c/2 < c, (7.11) (with the new (j,j+l)—entries, j=l,2,...,s—l) is the canonical decomposition of T E N (K) and (by Lemma 7.14) ran T is 5 C C closed for all j=l,2,...,s—l. By Theorem 7.11, is similar to a Jor dan operator. It is completely apparent that TS1 $ 0 and, since the (h,h+i) -entry of T is Vh(Hh+C/2) and ran Vh(Hh+C12) =ran Vh has infinite codimension, it follows from Corollary 7.25 that TES cannot Let T. —
.
be invertible.
Assume that 5 2 and (ran has finite codimension forali j=l,2,...,s—1, and let h be the last index such that ran P isnot :(,J
closed. Since T
it is not difficult to see that T1
be compact. Hence, there exists dEh
0 <
h+l
cannot
< c12, such that if Hh= 1
(spectral decomposition) then Eh([O,n)) =Eh((O,n)) and Eh([fl,Go)) are
both of infinite rank. In this case, we define T' as the operator de-
replaced
fined by (7.10) with T.
for j1,2,
by P'
...,h-1 and jh+i,h+2,...,s—1 and Thh+l replaced by A dEh). It is easily seen that
Tts=O
T-T'II=max{c/4,n} < El2,
and
0. Furthermore, (use 0
.
0
.
.
N1
.
*
=
:
0
Since it
is
N5
is a Fredholm operator for all
3,3
(polar decomposition), then nul W1==. 0
A12 A13 0
A
if
Let
ker
ker T'2eker T'1
23 0
=
==. Thus,
.T'1
not difficult to check that nul
T1+Tl*=W1R1
and nul T
C3
*
ker
.
C2
T' 9ker T' C
C
C
0
. 0
A
ker 187
be the canonical decomposition of T'. Since nul Corollary 7.25 that (ran A1
i
it follows from
has infinite codimension for some i,
1 s-i. Thus, we can proceed exactly as in the first part of the proof in order to find an operator similar to a Jordan nilpotent of order s such that IT'—T < c/2, 0 and is not invert ible.
It is completely apparent that
< €. Since
E
>
chosen arbitrarily small, we are done.
0
can be El
REMARK 7.28. It is convenient to observe that, by using translations, direct sums and similarities, all the results of this section can be rephrased as results about compact perturbations of similarities of nice (not necessarily nhlpotent!) Jordan operators, etc.
7.7 Notes and remarks
The theorem about the structure of a polynomially compact operator (Theorem 7.2) is due to C. L. Olsen [159]. The auxiliary result Lemma 7.1 is essentially contained in [139] (see also [43]). Corollary 7.6 is a trivial consequence of C. L. Olsen's theorem and Corollary 7. 7 is a remark of D. A. Herrero. Lemma 7.8 is an unpublished result of C. AposLol and D. Voiculescu 1341. (A particular case of this result, for a = 1, can be found in B. Charles' article [65,Theorem 7].) Lemma
7.9 and Proposition 7.10 are unpublished observations of J. Barria and D. A. Herrero [43] and D. A. Herrero, respectively. Theorem 7.11 has been simultaneously and independently discovered by C. Apostol and J. G. Stampf ii [30], L. 3. Gray [109] and L. R. Wil-
liams [197]. Theorem 7.15 is due to L. 3. Gray [110], but the proof given here is based on C. Apostol's Lemma 3.2 of [12] (Lemma 7.14, above).
The results on similarity invariants of Section 7.5 have been taken from L. A. Fialkow and D. A. Herrero's article [93] and the notion of "nice Jordan operator" is also due to these authors [94]. The key result given by Lemma 7.20 is an unpublished result of C. Apostol and D. Voiculescu [34]. The statement given here is a mild improvement due to L. A. Fialkow and D. A. Herrero and Proposition 7.22 and Corollaries 7.25, 7.26 and 7.27 also correspond to these authors (unpublished results). On the other hand, Corollaries 7.21 and 7.24 are contained in [34].
188
8 Closures of similarity orbits of nilpotent operators
With the help of Lemma 7.1, the problem of characterizing S(T) for T a polynomially compact operator can be easily reduced to the case when T is nilpotent.
This chapter is an account of what is known and what is unknown about this problem. The results include some information about quasinil potent operators, the complete solution of the corresponding problem in the Calkin algebra and several conjectures and examples that illustrate the difficulties of the still open part of the problem.
8.1 Universal operators W Given a class W of operators acting on if, we shall say that R is a universal is a universal operator for W if W c S(R). Namely, operator for (k=i,2,...), but U(H) (H an infinite dimensional space) and A c L(U) do not contain any universal operator.
8.1.1 Universal quasinilpotent operators k
L(H) be a quasinilpotent operator. If Q =0 and ran Q Let Q (Leincontains a subspace of dimension n (1 n =), then Q (0 for some space H of dimension mas 7.9 and 7.8), where R N (H )
=). By Rota's corollary (Corollary 3.36(i), R slut q1 (= the 0 operator acting on H6). A fortiori, (by a new application of Rota's corollary) we have slm
slm
(8.1)
Qkx for all k=l,2,..., and let E H: is a closed nowhere dense subset of = 0}. It is easily seen that H. Hence, by Baire's catagory theorem, H\ukl Rk(Q) 0, i.e. there Qky for all k=l,2 H such that 0 exists a vector y Let N =
Assume that
Vk>o
and let
189
M* M 0 L
Clearly, a(Q) ={O}=c(M)ua(L). Thus, by Lemma 3.36 and Corollary sim NOL slm Nb, where M=QIM is a cyclic quasinilpotent
for all k=l,2
operator such that LEMNA 8.1. If Q then S(Q)
L(HJ is quasiniipotent and
for all k=l,2,
contains all compact quasinilpotents.
PROOF. By Lemma 2.5 and Proposition 2.33, it suffices to show for all k large enough and, by our previous observa-
that
tions, we can restrict ourselves to the case when Q has a cyclic vector y.
Let quence
be the Gram-Schmidt orthonormalization of the seSince y is cyclic, it is straightforward to check
that the matrix of Q with respect to this ONB has the form q11 q12 q13
.
q21 q22 q23
.
q32 q33
.
q43
.
•
q31( .
Q= (flj31
0
Moreover,
for all k
2 and
2}0. k -* sp(Q) =0 (k ÷ co), so that Fix k 2 and let W be the diagonal normal operator
11Qk11l/
w2,...,wk_2,wk_l,wk_1,wk_l,wk_l,...})I where 2,...,k—l. If t4= V{el,e2,...,ek}, then we have
WQW where
190
ABN
1
=
j=l,
1
1
q11 q12j—
1
1
1
w3
w3
1
q34—
1
1
Wk_2
o
1
Wk_l
0
0
C
0
and
D =
=
It IICII
follows that
1Q11
Since
and
IIDII
<
hA1Q11.
we conclude that
By Rota's corollary (Corollary 3.35(u)), there exists a sequence {V}n:1 such that IV QV 111< 1/n (n=l,2,...). It is completely appar ent that VnQVn1 is also a cyclic quasinilpotent. Thus, we can apply 191
the above argument to V QV n
in order to obtain a sequence {W nn=i }
n
in
GUI) such that n
QV
1)W n
n
oo). Hence,
n +
V )Q(W V
II(W
< 4k/n ÷ 0
for all k=2,3
stm
0
From Lemma 2.5, we obtain the following auxiliary results:
(k—2)
(k—i)
+ sim
(00)
for
all k
(cc)
(co)
sim g1 >
(k) for all k
(k—i) ÷ sim
COROLLARY 8.2.
>
(cc)
sim
>
1.
2.
1 and all k
for all j
1.
s—i
If
(iv)
k
k
and rank
<
e.j
rank
j=s,s+l,...,k—i, then there exists a non-negative integer T sLi all k
k—i
If
(v)
T
=
then
then
S(q1
[S
k—i
(T
<
for
such that (co)
j1
for
> 1.
(vi)
If k > 1,
(cc)
(oc)
D
Nk_i(It)
is defined by (7.9)). PROOF.
(i)
Since k(k-l) =
(k—i)k, both operators act on the same
finite dimensionai space. On the other hand, (k)
rank
]
j =k max(k—i—j,0}
(k-i)inax{k--j,0}=rank
(k—i)
j
(j=l,2,...,k—i). Now the resuit follows from Lemma 2.5. This can be proved by using the same arguments as above.
(ii)
By (i),
(iii)
(ii), Lemma 2.5 and an obvious inductive argument,
we have (cc)
k
(j)
(cc)
(cc)
+
sim
(cc)
÷ q sim
(k)1(O0)5q(00) = (k—i) (cc)
=q
+ q1 (j) sim
(cc)
(cc)
(k—2)
(k—i)
(cc)
(co)
sim
(cc)
(iv)
rank
for all j=s,s+l,...,k-l, it is notdif ictjsT ficuit to find non-negative integers e, t such that k If rank
By Lemma 2.5, q(Tj)]+
A
q
(e)
(a)5[5k
fortiori, T sim q(CL)GA.
(v) 192
By (iii), q1
(00)
Sq k
÷ sim
[5 j=k
g
(tk)],
(cc)
and, by Lemma 2.
q(Tj) (j
q1
(oo)
(no) .
(vi)
= 1,2,... ,k—l)
k—i
Hence,
(T.)
,
()
q(Tj)
so that
#q1
]eq(Oo)+
(oe)
By Corollary 7.21,
c
Let
.
A
Nk(H)\Nk(H) and let a (1 < s s k) be the largest exponent such that
If s=k, then it follows from Corollary 7.27 that for a sequence in Nk(H) such that
÷ =)
T
that
0
(n
it readily follows
#q1
A fortiori, A If s < k, then it follows from Theorem 7.15 and Corollary 7.12
that IA- Sn!1
+0
(a.) flJ
k
=) for
(n
where
,
a sequence {Sn}ni in Nk(K) such that By (v) and (iii),
and
qj(O
Proceeding exactly as in the first case, we
which is sirctilar to
conclude that q
-t
k
1
A.
sim
+
Since the inclusion Nk_l(H) c Nk(tfl\Uk(fl) is trivial, we are done. 0
PROPOSITION 8.3. The quasinilpotent operator Q
L(H) is a univer
sal quasinilpotent (and therefore c
a(A) and
(J3QT);
oe(A) are
connected and (8.2)
contain the origin)) if
and
OnZ!/ if
is not a nilpotent.
More over,
(def) c
{Q L(I1):
LU-I):
Q
is a universal quas-inilpotent}
uk)J (8.3)
Q is
in LU-I), dense in the set QjfI) of all quasinilpotent operators.
is a
PROOF. It is completely apparent that if Q 1[Wk(AUIJ is also contained in the (closed) set some k 1, then .S(Q) (H) J
,
so that the condition is necessary.
Assume that Q is quasinilpotent and
for
all k=1,2
By
for somefaith
Voiculescu's theorem (Proposition 4.21(iii)), if
ful
1] for
(co)
-
unital *_representatxon p of C*(Q), then QWR A fortioU(Q) and it is completely apparent that 0(R) ={0} and ri, Q sIn 0 for all k=l,2 By Corollary 3.35(u) and Lemma 8.1, .
193
for each k, k=2,3 S(Q)
Hence, by Corollary 8.2(vi) Uk:2 Nk_l(H) =N(KJ.
Uk:2
(Theorem 5.1), we see that S(Q) Since Q s W(H) we obtain formulas (8.2) and (8.3).
=
U(t{),
whence
It is completely apparent that is dense in Q(H). Namely, if V is the Volterra operator, defined by Vf(x) =fX f(t)dtinL2([Ol] dx),
then
V is a compact quasinilpotent and
c Q(K) and
It readily follows that
0 for all k=l,2 is dense in Q.(K).
in L(H). Since n1[Nk(A(K))] is a is each k=1,2,..., it follows that
By Corollary 1.3(u), Q(H) is a
for
closed subset of L(K) also a
8.1.2 Universal compact quasinilpotent operators Let K s
K(fl)
be a compact quasinilpotent operator such that Kk
O for all k=l,2 By Lemma 8.1, K is a universal operator forthe class of all compact quasinilpotents. By using the same arguments as in the last part of the previous proof, we obtain the following PROPOSITION 8.4. The compact quasinilpotent operator K
L(H) is
a universal compact quasinilpotent (and therefore
S(K)N(fl)nK(tI)
C
a nilpotent.
K
The set of all universal compact quasinilpotents is a
in L(H),
dense in the set of all compact quasinilpotent operators.
0.2 Compact perturbations of not nice nilpotents Assume that T but
L(fIJ
is
an essential nilpotent of order k (i.e.,
0) such that +
If hA
0
+
is not invertible. then
+0
(n +
and
)
it
follows from Corollaries 1.8 and 7.21 that A is an essential nilpotent of order at most k such that S(T)
c
is not invertible; therefore
(8.4)
This observation reduces the problem of characterizing S(T) 194
to
that of characterizing those operators in the second member of (8.4)
that can be uniformly approximated by similarities of T. Since Oe(A) =
{0} for all A in S(T), it is not difficult to see (use Corollary 1.6) that a(A) =o(T) and dim =dim H(X;T) for all A o(T); furthermore, TItI(A;T)
for
sTm AII1(A;A)
each A in (1(T)
(use Corollaries 1.8
and 2.8).
By using these results and Corollary 3.22, the problem of characcan be reduced to consider two cases: a) the case when T is a quasinilootent, and b) the case when 0(T) is the union of {0) terizing S(T)
and a sequence of normal eigenvalues converging to 0. This will be done in two propositions. L(H) and as8ume that Q is quasinilpotent, is not invertible; then
PROPOSITION 8.5. Let Q and S(Q)
0,
=
fA
but QjI1)
rank
rank
PROOF. Let R=
=0
:
(Q)
for
is not invertible,
for
=—,
k}.
all j
some faithful unital *_representation p of
then we have Q stm
stm
F0q1 1
sim
sim
k—2
•1
+
sim
0
where F is any Jordan nilpotent acting on a finite dimensional space
such that rank F3
rank
Q3,
for
k.
all j
(To see this, apply:
1)
Proposition 4.21(u); 2) Lemmas 7.9 and 7.8; 3) Lemmas 7.9, 7.8 and 8.1; 4) Corollary 8.2(u); 5) Corollary 8.2(i), etc.) Assume that A isa quasinilpotent such that is not invertible and rank A3 rank Q3 k) be the exponent such that
=—,
for
all j
>
k,
and let t (1
t
Atl
0. If either t>l and A +A* is not invertible or t=l (i.e., A is compact), then it follows from Corollaries 7.6 and 7.27 that A can be uniformly approximated by a sequence fAn}n:i c NIH) such that A0 (for suitably chosen operators Fn acting on finite dimensional spaces, such that rank F03 rank for all j k; n=l,2,...). As in the proof of Proposition 8.3, we conclude that Q stm A. If t > 1 and At_l+A* is invertible, then t < k, A is similar to a compact perturbation of (for some h, 0 h < t; use Proposition 7.22). By Corollary 7.6, A can be uniformly approximated by a sequence
chosen operators
c
but
such that Bn
(for suitably
acting on finite dimensional spaces, such that 195
rank that Q
rank
t
QJ,
k; n=l,2,...).
for all j
conclude as above
A.
S3ifl
Conversely,
if Q sTm A, then it follows from our previous observations (at the beginning of this section) and Corollary 1.8 that A is
quasinilpotent, A =0 and A PROPOSITION 8.6. Let T
L(tI) and
e
0 for
all n,
={0)u{A m and Am if n
An
the set of all those A
invertible, a(A) for
0,
infinite set
i8 an
0, as n +
); then
Lull such that =dim
=o(T), dim
all n
0
assume that
not invertible and a(T)
is
(where S(T) is
+A* cannot be invertible.
and
stm
=1,2
PROOF. Let p be a faithful unital *...representatjon of
By Propositions 4.21(u) and 8.5, T
let
and
T$R('°)sTm
By Corollaries 3.22 and 3.35(u),
Let
where Rn=An+IAnIJn and 3 potent operator {IAnI:
not
is
T
is the Jordan form of the niland
(n =l,2,...),a(T )
<
2
max
n > rn]. By taking suitable limits, we conclude that
s =q1
T stm
Rn]etOn;+i AfliM ], where
Clearly, S
of
denotes the identity on a space
(n=m+1,m+2,...).
dimension equal todim
be a unit vector in
Let
j
then in
(co)
Sm
q1 1
1
m+p-l
Am+p
whereFin
.1
=1
Since An +
Mm+j e 0
and
V [em+j }
< 2IAnI + 0
Hence, S
and Proposition 2.33, S 196
-*
=),
it
is easily seen that
of operators such that
tnere exists a sequence
rn=1,2,...) and
(n
0
(for all
(m
for all p =1,2
By Lemma 2.5
where Ku is a universal compact
quasini ipotent.
Assume that T sim A. It follows from our observations at the beginning of this section that: (1) Oe(A) =Oe(T) ={o), 0(A) =0(T) and dim for all n=l,2,...; (2)
for each n, n=l,2,...; (3) and is not invertible. Suppose that A E L(H) satisfies (1), (2) and (3) and let B=p1(A) for some faithful unital *_representatjon p1 of C*(A). Then B
k-l
On
+B* is not invertible and (by Proposition 8.5) q1(
the other hand, by Proposition 4.21(u), Thus, given
provided
IICU
0
> <
A.
there exists W invertible such that
c/2.
—B('°)jj
By continuity,
<
Let
<
A11 A12
.
A22
.
Al
Bl N1 B2 N2
.
'BN
o
N,,
o
and let
where N1=H(A1;A) and A11 A12
.
Aim Aim+l•
A22
.
A2,m+l•
A mm
.
.
N1
.
.
m,m+l
m
mn
A
nn
-A
n
n
N ni
i
n
o
It is easily seen that n > m}
Cm
is a normal compact operator, 0
(m + =)
and
A11 A12
.
Aim
A22
-
A2
.
197
is quasinilpotent,
where
is not invertible. By =0 and A fortiori (use properties (1) and
Proposition 8.5, (2)),
m [0n=1
RIOKO n u -k
("o)
sim
C
m
Choose m large enough to guarantee that ljA_Cmlj <
then
and there exist V1 and V2 invertible such that V2
<
W and
-
=
maxflIVlUOnmi
maxfllv1uom1 IIW(q1
(co)
(co)
)W m
Since UIn=l
—l
—B (a') (a')
+
c/2}
}0iq1 (a,)
<
=
(a,)
0(Onm+i
and c
can be chosen arbitrarily small, we conclude that T srm S stm REMARK
8.7. By Lemma
2.5
stm AOB
A.
0
and Remark 2.6, the condition
can be replaced by ttnul j
sTiu
nul
(Xn_T)J for all
=1,2,... ,dim
8.3
Quasinilpotents in the Calkin algebra
8.3.1 General quasinilpotents
L(tf)
Now assume that Q that
0 and
is
an essential nilpotent operator such is invertible. Proceeding as in the
first part of the proof of Proposition 8.5, we can find R such Hence, that QOR so that stm Combining this fact (7.9) and Propositions 7.22 and 8.5, we conclude that if Q
k
2
is
and 0
similar to a compact perturbation of h
<
k,
then S(Q)
?(Q) 198
where
is the disjoint union of
(8.5)
which is open and dense in S(Q). As a first step toward a description of the structure of s4(Q), we shall consider similarity orbits in the Calkin algebra. Let us re-
mark from the beginning that we have two kinds of similarities, namely
G(A(HJ):
those corresponding to invertible elements in ir[G(ff)]=
md W=O} and those of arbitrary index. Accordingly, we shall write A A(HJ if there exist invertible elements in s—im A for A(H) such that md for all n=l,2,... and + 0 (n and, respectively, A there exist invertible elements + ,
if
G(A(H)) such that
+ 0
LEMMA 8.8. Let T and
(n + co).
be two faithful unital 'i-homomorphisms of
L(&') into A(H); then (i)
There exists a unitary element tJ
A(H) such that tYT(X) =
T1(X)U for all X (ii)
There exists an integer h,
{ind W: (iii)
h
0
k—], such that
G(A(HH}=h+k7Z. There exists an integer h,
0
h
k—i, such that
t1(X)L for all X
{ind W:
PROOF. (i) Let be a partial isometry (i.e. such that W*W=T(e10e1), Then
is a
is a unitary element of A(H) and tir(X) =r1(X)tl for all X in (ii)
It is clearly sufficient to prove that
{ind V:
G(A(tl)),
commutes with
=k7Z.
Furthermore, it follows from (i) that T can be replaced by any because of other faithful unital *homomorphism of into the invariance of the index under conjugation by unitary elements of A(H). Thus, we can assume that there exists a unitary operator ,i(U(Xel)U*). If commutes with it is easiH such that T(X) ly seen from the matrix form of that L(I1))
and, since the diagonal part of the triangular operator matrix
it follows that l€køA0 is Fredhoim and md
is equal to
md
k—i
k (md A) (mod k). 0 L(H) a Fredholm operator of On the other hand, given m and A and of G(A(H)) commutes with index m, the element
V =
=
md
A)
(iii) show that
Use Ci)
once again (as in the proof of (ii)). We have to
{ind V:
commutes with t(X) for all X
G(A1II)),
Now it is easily seen that
in case t(X)
if and only if
commutes with is invertible if
and hence,
and only if A is Fredholm and md V=k(ind A).
Let
where h = dim
0
k-i (k
2) and
K
for j=l,2,...,k. Let
01 01
H2
0
0
ThO
(8.6)
K0 0
0 1
h
0
then 0 Sh
01 0
0 .
Th= 0
0 1 0
where
is
+
an isometry of corank h. By Lemma 7.23,
Tk_l in A(HJ; indeed, straightforward computations show is unitary
T2
that
in A(H) and ind(Shelwlo...Sl) =-h, h=0,i,2,...,k—1. It is easily seen that =Th uniquely determines a faithful unitai *_homomorphjsm from
into AIK) for each h, h=0,1,2,...,k-]..
e G(A(UH, md V=n such that
However, by Lemma 8.8, there is no
unless h-r-n % 0
(mod k), then
Given k
0
is"far"from
(mod k). Furthermore: (Corollary
If h—r-n below)
2, define T is similar to a compact perturbation of Th}
(h0,l,2,...,k—l) and Nk,h(A(II))
By (7.9), 200
(7.8), Proposition 7.22 and Lemma 8.6,
is the disjoint union of the sets
resp), h=0,l,
2,...,k—i.
identity 1
LEMMA 8.9. Let C be a Ct—algebra with -,
be a faithful unital *_homomorphism
C
and
let
Assume that x c Nk(C) and let
S = Then
xs and
=
us—lu
PROOF. Indeed,
k
k—i
j—1
j =sx,
k—i
=
and, proceeding as in the proof of Proposition 1.10, r(e30e1)x j-i
k
lls—1l1=
have
we
k
T
T
T
lxii' LEMMA
8.10.
If
(i)
Let k
>
2
and
then
j
h (mod k), then
j
G(A(H)) and md
If
(ii)
j
/
h
(mod k), then
G(A(HH and md
(iii)
=0.
are
The subsets
open in
(A(H)L
Indeed,
these subsets are the components of PROOF. By our previous observations, it only remains to show that
Nkh(A(KH is connected and open in Nk(A(HJ). The connectedness is an immediate consequence of (i), (ii) and the fact that G(A(K)) : md —-4-
—-4-
V=-h} is connected for each h, h=0,i,2,...,k-i. (4(H)) and
Let 0
(n
Lemma
(for
md
V=-h, be such that
oo). Then we have lim(n -*
8.9,
there exist elements
and therefore, by c
G(A(H)) such
that
111 ÷ 0 (n • =). Hence, for n large invertible, md S=O, and
all n1,2,...) and iiSn
is
enough,
xn
so that
=
X
Therefore,
is open in
.
0
201
From Lemma 8.10, Corollary 7.21 and comments preceeding it, we immediately obtain the following COROLLARY 8.11. (i)
Nk(A(HJ} and
If
<
(ii) If
0
md V = -h then
=
h
(0
e
and h—r
Nk,hA(KH.
(mod k). then
-
(I 1W Il.l1W'll)I.
I)
11.1
PROPOSITION 8.12. Let Q be a quasinilpotent element of A(H) and
let A
A(U); then
E
(i) If
and
Q
s4im A
(ii)
w-im A <=>
If Q
Ukl
0
h
k—i, then
Wk,jMUI)J e
NkIA(KH.
<=>
Ac U(A(flH.
We have
(i)
s—im A
0
Nk(A(K1), then
w4imA
<> PROOF.
2, and h,
A
A <=> A
If Q
then
A
for some k
A <>
(iii)
0,
>
Q
w4im A
A
>
(The last implication follows from the fact that the non—invertibility for X
of
Nk(A(H))) is a similarity invariant; see Lemma 7.20.)
On the other hand, if A
then (Theorem 7.2)
QJtII such that Q0k=A0k=O, A. By Proposition 8.5, Q0 stmA0• A fortiori, there exists Q0,
X.
s4im
0 < h
Assume that
(ii)
k—i. By Lemma 8.10,
{WQW1: We G(A(KH, md and V E
G(A(KH}
k-i,
0 r and r On the other hand, if A it readily follows from Lemma 8.10 (or Corollary 8.11(u)) that
h,then
0(4(H)) such that md W = 0.
All is bounded away from zero for all * Hence,
It
s-tim A. is clear that
s4im
w4im
>
A
and, by our previous observations and Theorem 7.2, it readily follows 202
Wk(A)Nk)).
A for each A
that Q
It is completely apparent (from Theorem 5.34) that
(iii)
s4im A
w4im
A => A
Conversely, if A N(A(tflJ then (by Proposition 3.45) there ex=A, is a guasinilpo— L(H) such that ir(Q0)
ists Q0, A0 =
It
is easily seen that for all k and (by Theorem 5.1) that A0 e t4(H). By Proposition 8.3, sim A0. A fortiori,
tent and a(A0) =oe(Ao) =a(A).
1,2,...
COROLLARY 8.13. (i) ion of the sets
2, Nk(A(H)) is the disjoint un-
For each k
Nk,k_l
(A(H))
Nkh(A(H))
(ii)
G(A(K)}, md W =
0},
for each h =
0,l,2,...,k—l.
(A(K))
(iii)
for all h=0,l,2,...,k'-1.
(iv)
))kh(AUflI =
Nk,h
for each h, h=0,l,2,...,k—l. As in Chapter II, let and
and iAi
c <
A(H):
means that
A(H): [A]5
T s4im
<
and
s-aim
means that
A
A, respectively. These two rela-
tions are partial orders in the corresponding quotient sets obtained from A(tl)
by using the equivalence relation
and
respectively.
If Q.(A(Hj) denotes the set of all quasinilpotent elements of A(H),then (Q(A(fl))/#5,<) is a lattice and
is a chain. Indeed, it
is not difficult to see (by using Proposition 8.12) that their structures are the ones depicted in Diagram (8.7) (below), where denotes an arbitrary universal quasinilpotent operator.
203
Js
['IT
LII
['IT
0q3
['it
)]
I w
S
(QJA(H) DIAGRAN (8.7)
8.3.2 Nice
elements of the Calkin algebra
In the case when T is similar to a compact perturbation of a nice Jordan operator with more than one point in the essential spectrum, the
analysis of S(T) present some subtlety that cannot be detected with the sole help of Lemma 7.1. The phenomenon will be better explained with an analog of Lemma 8.8 for this case. The proofs of Lemmas 8.14, 204
8.15 and 8.16 below follow the same lines as those of Lemmas 8.8, 8.9 8.10, respectively, and will be omitted.
LEMMA of
8.14.
be two faithful unital *_homomorphisms
Let -r and
into 4(H); then There exists a unitary element U .
(i)
4(11)
such that UT(X) =
-r1(X)U for all X
(ii)
There exists an integer h,
h
0
where
k0—l,
k=GCD{k1,k21...,k},
x1, k21•1
such that if
{ind
are m distinct
complex numbers,
W:
E J
•
J
(iii)
There exists an integer h,
{ind W:
WT(X) =T1(X)W, for all X E
3)
0
< h < k0—1, such that
WEG(A(H))}
8.15. Let C be a with identity 1 and let be a faithful unital i-homomorphism. Assume that A1, A21...
LEMMA L(Q
C
•l Am are m distinct complex numbers and let
nomial p.
C with minimal poly-
Then there exists ss(x) cC such that
p(A)
])s=sx and
J-i.
3'...
J
j=l k.
11sllI is a constant depending only on p.
where
8.16.
LEMMA
Let
be in
distinct
complex numbers and let e
where
then
h
0
LU-I);
k0—l,
Then S(T0) has exactly k0 components:
={WT0W1:
G(A(N))I mw
W
h (mod k)},
h=0,l,2,...,k—l.
8.4 Compact perturbations of nice Jordan operators From our observations (at the beginning of Section 8.3.1) and Proposition 8.12(11), we obtain the following COROLLARY 8.17. bat-ion
of
disjoint union of
(1)
If
Q
LU-I)
where k > and
2
-is similar to a compact perturh k—l, then S(Q) is the
and 0
(defined by (8.5)). 205
is an open dense subset of 3(Q).
(ii)
(iii)
I].(H) = 0 for
JH) and
c I
(iv)
h
0
all
k—i, and
j
h,
0
j
intersect
Wk(Hl vn open sets; furthermore,
is
dense in Nk(H).
is
Let Q be as above. If Q is quasinilpotent, then completely described by Proposition 8.5. If c(Q)
is
a finite set, then Q - QOF, where Q0 is a quasinilpotent and F
acts on a finite dimensional space; in this case
can be
easily described by using Lemma 7.1, Proposition 8.5 and Theorem 2.1.
If cl(Q) ={O}u{A}1, where An + 0
co),
(n
then
0 for all n, An Am if n m, and A also admits a simple description, given
by Proposition 8.6.
3(Q)
On the other hand, it follows from corollary 8.17 that +
nN1.iK)
= S(Q)
+
JU). Since every element of
+
=
JII) is similar
to some operator of the form
K(K),
C
(8.8)
we can directly assume that
Q=q(h)eq(oo)+K
k
2, 0
h s k—i,
K(Il).
1<
(8.9)
Now it is clear that the structure of S(Q)
will be completely determined if we can specify which operators of the form (8.8) belong to S(Q). Furthermore, by our previous observations, we only have to analyze the case when Q is quasinilpotent and the case when 0 is not isolated in c(Q). There are only partial answers to these problems. However, these partial results are sufficient to exhibit the complexity of the general problem. The case when 0 is not an isolated point of
will
be considered in Section 8.45. Sections 8.41 through
8.44
will be devoted to the case when Q is a quasinilpotent. A conjecture will be presented concerning the classification of the closures of sirn ilarity orbits of these operators.
8.4.1 Nice Jordan nilpotents 8.18. Let
be a sequence of nice nilpotent operators, —
and let. o where Fn c Jordan nilpotents. Then the following are equivalent. (i)
There exists operators Wn in GUI)
lIWnQnWn1_QI! + 206
0
such that
(n +
F
are
There exists a natural number p auch that Q
(ii)
sIn Q for all
p;
n
There exist a natural number p and non-negative integers
(iii)
(n ? p) such that
and
PROOF. (iii)
>
=> (iii)
(i)
(ii)
>
and •F. sim qJ( (i) are trivial implications. ff(n)
let
If
subspaces corresponding to q
and F
and
denote the
respectively,
,
denote the orthogonal projection of ft onto K.
and let
j=1,2.
(n)
Similarly, = H10K2, where is the space of and is the space of F, and let P be the orthogonal projection of if onto Ii., j=l,2. Let Q = ,
let
—iJ X
WQW
,
and
=
VQXnV••• vQk
Define Yn=X
()
X:
{x
are pairwise orthogonal and
isometric (for j=O,l,2,...,k—l), the algebraic sum will be closed and direct for all n large enough; moreo-
ver,
II
-*
0
(n
oo).
{0}, it follows that
)
that
QkX
Furthermore,
and
so
and, for n are similar via
and
we can use Lemmas 7.9 and 7.8 in order to
c
find an invariant subspace Rn c (Xn)
111(fl);
has finite codimen—
since
sion, Zn also has finite codimonsion. Since
(dim
c
= {0} and therefore Zn is invariant under
large enough, Q'IZ W
n
x0}.
and
Since the spaces X,
is
—l
ft(n)
<
it
Since
a non-negative integer
such that
of
is easily seen that there exists
such that QnIRn
On the other hand,
reduces Q and therefore there exists a non-negative integer
such
that
OF.
QIHevn=PKey Since
+ 0
unitary operators U
(n-i. co), for all n large enough we can find
L(fI)
such that
0
and U*PZU=Py.
Hence, we have: lim(n + =)
= 0.
IIPifeyQPKev
But n ()
(8.10)
(ci) n OFn•
Hence, (8.10) implies the existence (for n large enough) of invcrtiblc operators
na +r
S:U n
n
n
n3 +r n
207
such that
liin(n
(cx)
(cx)
•F)S
+
•FIV=O.
Now it is easy to derive from this relation that, given j
then
(as)
•F) for all n
1,
(8.11)
n(j)
for some in k, it follows that (8.11) hdds Since for all j=l,2,..., and for all n p. (cx If n p, it follows from Lemma 2.5 that WF •F. sIm )
U
be similar to a nice Jordan nil-
PROPOSITION 8.19. Let T e potent: (t .)
T =
Then
in
iTjdT <
k,
such that
is the set of all A
q.
A =
and there exist non-negative integers
'k'
(8.12)
<
cxk such that
0, klk+dT=kcck+dA, and
min{tk,czk} =
r
m
(8.13)
r
(i-i)
m
I
11
for all r=1,2,...,m—1. Furthermore, there exists a constant
S(T)
= dA
ja1
>
0
such that if
B
and (T.)
3'
B—
<
k,m'
'-
q1(Tj)1
is an open subset of S (T). PROOF. It readily follows from Lemma 2.5 that if A satisfies (8. 12) and there exist 'k' cxk 0 such that (8.13) hold, then then B
In particular, S(T)
T.
dense
(T)
j
j=1 '1j and,
a fortiori, T
A)
(814)
j=1
A.
sum
Conversely, if n}nl in 0(H) , then
(cx.)
+
Sin'
-All ± 0
it
(n
co)
for
a suitable sequence
follows from Lemma 8.15 (with Q
that there exist non-negative integers Tkl cxk
and (8.14) holds. Since,
by Lemma 2.5, this is equivalent to the
validity of the inequalities r
(a.)
q1
it 208
readily follows that
T and Q =
such that
I
r
(T.) <
]
,
r1,2,...,m—1,
ak can be chosen so that min{Tk,cik} = 0.
0 (n + so). Since Assume that {Bn}nl c S(T) and (Corollary 8.17(u)), B must be similar to S(T)
a compact perturbation of T for all n large enough. Furthermore, since rank
<
rank
for all n and rank Tk is finite, it follows from is similar to a nice Jordan nilpotent:
Corollary 7.24 that
m
By Lemma 8.18, there exist a natural number p and non-negative in and such that tegers Tnk (n p) (T.)
m
r
r
in
]}
for all r=l,2,...,m-l, and for all n p. But, on the other hand, the first part of the proof indicates that the reverse inequalities also hold in (8.15), i.e., these inequal p, then ities are actually equalities. It readily follows that if n T for all for all j=l,2,...,k—1,k+l,k+2,...,m. I.e., B From this result and the proof of Lemma 8.18 we deduce the existk and m (m k), such ence of constants about
that the intersection of the open ball (in LIUI) of radius
q(Tj)j J
iJr
with .S(T)
it is easily seen that if
only contains points of S(T). Now m —l — T, then q.'
j
J—
—1
the intersection of the open unit ball of radius
about R with S(T)
is also included in S(T). Hence, S(T) is an open
(and obviously dense) subset of S(T). From Proposition 8.19 and Theorem 2.7, we obtain the following
COROLLARY 8.20. For each k ({T
N1h(H) :
2 and h,
0
h
k—l.
T is a nice Jordan nilpotent}/#,<)
is a lattice. (T.)
If T I
]
and A -.
(
)
in
h(H) are nice Jordan nilpotents, then there exist non—ne-
gative integers
ak such that W(ccd)
If
are the Jordan nilpo—
tents determined by Theorem 2.7 such that ],EC] =
[B] =
q(Tj) ]A[Wjin1
then it readily follows from Proposition 8.17 that [T]vfA] =
and
[T]A[A] =
209
We shall see later that this lattice is not closed under "sup' or "inf" (for denumerable subsets). From this result, Propositions 8.3, 8.4 and 8.5, and Corollary 8.17, we obtain COROLLARY 8.21.
T is a nice Jordan nilpotent}/#,<) is a lattice.
This lattice contains the infimum of each of its nonempty subsets, but it is not closed under "sup".
is
CONJECTURE 8.22. (QJH)/#,<)
a lattice.
We shall say that a similarity orbit S(T) is locally closed if (1) there exists c (in LUll)
such that the intersection of the closed ball
0
>
of
radius c about T with S(T) is closed in LUll.
This notion admits several equivalent formulations; namely:
3(T) there exists e(A)
Given is closed in L(H)
>
0
(3) There exists c
(Take c(A)
such that 8(T,s)n3(T)
c S(T);
its closure. (Take 0
<
(4) Given A
cS(T);
such that 8(A,c(A))nS(T) c
<
(2)
such that 8(A,c(A))nS(T)
to
0
>
3(T) there exists c(A) >0
(5) S(T) is an open dense subset of show that (5)
=>
(1).
The converse
implication is obvious.)
COROLLARY 8.23. (i)
If T is a nice Jordan operator, then 3(T) is
locally closed. (ii)
is
If T is a nice Jordan operator, then
locally closed
in 4(H).
PROOF.
(i)
By using the same arguments as in the proof of Lemma
7.1, we can reduce the problem to the case when
=
{0}.
If rank T
is finite (T acting on a finite or an infinite dimensional space) ,then
the result follows from Lemma 2.5, Corollary 2.8 and Proposition 1.12 then it follows (i). If T is a nice Jordan nilpotent and rank T= from Proposition 8.19. (ii)
This follows from Corollaries
PROPOSITION 8.24.
LUll:
A#T} (ii)
PROOF.
all T
210
(i)
and 8.13.
D
If S(T) is locally closed, then [T] = {A
£
coincides with 3(T).
If (i)
LUll.
is
locally closed, then [TjS(T).
It is completely apparent that 3(T) c [TjcS(T)
for
inSA) such that If A#T, then there exists a sequence If s(T) is loand A e S(T) for all 0 — P for all n large enough. A cally closed, it readily follows that — T (for all n large enough). Therefore, A fortiori, A — S(T). The second statement follows by the same argument.
8.4.2
0
Nilpotents of order 2
then either Tis nice and similar to
If T e W2(H) ,
for some j,
j
0
or
<
eq2
,
or
P is compact. The struc-
ture of the poset (N2(H)/#,<) is the one given by the following diagram:
Eq1
3
[q
(co)
eq Eq
]
(6)
eq
('o) ]
1
[q1
(oo)
(00)
q2
[qi (oo)
(3)
[q1(°')
(2)
[03
DIAGRAM (8.16)
211
8.4.3 Quasinilpotent perturbations The next lemma provides a device to construct quasinilpotent oper ators T such that
8.25.
(k
Let
operators 8UCh that
stm and a quasinilpotent T
G(HJ
and
L(ffJ
2; n=l,2,...) be quasinilpotent for all n=l,2 Then there exists such that
-*
0
>
be such that
We shall recurrently construct
c.
1T1—AII
is invertible for any A e n
where S, R. (:1) are invertible operators and S11, c1c/2 and for all n > 1, we have:
£11? 1)
2)
0
S(T1.
PROOF. Let c
with
2).
k
and
>
j—
j
0, such that
<
IS nTnSn
n—lTn—l n—l
S _lII
e
n—i
/2; <
>
<
This
<
stm T1, it is easily
will be done as follows. Since
seen that we can find S choose
(j=l,2,..,n-1)
1/n
e G(H) satisfying 2)
(n=2,3,...). Next we
such that 3
—l
T S nnn
3
—T.II < 3
1/n,
3
j=i,2....,n-l. Clearly (by continuity), we can find
such
that 1)
and 3) are satisfied. is a Cauchy sequence in
Then its limit. Since IIT—T111
c, it follows that
<
vertible. It is also clear that =
lii
(0}; moreover,
for
j
3
Hence
T
1
let T denote
<
t
<
I
so that (by using 3)
— T.II < i/fl.
3
T. for all j=l,2
0
j
Some extra notation will be needed. If
T
n+lsim n c G(KJ such
and T
that
sint
T
n
is a sequence of
k-l) such that n=l,2,..., and there exists {W nn=l 0 (n ÷ =), we shall write: h
quasinilpotent operators in T
is in-
and (by Corollary 1.2(u))
for all +
<
}
T S
The II
following result summarizes the most immediate properties of
+
S mm
PROPOSITION 8.26. (i) 212
Given a quasinilpotent R in
exist a sequence of nice Jordan nilpotents quence
in GUI) such that
0
co) and
(n
for all n1,2
T
For every sequence
(ii)
sim
T
-* n+1 sim Tn sim
such that T
+ T,
fT
2 sim
1
then
S(T)=
(iv)
T
$ T.
rf T
(iii)
aim
there exists a quasinilpotent T
of quasinilpotents in (H)
and a se
Tn+T
.
(8.17)
fora sequence
many n's, then S(T) is not locally Tn+l for Tn closed. This holds, in particular, if is similar to a nice Jordan nilpotent for all n, but T is not similar to a nice Jordan nilpotent. If
(v)
PROOF.
+ T and A sin (ii)
T
for all n1,2,...., then A stm T.
is an immediate consequence of Lemma 8.25 and (iii),
(v) and (vi) follow trivially from the definition of " S im From Corollary 7.6, Proposition 7.22 and Corollary 7.24, c there exists a sequence h(H) of operators similar to nice Jordan nilpotents such that IR_RnII 0 (n +
(i)
Define T1=R1. By Corollary 8.20, there exists a nice Jordan potent T
such that [TJ= [R11v[R2 ]v...v[R
t ly seen that T n+1 sim T n sim Rn '-
(n=2,3,...). It is easi(n=l,2,...), whence the result fol— I
lows. (iv)
Observe that if Tn
T
the Tn'S• This is true, in particular, if
Tn is similar to a nice Jordan nilpotent, but T does not form. Clearly, 3(T) cannot be locally closed.
that 0
REMARK 8.27. Observe that Proposition 8.26(i) and (ii) guarantee that, if R e (T
n n— —1.
is a quasinilpotent, then there exists a sequence of nice Jordan operators in T for satisfying T ,
quasinilpotent T in
h(H) such that T aim R. However, this proposi-
tion does not guarantee that the Tn'S can be chosen so that T#R, i.e., R. This is, precisely, the main gap to be filled in order to comim pletely determine the closures of these similarity orbits. T
S
+
Nkh(H) is similar to a nice Jordan nilpotent, then
acts on a space of finite di
213
mension equal to
for some r
0. Define the number
Stkm(T), the m—th stable kernel of T, by
(m=12,)(818) The following assertions are easy consequences of Lemmas 2.5 and 8.1
and Corollary 7.24:
does not depend on the particular operator
a)
similar to T. Hence, it is a well-defined similarity invariant; are similar to nice Jordan operators,then
If T1, T2
b)
for all m=l,2,...;
T1 slm T2 if and only if Stknt(T1)
e Nk,h(ii) are similar to nice Jordan nilpotents and
If T,
c)
÷0
co), then
(n
cc)
lint sup(n +
for all m=1,2,...
Now we shall extend the definition of m-th stable kernel for an arbitrary quasinilpotent T in Stkm(T) =
inf
(sup{Stkm(T'):
is similar to a
T'
(8.19)
nice Jordan nilpotent,
<
By c), it is easily seen that if T is actually similar to a nice
Jordan nilpotent, then the new definition (8.19) coincides with the previous one (8.18). It is also clear that this extended m—th stable kernel is a similarity invariant. PROPOSITION 8.28. Let T,
R,
be quasinilpotent
e
operators and a8sume that Tnj Rn are similar to nice Jordan nilpotents
for all n1,2,...; then (i)
+ 0
If lint
(ii)
(iii) (iv)
If T
If T
sup(n
(n +
°°), then
Stk"(T), for all m=l,2
cc)
R, then Sthm(T) + T, then Stkm(T) =inf n nsim
sim
for all m=l,2,...
and only if Stknt(T)
T
T
If
for all m1,2
sim
Stknt(R) for all ml,2 PROOF. (i) follows immediately from c) and (ii) is a consequence of (i) (iii)
Since T slm T
for all n, it follows that Stknt(T)
(T) (n=l,2,...; use (ii)). Since T is the limit similarities of the T's, we have lim sup(n and 214
cc)
Stkm(T)
of a sequence of
(m=1,2...)
is non—increasing because Tn+l sim Tn for each n=l,2,...
(m=l,2,...).
Hence, Stkm(T) =inf Stkm(T )
R, it follows from (ii) that Stkm(T)
If T
(iv)
=
(m
1,2,...). Conversely, if Stkm(T)
nilpotent of order
for all m=1,2,..., and R
<
Stkm(Rn) (m=l,2,...) and to show that there exists
for 1
(T5)
this guarantees that T ...). Since R
is a
then we can use (iii) to see that Since
at
S
+
sini
R
n
such that
is a nilpotent of order
and therefore T
R, it readily follows that T nsjm
R (for all n=l,2, n
t
sin
R.
+
slat
0
It follows from Proposition 8.28 that the conditions Stkat(T) Stkat(R)
Rand, in the case
(m=l2,...) are necessary to have T
when T
of t T and R t R for suitable sequences {T } and tR } n n=l n n=l nsim nice Jordan nilpotents, these conditions are also sufficient. So it is
natural to raise the following are quasinilpotent operators CONJECTURE 8.29. If T, R E and Stkm(T) < R. Moreover,given for all m=l,2,...., then T +
T as above there exists a sequence {Tfl}nl c
of nice Jordan
operators such that Tn + T. S im
An affirmative answer (together with the previous results) could be regarded as a complete solution to the problem of characterizing the closures of the similarity orbits of quasinilpotent operators. By Proposition 8.26, an affirmative answer to this conjecture would also provide an affirmative answer to Conjecture 8.22:
is a denumerable family of quasinilpotents in 1,2,..., then we can define T
and P
1 Am,m=
be [Tn]= VtITiji1
where [TJ=
T E sim
T
Indeed, if
Another consequence would be that S(T)
(T as above) cannot be lo-
cally closed unless P is similar to a nice Jordan nilpotent.
8.4.4 Universal operators in Let T E Nk,h(H) be a quasinilpotent operator. If T is not a nilpo tent, then it is easily seen that rank T3=oo-
j
for
all j
k. If T =0,
for j=k,k+1,...,r-l and rank is finite for r rank t-l, then it follows from Lemmas 7.9 and 7.8 that T —
215
q(Tj)], where d=
j
=k,k+l,
k+2,...,r—l; moreover, by (2.2),
T3=rank
0, for j=r+l,r+2,...,t.
1—2rank
(A finite sequence of finite ranks of operators will be called compati ble if the above conditions are fulfilled.) It is also clear that R
huH0) ,
where
is a subspace of co—
H
k-i, satisfies d+h' 0 < h' h (mod k). The following results provide some insight about the structure of
dimension d and h', the poset
PROPOSITION 8.30.
(i)
Given k
a compatible finite sequence
and
such that
(H),
rank T3
for 1 for k
=
k—l
j
<
m., for r < and T
k+l
2, h (0 h k—l), r > of ranks, there exists
j
r—l
j
< t—l
(8.20)
jjr t—l
+
is a universal operator for the class
,
{rank A3}
N4j11) nN1
(ii)
satisfies (8.20) }. +
Besides the maximal element [Tab
{A
t—l
whose supremum is
order-isomorphic to
contains a chain
) =
equal to [Ta]. (iii)
If k > 2 and r
k+2, then
con-
and
tains at least two "parallel" chains isomorphic to
The supremum of each chain is [Ta], Tm
if and only if m
any pair (m,n) and Tm 2 sim
2
for
for the case
(This partial structure of when k > 2 and r
Tn
1
n.
k+2 is depicted in Diagram (8.21).)
[Ta]
,,_[T2121 1T
1,2 I
I
I
IT'
0,1
DIAGRAM (8.21)
We shall need the following auxiliary result. 216
..—1o—
IT
u
I
I
r
IT u
I
LEMMA 8.31. (i)
2,
If K is compact, k
01 0
0
k,
p
1
cind
if1
1
000.
H 0
000. Qk(K;p) =
K
00
.
.01 .
0
tIp_i
01
(8.22)
,
000..000i
11+1
000. .000. Qk(C;p) for
then Qk(K;p) sTm
(ii)
k,
0
with
K
If p <
fined by (8.22)
C. each C such that K j < k—p—i and Qk(K;p,j) is the operator dein the (p,1)-entry replaced by 0 and the 0 in
the (p+j,j+1)-entry replaced by K, and K
is
quasinilpotent, then
p,j) #
(iii)
for
If K5 is a compact nilpotent of order s j=1,2,...,s—1., (namely,
2
such
rank
that
then
and k+p(s—1)
0, for m
for
stkm(Qk(Ks;p)) =
0, for 1 (In particular, Stkm(Qk(Ks;k))
(iv)
• n=l ce
1
—q) n n
If K
is
—ce
k+p(s—l) k—p.
k—p < m
for
in
1
(8.23)
<
ks.)
m <
a universal compact quasinilpotent (namely, K
U
=
then
Qk(KU;p) and in
1_ce, for
Stk (Qk(KU;p)) = (In particular, Stkm(Qk(Ku;k)) =.. PROOF.
If W E
for
for
in
> k—p
(8.24) 1
all in
G(H1), then W' 'Qk(K;p)W'
1.) 1
;p).Now the result follows by taking limits as W runs over suitable sequences (i)
'
in G(111)
Similarly, K
C implies QkU(;Pli)
Since Sim K is a compact quasinilpotent, it follows from Propositi3n 8.5 that (ii)
217
is
S(K)
uniquely
tains. If
determined by the finite rank nilpotents that it con
m
] is a finite rank nilpotent, then Qk(F;p;j)
and a straightforward analysis of these operators shows that (8.25)
for 0
for 0
Hence, Qk(F;p,j) follows.
<
j
k-p-i, whence the result (n)
The formula
(iii)
j
is an
/ S im
obvious consequence of (i) and (8.25). Now (8.23) follows from Proposi tion 8.28(iii) and some straightforward computations. This follows by the same arguments as in (iii).
(iv)
PROOF OF PROPOSITION 8.30. the operators in
(t.)
J=r+l q.
,
to 0) and R
By our previous observations, all
(i)
have the form T
where F =
(j=r+l,r+2,...,t;
is definedequal
j j—r
'
0
Nk,h,(H';r;ø).
By Lemma 2.5 (see also the proof of Corollary 8.2(i), there exists a sequence
q(k) and
such that
in
+0 (n÷oo). Define Ta— It is completely apparent from this definition (and Lemma 2.5) (kfl)o.q (h').aF
that
r
and therefore, by Proposition 8.26(iii),
+
sim
S(T)
(8.26)
Let A be an arbitrary operator in
h(H;r;{mJ}j=r). By Proposiof operators similar to
tion 8.26(i) there exists a sequence nice Jordan nilpotents such that IIA-A.Il-*
1,2,...) and rank =
A)
A.+i
Clearly, =r. for j=r+i,r+2,...,t
(a..) j,
(for all i
where i
)
such that
t.
t
q
.
r+1
and <
for all p=l,2,...,t—l. (Observe that 218
(i=
h (mod k),
and a.k=O for all i1, r ]P 1 for all p=l,
(k) Fix i i0. Since 2,...,r—l, it is not difficult to see that there exist n. 2
0
stm A.
rank A3 for all j.=l,2,...,t—l, and for all i
1,2
a.. 13
(i +
0
1 and
2.5,
By Lemma
it follows that
By (8.26), we conclude that T
fortiori,
that T
+
A.
si.m
i
A. (for all i=l,2,...) and, a
U SlUt
1
u sim is universal for
Hence
A
}t_l j
Define
(ii)
T01
q1
(h')
•FGQk(Kk+l;l)I
where Qk(Kr_k+l;l) is the operator given by (8.22) with Xr_k+l as in (iii) of Lemma 8.31 and (nk)
fT0
T
,
if
n,l
n > 0,
if n < 0.
By Lemma 8.3l(iii), T
T
T
n
T
U aim n+1, that T T n,l sim n+ 1,1 for any n. Formula (8.23) implies that
and
(n
It
e
only remains to show
In(r_k)(k_l), if n
_ck_l(F) —
ink, if
0,
(8.27) <
(where ck_l(F) is a constant that only depends on F).
Now the result follows from Proposition 8.28(iv). (iii)
Assume that k > 2 and r—k
2. Define T n,l as in (ii) and
TfllOQk(K2;2).
By Lemma 8.31(iii), T.Oq 1
•q
r
(n
+
0)
sim
and (i)
sim
2
+
Since that
+ T sim
aim T
n+1,2 sim
T
t
n,2 aim
T
n,l
< 0).
(2)
Lemma 2.5), we can see for each n
for all n e Since (use (8.23)) from (8.27) and Proposition 8.28(u) that T T m,l sint n, of integers m, n.
it follows 2
for any pair
By (8.23) (n(r—k) (k—2) ,
n,l
)
if
n
0,
=ck..2(F) -
n < 0
(where ck_,(F) is a constant that only depends on F). By Proposition
8.28(u) 219
Tni
Tn_12
for any n
On the other hand, since
72.
>
1
0 for all
n, Proposition 8.28(u) also implied that
Tn2 8.32.
REMARKS
for any n
C]
Let k, h, r and
(i)
be as in Proposition
8.30. The same argument of the above proof can be used to show that actually contains s "parallel chains"
jj—r
E 72), order-isomorphic to 72, where s =niax
{[Tnsl}
{r—k,k—l). The suprernum of each chain is equal to [Tu] and Tns stm
and only if n > n' and s s'. The chains of Proposition 8.30(u) and (iii) are not maximal (except for the case (ii) when k=2): If k 3 and n 2, then if
(ii)
stm
sTm
sim
and none of these arrows
sim
can be reversed. However, (iii)
1 stm
sTm
#
Besides these chains,
contains
many other elements; namely, I and
I
elements of
are two
The same phenomena appear in the case when we consider quasinilpotent non-nilpotent operators. The proof of the following proposition follows exactly the same lines as the one given for Proposition 8.30 and will be omitted. PROPOSITION 8. 33. Given k
2 and h,
h
0
k—i, the operator
is universal. for the class wkh(I•I,00)
={A
rank
e
for all.
k}.
and satisfies
for
all m=l,2
If
(n > 0)
and
(n < 0), p=l,2,...,k—l, then
sthm(T
if in )
=
k-p
—ra(n+l), if 1
—n(k—m), if 1 and
220
stm Tn,p if
and only if n'
m
m n
<
k—p
and n
0
<
k—p
and n <
0
and
P'
P.
is
(This partial structure of
depicted in Dia—
gram 8.28.)
ET1)
—t!Z,
(Ta]
.1
I
U
IT] IT
IT —2,1
]
-l DIAGRAM (8.28)
8.4.5Ageneral criterion for universality All universal elements (for the above studied auasinilpotent clas-
for 1 m k, and this seems to be the"ab— ses) satisfy characterization of universality. It will be shown that, if T then T is a universal operator for its own
if TsTm
class
if and only
In order to make it more precise, we shall estab-
lish the result (and proof) as three separate propositions. In the case when T is a quasinilpotent, this result provides some support to the above conjecture about universality. We shall need two auxiliary results.
221
LEMMA 8.34. Let
If
so(T)
<
(k
0),
(for some
e
0
2,
k—i, K Kill)). GIN) and m=m(e)
h
then there exist
only
<2€,
such that
depend8 on
=maxflj(X—T)
lxi =r)
II:
a).
(r >
PROOF. Let Pm denote the orthogonal projection of li onto the f ig1(h)
and the firstmcopies
nite dimensional subspace corresponding to then
of
A
(m)
A
11 (m)
=ran Pm m
Since
N
m
+A22
21
where li
(m)
12
it
m +
m
readily follows that ÷ 0
(m
and (by the upper semicontinuity of the spectral radius; Corollary 1.2 Cl)) SP(A11(m)) < (3/2)c for all in > m0(c). Furthermore, it is also clear that
m1(c)
>
l(X
< 24(r)
_All(m))
for all r >
(3/2)c
and all m >
rn0(c).
By Rota's construction (see (3.5) in Remark 3.34) there exist invertible operators Vm
such that
G(Hm)
<
GIN) by W
Define
—l — in
m
<
M, where M only
(Clearly, we can
depends on •((3/2)a)
A21
(3/2)a. then
H =V,
VA m 11
-
<
(m)
m (in)
V
-l
m 12 (a')
+A22
Cm)
-q1 (h+km)
,
•
(3/2)c+M1l2Tlm < en), we can find m(e) > m1(a) such (m +
where E(m;c) is a compact operator such that ilE(m;c)li (in
> m1(e)).
that
Since <
0
c/2 for all m It is completely apparent that
Define
and m =m(c)
satisfy all our requirements. LEMMA 8.35. Let
U
be the operator defined by
is a Jordan where Rn=Xn+iXniJnI (n1,2,...) and nilpotent acting on a space of (finite) dimension {Afl)nl is a sequence of distinct non-zero complex numbers converging (8.22) with K replaced by •n:l
to 0.
It
is a universal operator for the class is a universal operator for the class where
then
PROOF. By Proposition 8.33 and Lemma 8.31, it is enough to show that 222
(oo)
sTm
for all m=l,2,... Clearly, for all practical purposes we can (and
shall!) assume that h=0, i.e., that the direct summand
is ab-
sent. Define
Ri n n 0
0
0. n 0
01 n 0
(k x
k operator matrix; in denotes the identity on a space of dimen-
j(mod k)}, j=0,l,...,k-l. clearly, itis for fl possible to write •n1 Sn), where E is a finite set} acts on a finite dimensional such that some j = space, Sn acts on a space of finite dimension krn, and sion dn) and let D. =(n:
=0 for all n, n', n", n'
(each Sn is the sum
of
k
Rm'S such that the corresponding dimensions dm are all congruent modulus k).
It is completely apparent that L can be written as L = where L0 = W1 and R0 and are defined in the same way as the operator matrix
This reduces our task to showing that (m =
stm
Let
where
1,2,...).
be a unit vector in the space of Sn such that is the eigenvalue of
of largest modulus (by definition of
for all r=l,2,..., and S
=
n
Lemma 3.36,
if r'
* i-i
0
s1;
sim
and it is straightforward to check that there exists such that 21111. Fix m and let p > 1; then (by Proposition 4.21)
$L0 and
Sn)(n=D+km+i Sb)] 2
q
(k
p+
rn)
223
•p+km
(=)
fl=p+i
—
(,P+km (50 -s'})eo]e[(e P1 n=p+1 n n fl
n=p+l
n
—RS000S...eo})
00
km-fl (co)
p
n=1
n
p+km n=p+1
..S0)S(e n=p+km+1 0)1 1 1
11p+2
0
1
n=p+1
p+km .00)00]0(diag n=p+1 is a finite rank operator such that n
2 maxfsp(S ): n
n > p}
0
(p
Hence,
stm
for all m=l,2
is a universal ooerator for the class
Since
it readily follows that LOT
ua
LSq
(eo)
k
ST
(km)
(km)
(00)
u sim
(m=l,2,...).
sim
The proof is complete now.
0
Nkh(K) and assume that aCT) is the
PROPOSITION 8.36. Let T
union of fO) and a sequence {Afl}nl of infinitely many distinct nonzero complex number8 converging to 0. Then the following are equivalent (i)
S(T) =iA
a(A) =0(T) and dim U(X ;A)
Nkh(K).
n
and
dim U(A ;T) n
T
for
a universal T
224
U
is
for
class
a universal operator for the class
T
(iv)
as in (iii)).
T (vi)
T stm
PROOF.
q(k)eT <=>
(i) =>
(iv)
=>
(iii)
<=>
(v) => (vi) are trivial
implications. On the other hand, it is clear that if T
s'Tm
then (use Proposition 4.21(u)) we also have P sim (m=l,2,...). By using
stm
Lemma 8.31 and Proposition 8.30, we see that sTm T$TU. Since (by Lemma 8.35) we conclude that (ii) - (vi) stm are equivalent statements. Thus, in order to complete the proof it only remains to show that TOQU is a universal operator for the class of T. This will be done in two steps. 1)
Clearly, there exists a unique (up to unitary equivalence)
R;l)
operator
such that a(L) =a(T) and
for all n=l,2 LOQu• Observe that sTm of Lemma 8.35)
It will be shown that TOQU and (with the notation
sTm
stm (=)
q1
" Cd° +k-h) (h) •R00(T00q1
(co)
where T0 =TIIl(c(T)\a(R0);T).
It is completely apparent that A=T00q1 TIfl(Xn;T) for all An problem has been reduced to showing that
A$TOq U
(a,)
k
co
1
sim
L=$ n=1 0
a(T)\cy(R)and
that our
S. n
For each m Cm =1,2,...), A that Am is similar to (m
it is not difficult to see
acts on a space of dimension
Since
(for some compact
and SP(Arn) + 0
co). Proceeding as above (see also the proof of Lemma 8.35), we m
see that (=)
m
r) n
q1
By Lemma 8.34, we can find a subspace Urn of the soace of m1r whose dimension is a (finite) multiple of k and Wm in-
vertible such that
225
m
Cm)
Cm)
Lfl=l
—l
•AmlWm
and
H
12
11 (co)
Cm)
+A22
m
(m)
m
2 SP(Am)• (k(k—1)c m
and, proceeding as
Let kcm=dim tim; then Tu sTm in the proof of Lemma 8.34, we conclude that m n-l
r
n
(k(k—l)c m
)
0
(m + co).
A fortiori, 0
Since
2
—L011
follows from the universality of
Cm -*
SP(Am) + 0 Cm ÷ co), it readily and Proposition 4.21 that
LGT
R;U&) = q
Hence, TOQU stm LOQU =
2)
Let B
C K(H), be any operator such that =dim H(An;T) for all a(T) and BO for each n, n=l,2 By Proposition 4.21, B
c(B) =o(T), dim
Given e > 0, choose m large enough to guarantee that dn / is a finite set) and lxi < c/4 for all n > m. Then
D.
H(CA}m1;B) B — —
HC{A}m1;B)l
0
and sp(B) < €/4. As in Proposition 3.45, we can find a compact normal operator
c/4,
<
such that
is quasinilpotent. By
=
Corollary 3.22,
AC C
0
so
M
that
C
A OM
c
,
and
AC 0
C
A
A
3k+h0-h
C
C
Then 3k-h0+h
be the dimension of >
k and
>
k,
stm
(use Lemma 2.5);
A
tion
M
C
0
Let
<€/4,
B—
and the universality of
the above construc-
indicate that
LOQU stm AcO(On=;+i ÷ sim
Since 226
A 0(0 e
n=m+l ñ n )G n=m+l Rn11
2
n>
m}
<
c/2, we see that
&i.st
IA C C
C
+
CO
Oq
10
< C/2.
k
Therefore, <
C.
Since c can be chosen arbitrary small, it readily follows that LOQU sim The proof is now complete.
U
CONJECTURE 8.37. If T is similar to a compact oerturbation of (for some k 2 and 0 contains infih k-i) and nitely many points, then T#LOQ, where L is the operator defined in
q(h)Oq(OO)
the proof of Lemma 8.35 and Q perturbation of
is
a suitable quasinilpotent compact
q(CO)
Wk,h(H;.r;{mj}jr).
PROPOSITION 8.38. If T
then the following
are equivalent (i)
(ii)
+ t—l T is a universal operator for the cla8s Wk,h(ti;ri{mJ}jr).
is a universal operator for the class
T#TOTU, where T sTm TOT, where
(iii)
T
the the first part of the proof of the previous
(iv)
proposition. (iii) => Ci)
Combining Lemmas 7.8 and 7.9 we see that, for each
m (m=l,2,...), T
where
is a nilpotent
acting on a finite dimensional space such that rank F3 = rank T3 for all i with 0 k—l, h' h' and Tm
r
(mod k).
Now we can proceed as in the proof of Lemma 8.34 and the first part of the proof of Proposition 8.36 in order to show that T sTm TOTu
sTm
Since, by Proposition 8.30 and Lemma 8.31, A a universal operator for the class
h(K;r;
R
,
where R
is
we are done.IJ
The argument for the proof corresponding to the third case is closely related to the one used in the proof of Lemma 8.1.
PROPOSITION 8.39. If T
Nkh(H;Co), then the following are equiv227
a lent
(i)
is
T
(ii)
T #T$QU, where
(iii)
has the form of (ii).
where
T stm
(vi)
is a universal operator for the class
where
T sTm
h(H;).
is a universal operator for the class
where
'P
(iv) (v)
a universal operator for the class
has the form of (iii).
'P
PROOF. Once again,
Ci) =>
two of the statements (ii) —
(ii)
(vi)
and the equivalence between any
follow exactly in the same way as in
the proof of Proposition 8.36.
By hypothesis, we can directly assume that T is a Ci) (co) for quasinilpotent compact perturbation of q1 such that (iv) =>
all j
=1,2 Let m
= 1,2,...
2
and let e1 be a unit vector such that
0 for all j
(see the introductory paragraph of Section 8.1.1). Let P5
denote the orthogonal projection of II onto the finite dimensional sub-
space corresponding to the direct sum of of
and the first s copies (s=l,2,...); then P3 ÷ 1 (strongly, as s ÷ be the Gram—Schmidt orthonormalization of the seLet
quence {Tnlel}n=l. It is not difficult to check that 'P admits a ma-
trix representation of the form t11 t12 t13
.
t21 t22 t23 t32 t33
.
.
e1
.
.
e2
.
.
* T
•
,
o
=
. 0
0
0
.
.
.
...P2
...L.
0
(8.29)
-
o 0
I
0 •
I
N
where p=km+h and N is the orthogonal complement of M = Since N is finite dimensional, it is not difficult to check that IPM(l—Ps)jI1-fl(l_Ps)PMII=Es + 0 (S + =). This means that N is "almost" 228
orthogonal to ker
for all s large enough. Fix r >
so that
in
<
then there exists an invertible operator W=l+G (for some finite rank Thus, for practical purposes (reoperator G) such that WM j ker placing, if necessary, T by WTW1) we can directly assume that M
ker P5 for all s
>
i
r.
Given c > 0, we can proceed as in Lemma 8.34 and Step 1) of the
proof of Proposition 8.36 in order to show that
(for some s(c)
>
r
AB
C
M
D
E
F
ranP s(s) 8M
0
H
+J ker P5()
depending on C), where a11 a12 a13
.
.
.
e1
a21 a22 a23
.
.
.
e2
a32 a33
.
.
.
e3
0
.
.
(i.e., the matrix of A looks like the (1,1) block of (8.29)), 0
0
0
.
.
. 0 0
0
(i.e., the matrix of D looks like the (2,1) block of (8.29)) and
ABC OHJ D
E
F
<
Now we can proceed exactly as in the proof of Lemma 8.1 in order to show that T
where E(m;c) is a compact operator such that IIE(m;c)jI On
the other hand (by Corollary 8.2(u)), given k2 )
< >
0
we can find
such that <
The definitions ofQ , T
and the above observation indicate that
229
T stm T$QU sTm TOTU sTm ((k—l)k[s(c)—m])0
•
Tmc =
}+E' (m;e),
E(m;E)$O.
where E'(m;c)
Clearly, we have
=
where E"(m;c) is a compact operator such that <
IIE"(m;c)II
Since
can be chosen arbitrarily small, we can assume that <
whence it readily follows that <
2n.
Since n can be chosen arbitrarily small, we conclude that T sTm Cm =2,3,...).
Tm
By Proposition 8.30 and Lemma 8.31, Tm
COROLLARY 8.40. If T E
K(N),
IIK1I <
whence the result
+
sim
follows.
h(H)3 then given £
9 >
0
there eciste K
such that T+K is a universal operator forthe class of T.
PROOF. We can directly assume that T <
(k
2, 0
k—i, C compact). By Proposition 4.21, there exists K1
h
c/2,
K(fl),11K111 Now it is easy to find a second com-
such that T+K1
pact operator K2, 11K211
< £/2,
whose action only modifies the direct
TOTU for some universal operator Tu for where K=K1+K2. Now the result follows from Propositions 8.36, 8.38 and 8.39.9
suxnmand
such that T+K
the class
8.5 Separation of isolated points of the essential spectrum affiliated with nilpotents Let P c L(ti) and assume that u
is
an isolated point of Oe(T)• If
r > 0 is small enough to guarantee that D(ii;r)Tha5(T)
and y =
9fl(i.t;r) (positively oriented), then =
is 230
dA
a non—zero idempotent of AUI) and can be lifted to an idempotent
L{H) . Let W
seen
where
,
U
U
is
be such that P
GUI)
tion and let H =11 •f11
=ran
H U
an orthogonal projec—
=ker
and
P U
U
P
.
U
It
is easily
that
AB K =
C
H1'
D
where B and C are compact operators, Oe(M =
{jj} and
cYe(D). Hence
where Q=A—U is an essential quasinilpotent. It will be convenient to introduce the following definition: T
Let
L(U)and A 0, if A
Oe(T)
n, if A is an isolated point of o =
=,
If 2
defined above) otherwise. <
and
(T) and
(as
(8.30)
is a nilpotent of order
we shall say that
is similar to
If either k(A; A is an isolated point of ae(T) affiliated with is not similar to we shall = 1 or 2 < = and (It is completely apparent say that A is affiliated with
that this definition is independent of the particular invertible opera tor W above chosen.)
From Proposition 6.19, we obtain the following
let let
COROLLARY 8.41. Let T e L(H), set
be a finite
be the component of ae(P) such of isolated points of either coincides with or is disjoint from 0h' if
that A
h) and there
e
assume that ind(A exists K e K(H) ,
—
T)
0 for all A
m
0 =
Given
>
0,
< e, such that T-I< and
A
are quasinilpotent operators. Furthermore,
order
tent of
of
j
=
if 1 but
k(A;T) =n.
=, thenQ.
<
can be chosen nilpo-
n—i cannot be compact for any possible choice i
1,2,... ,m.
PROOF. Assume that ind(A-A1) >
for all A
0
By Proposition 6.19 there exists a compact operator K1, 11K111
<
s/3m, such that (8.31)
R
where o(R1) = 0 for all A
{A1}' A1 I
J
and nul(A -A1) 0
for all A
ind(A —T) and 01, then we canapply 231
Proposition 6.19 to T* in order to obtain the same decomposition with
={A1}, A1
and nul(A—A1) =0 and nul(A_A1)*=_ind(A_T)
c11.)
for all A
By Theorem 7.2,
where (Ri) 1=0 and Kj is compact. Now we can repeat the argument of the proof of Proposition 6.19: First replace by a suitable finite rank operator so that IIKj-FjII <
s/3m and *
F1'
01
(8.32)
R"
1
where
acts on a finite dimensional space,
c
and
is a
nilpotent of order n1. Thus the operator A1 * *
(8.33)
*
0 0
0
differs from T by a compact operator whose norm is less than 2t/3m.
By Theorem 3.49, there exists a compact operator
< c/3m
such that *
A
K3
01
satisfies the properties: all A
(1)
and (2)
A1
for
e
It follows from Corollary 3.22 that, if
corresponds to the decomposition (8.31) and
(K3e0)2 (where
corresponds to the decomposition (8.33)), then C1
K(IIJ,
c/rn and A" *
T-C1=
It
01
Rj
—
rlçl
is completely apparent that (Ri)
cannot be compact. Now the
result follows by an obvious inductive argument.
fl
PROPOSITION 8.42. Let T E L(I1),
be a finite let A_ set of isolated points of ce(T) such that A c interior a(T) and assume for each j=l,2,...,m. Given c > 0 there exthat 1 k(A.;T) =n. < ists K
K(K),
IKII
<
c,
such that T-K — T1=A,[,.m1
(A.+Q.)],
rnin.ind(A_T1)k=rnin.ind(A_T)k fr all A ps_F(T) for each j=l,2,..., and all k1,2,..., is a nilpotent of order m, and, moreover where
(i)
232
If
/
IP:_F(T) 1, then
is quasinilpotent,
If
(ii)
o(A.+Q.)
then
c
{A.}ij(A
pS_F(T):
If
is affiliated with 41 [p_F(T)] and Q. is a universal operator for — nJ, J ('°) If (iv) is affiliated with and (iii)
then
n}n:l
only accumulates at
(de-
then
fined as inPropositition 8.36(u)) for suitably Jordan nilpotents
Furthermore, there also exists R
L(UJJ
T-K-R
IIRII
< c,
11
l.
such that
(A.+R.)),
where
then R.Q.;
If A. is affiliated with
(v)
(vi) (vii)
is affiliated with q1
I[ps_F(T)] and
q1
then R.
—
S
If
;
and only accumulates
[P:_F(TYI, A. is affiliated with q1
If
=
then
at
(oo)
j
R.
for
(a.)
(on)
{e
]}eq fl. n=l [j.A.j,n +Ii.'.j,n IJ. j,n n=l,2
suitably chosen Jordan nilpotents
PROOF. As in Corollary 8.41, we can confine ourselves to the case when A ={A1}(The general case follows by an inductive argument). Fur-
thermore, we can also translate the operator and assume that X10 is an isolated point of o(T) such that 1 =k < a..
Let T =
T
T12 T13
0
T0
0
0
T23 t10(T)
be the triangular representation (3.8) of T. Applying Corollary 8.41 to Tr and
c/3,
we can find compact operators Krl Ki, IIKrIl
<
11K2,II <
such that
Tr
—
Kr
=
A
r
*
(B
and Br
TL -
=
where Br and Bt are nilpotent operators, Br and of nilpotency, Bt and
Tr_Kr — ArSB K1 is compact, 11K111
<
*
have the same order
have the same order of nilpotency, 0
and Ti_Ko
Thus, if
41
Oe(Ar) then
and (by Corollary 3.22)
233
A
T-K1
r B
=
*
r
B
To *
T0
Bi
o
*
*
r
0
(8.34)
0
A
where 0 is an isolated point of the essential spectrum of the second direct summand, not contained in the interior of the spectrum of this operator.
If 0 is an isolated point of the spectrum of the second
(iii)
direct suinmand of (8.34), then it follows from the Riesz' decomposi-
tion theorem that T-K1 nilpotent such that
where 0 / and Q is a quasi 0. If 0 is an isolated point of then it is easily seen that is in-
(A) affiliated with
vertible some h,
and Q is similar to a compact perturbation of q1 0
h
(h)
(ce)
for
k-l. Clearly, we can directly assume that Q =
(where C is compact).
Assume that 0
and let be the orthogonal projection in the space of Q onto the span of the subspace corresponding to q1th)
and the space corresponding to the first n copies of 0
110-
and P QP
then
(n
acts on a finite dimensional space.
n By Theorem 3.49, there exists a compact operator I( n
of arbitrari-
ly small prescribed norm such that
is smooth. It readily
follows that we can find a compact operator K2,
11K211
<
such that
T-K1-K2 where A' is smooth.
By Corollary 8.40, we can find a compact operator K3,
IlK3
U<
such that
T-K where K=K1+K2+K3
K(H),
IIKII
<
a universal operator for we can apply the above argument to (i) and (vi)
AeQU,
(If
0 /
c(Ar),
then 0
and Q
is
o(Ai)
and
If 0 is isolated in the spectrum of the second di-•
rect summand of (8.34), but it is also an isolated point of ce(T) affiliated with
then we can use Theorem 7.2 in order to ob--
tam the decomposition Q=Q1+C1, where and C1 is compact. By Corollary 7.6 we can find a compact operator K2, 1R211 < such that
T-K1-K2 vertible, c(F2)
234
where
is not in
a finite dimensional
Space. By Theorem 3.48, there exists a compact operator K3,
such that T—K
AWQ2, where K=K.+K2+K3 E K(HJ, I1KII < and (where is a smooth compact perturbation of in the case when 0 or A = is a smooth com(where pact PerturbationofF2GAf in the case when 0 f o(Ar) and 0 ). Since is not invertible, it follows from Proposition 8.5 that there exists such that < €, whence we obtain (vi). If 0 is not an isolated point of the spectrum of (iv) B
*
10
0
*
8= let (p}1 be an enumeration of the normal eigenvalues of 13 contained in D(0;r), where r=¼distfO,o (T)\{O}) and let J be the canonical Jordan form of the nilpotent B0eB1, where c(B) =c(B)nD(0;r), a(B1) =o(B)\D(0;r) J for all n=l,2
B
Clearly,
and
0; moreover, 0 is an isolated point of
if
affiliated with equivalent to q(h)eq(=)+c
and only if B0 can be chosen unitarily
where 0
<
h
k-l
and C is compact. By
Proposition 4.21 and Corollary 8.40, there exists a compact operator <
e/3, such that
It is
q
where
#
36
thepxoof
Qk(fl1
and that (by using Lemma 2.5)
ator K2, 11K211
<
easily seen that of Proposition
8.
can be replaced by a compact oper
c/3, such that (co)
T - K1 - K2
ArW[
Now the proof follows exactly as in case (iii).
(If 0 E O(Ar)i
is "absorbed" in Ar via a small compact perturbation; if 0 ci(A), then 0 (vii)
and
is
"absorbed" in As.)
Assume that 0 is not isolated in 0(B), but it is an isol-
Then it follows from Hence, there exists
ated point of Ge(T) affiliated with Proposition 0.6 that B0#
such that
<
c.
It is completely apparent from the above constructions that
min.ind(A-T1) k =min.ind(A-T2) k=min.ind(A-T) •
for all A
•
k
ps_F(T) and for all k1,2
The proof of Proposition 8.42 is complete.
II
235
Let T, K, e be as in Proposition 8.42 and let
T-K-T 1 =A.Ie.m j=1
3
i
KIN), IlK II
be the decomposition provided by that proposition with K where the Q.'s satisfy the conditions (i) — 3
if
<
(iv).
is similar to a compact perturbation of
2),
/
then by a formal reoetition of the arguments of the proof we can find a compact operator C. of arbitrarily small prescribed norm such that :i
where Q is a finite multiple of
acts on a space whose dimension
',
and
c c(A). If
does not have that
form, then we can find C. as above such that Q. - C. 3 3 acts on a finite dimensional space, c 0(A) and Hence, there exists K' e KIN), IlK' II
T—K—K'
c/3,
<
where F. 3
# q,'
ti3
such that -
—
Finally, by Theorem 3.48, there exists K"
KIN), IIK"ll <
that
T-C where CK+K' +K"
KIN), 11CM <
and A' is a smooth compact perturba tion of A. Thus, we have the following COROLLARY 8.43. Let T E
c
tIN), tat
set of isolated points of 0e(T) such iste C E
for
=n. <
that I
K(H),
jCIl < e,
that
be a finite A c interior oCT) cind assune
each j=l,2,...,rn.
Civen
c
0 thore er-
such that
T—C whar'e A
smooth and etther n. or
n—l 3
2 and Q.
3
ie not invertible and
+Q.k
and
#q1
8.6 Notes and remarks
Proposition 8.3 has been independently (and almost proved by C. Apostol £131 and by D. A. Herrero [138]. In fact, the resuits of 1137i and 11381 provide a complete characterization of the set of all universal quasinilpotents for L(H) and for each closed bilateral ideal of L(fl)
strictly
larger than KIN), for the case when H
is not necessarily separable (orovided the ideal admits some universal quasinilpotent.) On the other hand, the characterization of the set of
all universal compact quasinilpotents (Proposition 8.4) was obtained by C. Apostol in 113]. The proofs given here combine C. Apostol's Lem-
236
ma 8.1, the technical Corollary 8.2 (essentially contained in [44])and an argument of [138].
The results contained in Sections 0.2, 8.3 and 8.4were not published before. In [44], J. Barria and D. A. Herrero extended their pre vious results about finite rank operators [43] to arbitrary nilpotents and nilpotent elements of the Calkin algebra. Unfortunately, the article contained several errors and the main result did not cover all possibil ities as claimed there. Correct proofs of the same results (Indeed, slightly better ones) were independently obtained by C. Apostol and D. Voiculescu in [34], by using a different approach. The proofs of Propositions 8.5 and 8.6 are essentially the same
as the ones given in [44], corrected with the help of the structural results (due to L. A. Fialkow and D. A. Herrero) given in Sections 7.5 and 7.6 of Chapter VII. The proofs of Propositions 8.12 and 8.17 given here have been taken from [34]. They are more illuminating and "coordinate free" than the
corresponding (correct) proofs given in [44]. C. Apostol and D. Voiculescu's approach is heavily based on their Lemma 7.20 and Lemma 8.8,
which can also be derived from [l82,Lemma 3.3]. Lemma 8.8 can be regarded as the link between the present problem and Brown—Douglas-Fillmore theory. In order to make it more explicit, it will be necessary to introduce some definitions: Let C be a separ-
able C*_algebra with identity. By an extension of K(H) by C we shall mean a pair where E is a C*_subalgebra of L(H) containing K(H) and the identity operator and 4,:E
C is a *...homomorphism with kernel
K(K). Such a pair yields the short exact sequence 0
where
-*
K(I1J
E
C
0,
is the inclusion map. Two such extensions
called strongly equivalent if there exists a unitary operator V such that the following diagram commutes:
v(T)
are
L(H)
(8.35)
Ext5(C) will denote the set of all strongly equivalent classes of extensions of K(H) by C. and (E2,q2) are weakly equivalent if there exists a Fredholm
partial isometry V c L(U) such that Diagram (8.35) commutes. Let
Ext(C) denote the set of all weakly equivalent classes of extensions of KIN)
by
C.
237
If (E,$) is an extension of K(H) by C, then is *_jsomorphjc to C and this defines a faithful unital *_homomorphjsm T:C A(H).Cofl
versely, if T:C
A(K) is a faithful unital *_homomorphjsm, then a extension of C. Thus, we can identify extensions with *_monomorphjsms from C into A(H). Two *_monomorphjsms T1 and are weakly equivalent (strongly equivalent, resp.) if and only if there exists a unitary element 0' A(KJ (a unitary operator U L(H}, resp.) such that For two *_monomorphisms 11 and from C into A(H) we define to be the *monomorphjsm from C into A(HOK) suâh that
defines
(T1+r2) Cx) =
this
Since H Ext(C)
(ExtS(C))
,
x
E
C.
defines a structure of abelian semigroup in
by:
[T1]+ [T2]= [t1+T2]
(['r1]5+ [T2]s
[T1+T2]5, resp.)
In fact, Ext(C) and ExtS(C) are abelian semigroups with identity and, in many cases, they are actually groups. It is also apparent that defines an epimorphism of semigroups (of groups, in case they are actually groups) from ExtS(C) onto Ext(C). The reader is referred to
[60], [61], [182] and, especially, to [40] and the ref er-
ences given in the different articles contained there. In particular, if and
,
(for some k 2), then so that the natural epimorphism +
is a trivial mapping [182,Lemma 3.3].
This interesting connection with the Brown-Douglas-Fillmore theory is not apparent from the proofs of Propositions 8.12 and 8.19 given in [44].
Corollary 8.23 and Proposition 8.24 are due to L. A. Fialkow and D. A. Herrero. These two results are related with the local analysis of the similarity orbits (see [16] and [941).
The results of Section 8.4.3 belong to C. Apostol and D. Voicules cu [34] and partially fill the gap in the main theorem of [44]. Sections 8.4.4 and 8.4.5 contain recent results of D. A. Herrero. (Actual is a where ly, the peculiar behavior of the operators universal compact quasinilpotent has been partially analyzed in [44, Section 7].)
Finally, Corollary 8.41 is Corollary 1 of [42] and Proposition 8. 42 can be regarded as a nontrivial improvement of Theorem 3 of that reference.
238
References
1.
G. R. Allan
2.
G.
3.
J.
4.
C.
5.
C.
6.
C.
7.
C.
8.
C.
9.
C.
10.
C.
11.
C.
12.
C.
13.
C.
14.
C.
15.
C.
16.
C.
17.
C.
18.
C.
19.
C.
20.
C.
21.
C.
22.
C.
23.
C.
On one-sided inverses in Banach algebras of holomorphic vector-valued functions. J. London Math. Soc. 42 (1967) 463—470. R. Allan Holomorphic vector—valued functions on a domain of holomorphy. J. London Math. Soc. 42 (1967) 509-513. H. Anderson, On compact perturbations of operators. Can. J. Math. 26 (1974) 247—250. Apostol Quasitriangularity in Hilbert space. Indiana Univ. Math. J. 22 (1973) 817—825. Apostol On the norm-closure of nilpotents. Rev. Rourn. Math. Pures et Appl. 19 (1974) 277—282. Apostol Matrix models for operators. Duke Math. J. 42 (1975) 779—785. Apostol Quasitriangularity in Banach space. I. Rev. Rouni. Math. Pures et Appi. 20 (1975) 131—170. Apostol Quasitriangularity in Banach space. III. Rev. Roum. Math. Pures et Appl. 20 (1975) 389—410. Apostol Hypercommutativity and invariant subspaces. Rev. Roum. Math. Pures et Appl. 17 (1972) 335—339. Apostol The correction by compact perturbations of the singu lar behavior of operators. Rev. Roum. Math. Pures et Appi. 21 (1976) 155—175. Apostol On the norm-closure of nilpotents. III. Rev. Roum. Math. Pures et Appl. 21 (1976) 143—153. Apostol Inner derivations with closed range. Rev. Roum. Math. Pures et Appi. 21 (1976) 249-265. Apostol Universal quasinilpotent operators. Rev. Rouni. Math. Pures et Appl. 25 (1980) 135—138. Apostol and K. Clancey, Generalized inverses and spectral theory. Trans. I½iner. Math. Soc. 215 (1976) 293-300. Apostol and K. Clancey, On generalized resolvents. Proc. Amer. Math. Soc. 58 (1976) 163—168. Apostol, L. A. Fialkow, D. A. Herrero and D. Voiculescu, Approximation of Hilbert space operators. II. (BostonLondon-Toronto: Pitman Publ. Inc., to appear). On the distance to biquasitriangular opApostol and C. erators. Rev. Roum. Math. Pures et Appl. 20 (1975) 261—265. Apostol, C. and C. M. Pearcy, That quasinilpotent operators are norm-limits of nilpotent operators revisited. Proc. Amer. Math. Soc. 73 (1979) 61—64. Apostol, C. Foias and L. Szidd, Some results on nonquasitrian— gular operators. Indiana Univ. Math. J. 22 (1973) 1151—1161. and D. Voiculescu, Structure spectrale des Apostol, C. opdrateurs nonquasitriangulaires. C. R. Acad. Sci. Paris (Sér. A) 276 (1973) 49—52. Apostol, C. Foias and D. Voiculescu, Some results on nonquasitriangular operators. II. Rev. Rouin. Math. Pures et Appi. 18 (1973) 159—181. and D. Voiculescu, Some results on nonquasiApostol, C. triangular operators. III. Rev. Roust. Math. Pures et Appl. 18 (1973) 309—324. and D. Voiculescu, Some results on nonquasiApostol, C. 239
24.
25.
26.
27.
28.
29. 30.
31.
32. 33.
triangular operators. IV. Rev. Roum. Math. Pures et Appi. 18 (1973) 487—514. C. Apostol, C. Foiaj and D. Voiculescu, Some results on nonquasi— triangular operators. V. Rev. Roum. Math. Pures et Appi. 18 (1973) 1133—1140. C. Apostol, C. Foias and D. Voiculescu, Some results on nonquasitriangular operators. VI. Rev. Roum. Math. Pures et Appl. 18 (1973) 1473—1494.
C. Apostol, C. Foias and D. Voiculescu, On the norm—closure of nilpoténts, II. Rev. Rouni. Math. Pures et Appl. 19 (1974) 549—577. C. Apostol and B. B. Morrel, On uniform approximation of operators by simple models. Indiana Univ. Math. J. 26 (1977) 427—442. C. Apostol, C. M. Pearcy and N. Salinas, Spectra of compact perturbations of operators. Indiana Univ. Math. J. 26 (1977) 345—350. C. Apostol and N. Salinas, Nilpotent approximations and quasinil— potent operators. Pac. J. tlath. 61 (1975) 327—337. C. Apostol and J. G. Stampfli, On derivation ranges. Indiana Univ. Math. J. 25 (1976) 857—869. C. Apostol and L. Szidó, Ideals in WA_algebras and the function of A. Brown and C. Pearcy. Rev. Rourn. Math. Pures et Appl. 18 (1973) 1151—1170. C. Apostol and D. Voiculescu, On a problem of Halmos. Rev. Roun. Math. Pures et Appl. 19 (1974) 283—284. C. Apostol and D. Voiculescu, Quasitriangularity in Banach space. II. Rev. Roum. riath. Pures et Appl. 20 (1975) 171179.
34.
35. 36.
C. Apostol and D. Voiculescu, Closure of similarity orbits of nil potent and quasinilpotent Hilbert space operators. (Preprint, not for publication, 1977). N. Aronszajn, Theory of reproducing kernels. Trans. Amer. Math. Soc. 68 (1950) 337—404. N. Aronszajn and K. T. Smith, Invariant subspaces of completely continuous operators. Ann. of Math. 60 (1954) 345350.
37.
38. 39.
40.
41.
42.
43.
44.
45. 46.
240
W. B. Arveson, An invitation to CA_algebras. Grad. Texts inMath. SpringerVol. 39. (New York-Heidelberg-Berlin: Verlag, 1976) W. B. Arveson, Notes on extensions of CA_algebras. Duke Math. J. 44 (1977) 329—355. W. B. Arveson and J. Feldman, A note on invariant subspaces. Mich. Math. J. 15 (1968) 61—64. M. F. Atiyah et al., K-theory and operator algebras. Athens, Geor (Berlingia 1975. Lect. Notes in Math. Vol. 575. Heidelberg-New York: Springer-Verlag, 1977). Rota's theorem for general functional Hilbert spaces. J. A. Ball Proc. Amer. Math. Soc. 64 (1977) 55-61. J. Barria and D. A. Berrero, Closure of similarity orbits of HilJ. Lan bert space operators. IV: Normal operators. don Math. Soc. (2) 17 (1978) 525—536. J. BarrIa and D. A. Herrero, Closure of similarity orbits of nil— J. Oper potent operators. I. Finite rank operators. ator Theory 1 (1979) 177—185. J. Barria and D. A. Herrero, Closure of similarity orbits of nil— (Preprint, not for publicapotent operators. II. tion, 1977) S. K. Berberian, Approximate proper vectors. Proc. Amer. Math. Soc. 13 (1962) 111—114. Index theory for perturbations of direct sums of nor I. V. Berg Can. 3. Math. 30 mal operators and weighted shifts.
47.
(1978) 1152—1165. An extension of the Weyl—von Neumann theorem to normal operators. Trans. Amer. Math. Soc. 160 (1971) 365—371. D. Berg On approximation of normal operators by weighted shifts. Mich. Math. J. 21 (1974) 377—383. A. Berger and B. I. Shaw, Self—commutators of hypo normal operators are always trace class. Bull. Amer. Math. Soc. 79 (1973) 1193—1199. A. Berger and B. I. Shaw' Intertwining, analytic structure, and the trace norm estimate, Proôeedings of a confer ence on operator theory, Halifax, Nova Scotia 1973. (Berlin-Heidelberg— Lect. Notes in Math. Vol. 345. New York: 1973). H. Bernstein and A. Robinson, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos. Pac. J. Math. 16 (1966) 421—431. The triangular representation of C. Apostol. Proc. Bouldin Amer. Math. Soc. 57 (1976) 256—260. Essential minimum modulus. Indiana Univ. Math. J. Bouldin (To appear). Brown, C.-K. Fong and D. W. Hadwin, Parts of operators on Hilbert space. Illinois J. Math. 22 (1978) 306—314. Brown and C. M. Pearcy, Spectra of tensor product of operators. Proc. Amer. Math. Soc. 17 (1966), 162—169. Brown and C. M. Pearcy, Compact restrictions of operators. Acta Sc Math. (Szeged) 32 (1971) 271—282. Brown and C. M. Pearcy, Compact restrictions of operators. II. Acta Sci. Math. (Szeged) 33 (1972) 161—164. Brown and C. N. Pearcy, Jordan ioops and decompositions of operators. Can. J. Math. 29 (1977) 1112—1119. Brown and C. ri. Pearcy, On the spectra of derivations on norm
1. D. Berg
48.
I.
49.
C.
50.
C.
51.
A.
52.
R.
53.
R.
54.
A.
55.
A.
56.
A.
57.
A.
58.
A.
59.
A.
60.
68.
L. G. Brown, R. G. Douglas and P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of Proceedings of a conference on operator theory, Halifax, Nova Scotia 1973. Lect. Notes in (Berlin-Heidelberg—New York: Math. Vol. 345. springer—Verlag, 1973) 58—128. L. G. Brown, R. G. Douglas and P. A. Fillmore, Extensions of C*_ algebras, operators with compact self—commutator and K-homology. Bull. Amer. Math. Soc. 79 (1973) 973—978. L. G. Brown, R. G. Douglas and P. A. Fillmore, Extensions of C*_ Ann. of Math. 105 (1977) algebras and K-homology. 265—324. J. Bunce and N. Salinas, Completely positive maps on C*_algebras and the left matricial spectrum of an operator, Duke Math. J. 43 (1976) 747—774. S. L. Campbell and H. Gellar, On asymptotic properties of several classes of operators. Proc. Amer. Math. Soc. 66 (1977) 79—84. Opérateurs linéaires sur un espace de Banach et moB. Charles dules sur un anneau principal. Symposia Math. 23 (1979) 121—143. Seminormal operators. Lect. Notes in Math. Vol. 742. K. Clancey (Berlin—Heidelberg-New York: springer-Verlag, 1979). Theory of generalized spectral operaI. Colojara and C. Gordon and Breach, 1968). tors. (New York: H. E. Curto, The spectrum of elementary operators. (Preprint,
69.
C. Davis and P. Rosenthal, Solving linear operator equations.
ideals.
61.
62.
63.
64.
65.
66. 67.
(Preprint)
1981)
241
Can. J. Math. 26 (1974) 1384—1389. Deckard, R. G. Douglas and C. M. Pearcy, On the invariant subspaces of quasitriangular operators. Amer. J. Math. 91 (1969) 634—647. Dixmier Etude sur les variétés et les opérateurs de Julia avec quelques applications. Bull Soc. Math. France 77 (1949) 11—101. Dixmier Les C*_algebres et leur (Paris: Gauthier—Villars, 1964) G. Douglas, Banach algebras techniques in operator theory. (New York and London: Academic Press, 1972). G. Douglas and C. M. Pearcy, Invariant subspaces of nonquasitriangular operators, Proceedings of a conference on operator theory, Halifax, Nova Scotia 1973. Lect. Notes in Math. Vol. 345. (Berlin-Heidelberg-New York: Springer—Verlag, 1973) 13—57. G. Douglas and C. M. Pearcy, A note on quasitriangular operators, Duke Math. J. 37 (1970) 177—188. DunfordandJ. T. Schwartz, Linear operators. Part I: General theory. (New York: Interscience, 1957). Dunford and J. T. Schwartz, Linear operators. Part II: Spectral theory. Self-adjoint operators in Hilbert space. (New York-London: Interscience, 1963). Edgar, J. Ernest and S.-G. Lee, Weighing operator spectra. In diana Univ. Math. J. 21 (1971) 61—80. B. Embry and M. Rosenblum, Spectra, tensor products, and linear operators. Pac. J. Math. 53 (1974) 95—107. Ernest Charting the operator terrain. Memoirs Amer. Math. Soc. Vol. 171. (Providence, Rhode Island: Amer. Math. Soc., 1976). A. Fialkow, A note on non-quasitriangular operators. Acta Sci. Math. (Szeged) 36 (1974) 209—214. A. Fialkow, A note on non-quasitriangular operators. II. Indiana Univ. Math. J. 23 (1973) 213—220. A. Fialkow, The similarity orbit of a normal operator. Trans. Amer. Math. Soc. 210 (1975) 129—137. A. Fialkow, A note on direct sums of quasinilpotent operators. Proc. Amer. Math. Soc. 48 (1975) 125—131. A. Fialkow, A note on limits of unitarily equivalent operators. Trans. Amer. Math. Soc. 232 (1977) 205—220. A. Fialkow, A note on the operator X -'- AX-XB. Trans. Amer. Math. Soc. 243 (1978) 147—168. A. Fialkow, A note on norm ideals and the operator X AX-XB. Israel J. Math. 32 (1979) 331—348. A. Fialkow, Elements of spectral theory for generalized deriva tions. J. Operator Theory 3 (1980) 89-113. A. Fialkow, Elements of spectral theory for generalized deriva tions. II: The semi—Fredhoim domain. Can. J. Math. (To appear).
A. Fialkow, A note on the range of the operator X + AX-XB. Illinois J. Math. 25 (1981) 112—124. A. Fialkow, Essential spectra of elementary operators. (Pre— print).
A. Fialkow, Spectral theory of elementary multiplications. Trans. Amer. Math. Soc. (To appear). A. Fialkow and D. A. Herrero, Inner derivations with closed range in the Calkin algebra. (Preprint). A. Fialkow and D. A. Herrero, Characterization of Hubert space operators with similarity cross sections. (Preprint, not for publication, 1978). A. Fillmore, J. G. Stampfli and J. P. Williams, On the essential spectrum, the essential numerical range and a
96.
97.
98.
99.
100. 101. 102. 103. 104. 105.
106.
107.
108. 109. 110. 111. 112. 113. 114. 115. 116.
117. 118. 119.
problem of Halmos. Acta Sci. Math. (Szeged) 33 (1972) 179—192. P. A. Fillmore and J. P. Williams, On operator ranges. Adv. in Math. 7 (1971) 254—281. C. Foias, C. ri. Pearcy and D. Voiculescu, Biquasitriangular opera tors and quasisimilarity, Linear spaces and approximation. (Proc. Conf. Oberwolfach 1977). (Basel: 1978) 47—52. C. Foias, C. M. Pearcy and D. Voiculescu, The staircase representation of a biquasitriangular operator. Mich. flath. J. 22 (1975) 343—352. c.—K. Fong Some aspects of derivations on 8(K). Seminar Notes. University of Toronto, 1978. C.-K. Fong and A. R. Sourour, On the operator identity 0. Can. J. Math. 31 (1979) 845—857. M. Fujii, U. Kajiwara, Y. Kato and F. Kubo, Decompositions of operators in Hubert spaces. riath. Japonicae 21 (1976) 117—120. T. W. Gamelin, Uniform algebras. (Englewood Cliffs, New Jersey: Prentice Hall, 1969). R. Gellar Two sublattices of weighted shifts invariant subspaces. Indiana Univ. Math. J. 23 (1973) 1-10. R. Gellar and L. Page, Limits of unitarily equivalent normal oper ators. Duke Math. J. 41 (1974) 319—322. F. Gilfeather, Norm conditions on resolvents of similarities of Hilbert space operators and applications to direct sums and integrals of operators. Proc. Amer. Math. Soc. 68 (1978) 44—48. I. C. Gohberg and M. G. Krein, The basic propositions on defect numbers, roots numbers and indices of linear operators. Uspehi Mat. Nauk SSSR 12 (1957) no. 2 (74) 43118; English transl. Amer. Math. Soc. Transl. (2) 13 (1960) 185—264. I. C. Gohberg and M. G. Krein, Introduction to the theory of line ar nonselfadjoint operators. Nauka, 1965); (rioskow: English transl. Transi. of Mathemetical Monographs Vol. 18. (Providence, Rhode Island: Amer. Math. Soc., 1969). S. Grabiner, Nilpotents in Banach algebras. J. London Math. Soc. (2) 14 (1976) 7—12. L. J. Gray Jordan representation for a class of nilpotent opera tors. Indiana Univ. Math. J. 26 (1977) 57—64. On bi—quasitriangular operators. Proc. Amer. Math. L. J. Gray Soc. 64 (1977) 291—294. D. W. Hadwin, Closures of unitary equivalence classes. Dissertation. (Indiana University, 1975). D. W. Hadwin, An operator-valued spectrum. Indiana Univ. Math. J. 26 (1977) 329—340. D. W. Hadwin, Continuous functions of operators: A functional cal culus. Indiana Univ. Math. J. 27 (1978) 113—125. D. W. Hadwin, An asymptotic double commutant theorem for C*_alge_ bras. Trans. Amer. Math. Soc. 244 (1978) 273—297. D. W. Hadwin, Approximating direct integrals of operators by direct sums. Nich. Math. J. 25 (1978) 123—127. D. W. Hadwin, Approximate equivalence and completely positive maps. (Preprint). P. R. Halmos, Introduction to the theory of Hubert spaces and the theory of spectral multiplicity. (New York: Chelsea Pubi. Co., 1951). P. R. Halmos, Invariant subspaces of polynomially compact ooera— tors. Pac. J. Math. 16 (1966) 433—437. P. R. Halmos, A Hilbert space problem book. (Princeton, New Jer243
sey: D. Van Nostrand, 1967). 120. P. R. Halmos, Irreducible operators. Mich. Math. J. 15 (1968) 215—223. 121. P. R. Halmos, Quasitriangular operators. Acta Sci. Math. (Szeged) 29 (1968) 283—293. 122. P. R. Halmos, Permutations of sequences and the Schröder-Bernstein theorem. Proc. Amer. Math. Soc. 19 (1968) 509—510. 123. P. R. 1-lalmos, Ten problems in Hilbert space. Bull. Amer. Math. Soc. 76 (1970) 887—933. 124. p. R. Halmos, Capacity in Banach algebras. Indiana Univ. Math. J. 20 (1971) 855—863. 125. P. R. Halmos, Limits of shifts. Acta Sd. Math. (Szeged) 34 (1973) 131—139. 126. P. R. Halmos and L. J. Wallen, Powers of partial isometries. J. Math. Mech. 19 (1970) 657—663. 127. R. E. Harte, Tensor products, multiplication operators and the spectral mapping theorem. Proc. Royal Irish Acad. 73 (1973) 285—302. 128. R. E. Harte, Berberian-Quigley and the ghost of a spectral mapping theorem. Proc. Royal Irish Acad. 78 (1978) 63-68. 129. J. H. Hedlund, Limits of nilpotent and quasinilpotent operators. Mich. Math. J. 19 (1972) 249—255. 130. H. Helson Lectures on invariant subspaces. (New York-London: Academic Press, 1964). 131. D. A. Herrero, Formal Taylor series and complementary invariant subspaces. Proc. Amer. Math. Soc. 45 (1974) 83—87. 132. D. A. Herrero, Normal limits of nilpotent operators. Indiana Univ. Math. J. 23 (1974) 1097—1108. 133. D. A. Herrero, Toward a spectral characterization of the set of norm limits of nilpoterit operators. Indiana Univ. Math. J. 24 (1975) 847—864. 134. D. A. Herrero, Erratum: 'Toward a spectral characterization of the set of norm limits of nilpotent operators", Indiana Univ. Math. J. 24 (1975) 847—864. Indiana Univ. Math. J. 25 (1976) 593. 135. D. A. Herrero, Norm limits of nilpotent operators and weighted spectra in non-separable Hilbert spaces. Rev. Un. Mat. Argentina 27 (1975) 83—105. 136. D. A. Herrero, Corrigendum: Norm limits of nilpotent operators and weighted spectra in non-separable Hubert spaces. Rev. Un. Mat. Argentina 27 (1975) 195-196. 137. D. A. Herrero, Universal quasinilpotent operators. Acta Sci. Math. (Szeged) 38 (1976) 291—300. 138. D. A. Herrero, Almost every quasinilpotent Hubert space operator is a universal quasinilpotent. Proc. Amer. Math. Soc. 71 (1978) 212—216. 139. D. A. Herrero, Clausura de las Orbitas de similaridad de operado—. res en espacios de Hilbert. Rev. Un. Mat. Argentina 27 (1976) 244—260. 140. D. A. Herrero, Closure of similarity orbits of Hubert space operators. II: Normal operators. J. London Math. Soc. 13 (1976) 299—316. (2) 141. D. A. Herrero, Closure of similarity orbits of Hilbert space operators. III. Math. Ann. 232 (1978) 195—204. 142. D. A. Herrero, A Rota universal model for operators with multiply connected spectrum. Rev. Roum. Math. Pures et Appl. 21 (1976) 15—23. 143. 0. A. Herrero, On analytically invariant subpaces and spectra. Trans. Amer. Math. Soc. 233 (1977) 37—44. 144. D. A. Herrero, On the spectra of the restrictions of an operator. Trans. Amer. Math. Soc. 233 (1977) 45-58. 145. D. A. Herrero, The Volterra operator is a compact universal quasi-
244
146. D. 147. D. 148. D. 149. D. 150. D.
151. D. 152. D. 153. T.
154. G. 155. G. 156. G. 157. G. 158. P. 159. C.
nilpotent. Integral Equations and Operator Theory 1 (1978) 580—588. A. Herrero, On multicyclic operators. Integral Equations and Operator Theory 1 (1978) 57—102. A. Herrero, Quasisimilar operators with different spectra. Acta Sci. Math. (Szeged) 41 (1979) 101—118. A. Herrero, Lifting essentially (G1) operators. Rev. Un. Mat. Argentina 29 (1980) 113—119. A. Herrero, Quasidiagonality, similarity and approximation by nilpotent operators. Indiana Univ. Math. J. 30 (1981) 199—233. A. Herrero, Unitary orbits of power partial isometries and approximation by block-diagonal nilpotents, Topics in Modern Operator Theory, 5th International Conference on Operator Theory, (Romania) Operator Theory: Advances and June 2-11,1980. Birkhausser, 1991)1Th-211). plications, Vol.2. (Basel: A. Herrero, The distance to a similarity-invariant set of oper ators. Integral Equations and Operator Theory. (To appear). A. Herrero and N. Salinas, Operators with disconnected spectra are dense. Bull. Amer. Math. Soc. 78 (1972) 525—536. Kato Perturbation theory for linear operators. (New York: Springer-verlag, 1966). R. Luecke, A new proof of a theorem on quasitriangular operators. Proc. Amer. Math. Soc. 36 (1972) 535—536. R. Luecke, A note on quasidiagonal and quasitriangular operators. Pac. J. Math. 56 (1975) 179—185. R. Luecke, Essentially (G1) operators and essentially convexoid operators. Illinois J. Math. 19 (1975) 389—399. Lumer and M. Rosenblum, Linear operator equations. Proc. Amer. Math. Soc. 10 (1959) 32—41. Meyer—Nieberg, Quasitriangulierbare Operatoren und invariante Untervektorraume stetiger linearer Operatoren. Arch. Math. (Basel) 22 (1971) 186—199. L. Olsen, A structure theorem for polynomially compact operators. Amer. J. Math. 93 (1971) 686—698.
160. C. N. Pearcy, Somerecentdevelopmentsinoperatortheory. CBMS 96. (Providence, Rhode Island: Amer. Math. Soc., 1977). 161. C. M. Pearcy and N. Salinas, An invariant subspace theorem. Mich. Math. J. 20 (1973) 21—31. 162. C. M. Pearcy and N. Salinas, Compact perturbations of seminornial operators. Indiana Univ. Math. J. 22 (1973) 789-793. 163. C. M. Pearcy and N. Salinas, Operators with compact self—commutators. Can. J. Math. 26 (1974) 115—120. 164. C. M. Pearcy and N. Salinas, Finite dimensional representations of C*_algebras and the reducing matricial spectra of an operator. Rev. Roum. Math. Pures et Appl. 20 (1975) 567—598.
165. C. M. Pearcy and N. Salinas, Thereducing essentialmatrical spectra of an operator. Duke Math. J. 42 (1975) 423-434. 166. C. N. Pearcy and N. Salinas, Extensions of C*_algebras and there— ducing essential matricial spectra of an operator. K-theory and operator algebras, Athens, Georgia 1975. (Berlin-Heidelberg-New Lect. Notes in Math. 575. York: Springer—verlag, 1977) 96—112. 167. C. R. Putnam, Commutation properties of Hubert space operators and related topics. Ergebnisse der Math. und ihrer (New York: Springer-Verlag, 1967). Grenz. 36. 168. C. R. Putnam, An inequality for the area of hyponormal spectra. Math. Z. 28 (1971) 473—477. 169. C. R. Putnam, Operators satisfying a G1-condition. Pac. J. Math. 245
84 (1979) 413—426. 170. H. Radjavi and P. Rosenthal, The set of irreducible operators is dense. Proc. Amer. Math. Soc. 21 (1969) 256. 171. H. Radjavi and P. Rosenthal, Invariant subspaces. Ergebnisse der Math. und ihrer Grenz. 77. (flew York-HeidelbergBerlin: Springer—verlag, 1973). 172. C. Rickart General theory of Banach algebras. (Princeton, New Jersey: D. Van Nostrand, 1960). 173. F. Riesz and B. Sz.-Nagy, Functional analysis. Ungar, (New York: 1955)
174. M. Rosenblum, On the operator equation BX-XA=Q. 175. 176.
177. 178. 179. 180. 181. 182.
183.
184. 185.
Duke Math. J. 23 (1956) 263—269. G.—C. Rota On models for linear operators. Comm. Pure Appl. Math. 13 (1960) 469—472. C*_algebras and W*_algebras. Ergebnisse der Math. S. Sakai und ihrer Grenz. 60. (Berlin-Heidelberg-New York: Springer-Verlag, 1971). N. Salinas Operators with essentially disconnected spectrum. Acta Sci. Math. (Szeged) 33 (1972) 193—205. N. Salinas A characterization of the Browder essential spectrum. Proc. Amer. Math. Soc. 38 (1973) 369—373. N. Salinas Reducing essential eigenvalues. Duke Math. J. 40 (1973) 561—580. N. Salinas Subnormal limits of nilpotent operators. Acta Sci. Math. (Szeged) 37 (1975) 117—124. N. Salinas On the distance to the set of compact perturbations of nilpotent operators. J. Operator Theory 3 (1980) 179—194. N. Salinas Hypoconvexity and essentially n-normal operators. Trans. Amer. Math. Soc. 256 (1979) 325—351. R. Schatten Norm ideals of completely continuous operators. Ergebnisse der Math. und ihrer Grenz. 27, 2nd. ed.. (Berlin-Heidelberg-New York: Springer-Verlag, 1970). A. L. Shields, Weighted shift operators and analytic function theory. Math. Surveys 13. (Providence, Rhode Island: Amer. Math. Soc., 1974) 49—128. M. A. Shubin, On holomorphic families of subspaces of a Banach space. Mat. Issled. (Kishinev) 5, vyp. 4 (18) (1970) 153—165; English tranl., Integral Equations and Operator Theory 2 (197 407—420. Indiana W. Sikonia The von Neumann converse of Weyl's theorem. Univ. Math. J. 21 (1971) 121—123. Mich. Math. J. 23 M. Smith Spectra of operator equations. (1976) 151—153. R. A. Smucker, Quasidiagonal and quasitriangular operators. Dissertation. (Indiana University, 1973). J. G. Stampfli, The norm of a derivation. Pac. J. Math. 33 (1970) 737—747. J. G. Stampf ii, Compact perturbations, normal eigenvalues and a problem of Salinas. J. London Math. Soc. (2) 9 (1974) 165—175. W. Szymanski, Decompositions of operator-valued functions in Hilbert spaces. Stud. Math. 50 (1974) 265—280. D. Voiculescu, Some extensions of quasitriangularity. Rev. Roum. Math. Pures et Appl. 18 (1973) 1303—1320. D. Voiculescu, Some extensions of quasitriangularity. II. Rev. Roum. Math. Pures et Appl. 18 (1973) 1439—1456. D. Voiculescu, Norm—limits of algebraic operators. Rev. Roum. Math. Pures et Appl. 19 (1974) 371—378. Rev. D. Voiculescu, A non-commutative Weyl—von Neumann theorem. Roum. Math. Pures et Appl. 21 (1976) 97-113. D. Voiculescu, A note on quasitriangularity and trace—class self)
186. 187.
188. 189. 190.
191. 192. 193. 194. 195. 196.
commutators.
ActaSci. Math.
(Szeged)
42 (1980) 195-
199.
197. L. R. Williams, On quasisimilarity of operators on Hubert space. Dissertation. (University of Michigan, 1976). 198. L. R. Williams, Similarity invariants for a class of nilpotent o erators. Acta Sci. Math. (Szeged) 38 (1976) 423—42 199. F. Wolf On the essential spectrum of partial differential boundary value problems. Comm. Pure Appi. Math. 12 200
(1959) 211—228. J. Zemánek and B. Au%,etit, Local behavior of the spectral radius in Banach algebras, J. London Math. $oc. (2) 23 (1981) 171—178.
247
Index
Calkin algebra, quotient Cauchy domain Closed linear manifold
A
Algebraic direct sum element of
5
the Calkin algebra 128 operator 114 Analytic Cauchy domain 2 contour 2 Jordan curve 2 Jordan curve, regular 2 Apostol-Morrel simple models 135 Apostol-Salinas approach 120 Apostol' s triangular representation 62,63 Approximate defect spectrum 42 42 point spectrum Approximately equivalent *_representations 97 unitarily equivalent operators 100 59 Arc—length measure Asymptotically similar operators 12
B
78 Bad properties 41,84—85,94 BDF theorem 237—238 theory Berger—Shaw trace inequality 86
span
1 2 4
22
Closure (in the metric topology)
2
Commutant of an operator Commutator Compact operator Compression of an operator Cyclic operator
85 84 1
45
14,35
D
Diagonal normal operator 70,76 Double commutant of an operator 85
E
Essential 143 minimum modulus 162 nilpotent reducing operator spectrum 106 5 spectrum 84 Essentially normal operator 177,183,237 Exact sequence Extensions of C*_algebras 237—238
F
Biquasitriangular operator 14 5—148 146 Block-diagonal operator
Browder spectrum 69 Brown-Douglas—Fillmore 41,84—85,94 theorem
Faithful *_representation Finite rank operator Fredhoim domain operator
97 13 9 9
C C
C*_algebra
81—82,84,97
C*_algebra 5,
irreducible
*_representations of Calkin algebra
97 98
97 Gelfand—Naimark—Segal theorem Gram—Schmidt orthonormalization 67
1
249
L
H
3,58,109 Hausdorff distance 76 Hilbert-Schmidt operator 5 Hyperinvariant subspace 85—86 Hyponortnal operator
I
2 Idempotent, Riesz 25 Idempotent operator 9 Index Infinite—dimensional essential reducing 107 operator spectrum reducing operator 107 spectrum Infinite direct sums of 130 operators Inner 81 derivation 76,13 product 20,35,124 Integral part Invariant 5,157 subspace under compact perturbations 11 1,11 under similarities under unitary 10,11 equivalence Irreducible 104—105 operator
*_represefltatjofl
Isomorphic Hubert spaces
17,209—210 Lattice Left 9 essential spectrum 45 resolvent 9 resolvent set 9 spectrum Locally closed similarity orbit 210 Lower semicontinuity of the rank 8
M Matrix 59 decomposition 92 models for operators 54 Maximal numerical range Maximal numerical range, 57 normalized 10 Minimal index 144 Minimum modulus 144 Minimum modulus, essential Modulus 146 of quasidiagonality 140 of quasitriangularity Modulus, 144 essential minimum 144 minimum 214 m—th stable kernel 85 Multicyclic operator Multiplication by A operator 59,89 85 Multiplicity
98
12 N
J
Jordan block curve curve, regular analytic form nilpotent operator
15 2 2
14
171 171
n-dimensional essential reducing 107 operator spectrum reducing operator spectrum 107 Nice element of the Calkin 2 04—205 algebra Jordan nilpotent 206,209—210 171,205 Jordan operator Nilpo tent
K
Kernel
250
4
element operator Nilpotent, essential Nilpotents, sums of two Norm Norm-topology Normal eigenvalue operator, diagonal restrictions
6
6,108 162 117 3
15 S
70,76 74—75,101
Normalized maximal numerical range n-tuples
Numerical range Numerical range,
57
100
54
maximal
normalized maximal
54 57
Operator—valued spectrum Orthogonal direct sum projection onto a subspace
Orthonormal basis
100,106 12—13
45 30
P 0
164,188 Olsen's theorem Operator 1 Operator matrix decomposition 59 Operator, algebraic 114 145—148 biquasitriangular block—diagonal 146 commutant of an 85 compact 1 cyclic 14,35 diagonal normal 70,76 double commutant of an 85 162 essentially nilpotent essentially normal 84 13 finite rank Fredholm 9
Hubert—Schmidt
76
85-86 hyponormal 104—105 irreducible 171 Jordan 85 multicyclic nice Jordan 171,205 nilpotent 6,108 polynomially compact 162 124 pseudonilpotent quasidiagonal 146 quasinilpotent 4,122—123,127 125,140 qua sitriangular rationally multicyclic 84—85 reducible 100 restrictions of a nilpotent 168 5 restrictions of an semi—Fredholni
smooth subnormal trace class triangular very nice Jordan Operator spectrum, essential reducing infinite-dimensional essential reducing infinite-dimensional
reducing
9
74
84—85 75
62—63 178
Partially ordered set 17 Perfect spectrum 111-112 5 Point spectrum 42 Point spectrum, approximate Polynomially compact operator 162 17 Poset Pseudonilpotent element of the Calkin 129 algebra 124 operator
Q 146 Quasidiagonal operator Quasinilpotent 4,122—123,127 operator 103—104 Quasitriangular operator 135,140 1 Quotient Calkin algebra
restriction
R 4 Range Rationally multicyclic 85—86 operator 2 Rectifiable contour 100 Reducible operator Regular 2 analytic Jordan curve 41 left representation point of the semi-Fredhoim domain 10 41 right representation
*....representation of a
algebra 106 107
107
n—dimensional essential 107 reducing 107 n—dimensional reducing 106—107 reducing
97
*...representat ion,
faithful irreducible unital Reproducing kernel space Resolvent Resolvent set Restrictions of nilpotent operators
97 98 97
59,89 2 2
168
251
operators
5
Riesz
decomposition theorem idempotent Riesz-Dunford functional calculus
153 2
Triangular matrix operator representation
62—63 62-63 62
2
Right 9 essential spectrum 44 resolvent resolvent set 9 spectrum 9 Rosenbium 79 corollary 41 operator 53 theorem Rosenblum-Davis—Rosenthal 53 corollary 61,82 Rota's corollary Runge's approximation theorem 86
S
Schatten p—classes Self—commutator Semi—Fredholm domain operator Separable C*_algebra
75 84
Hubert space
1
10
97
Unitary equivalence group orbit Unitarily equivalent operators
97
10—11 84
11,84,100
*....representatiofls
12 97
Universal compact quasinilpotent operator 194 59 model 189,215 operator 193 quasinilpotent operator 221 Universality, criterion for
V
Singular point of the semiFredholm domain Slight twist Smooth operator
very nice Jordan operator Voiculescu's theorem
178 40,98
12 12
1,11 Similarity orbit 1,13—14,113,114,189 135 Simple model 10 22 74
Spectral characterization of 144 quasitriangularity 77—109 measures 13 radius 2 Spectrum Spreading spectral measures 77—109 214 Stable kernel Staircase representation 147—148 Strongly equivalent extensions 237 4 Subspace
T
252
Unital *...representation
9
Similar, asymptotically Similar operators Similarity—invariant
Trace Trace class operator
U
26,35,75,86 75
w Weakly equivalent extensions measurable square integrable functions Weyl spectrum Weyl-von Neumann theorem Weyl—von Neumann—Kuroda theorem Wolf spectrum
237 59
69 41
104 9
Symbols and notation
A'
43
A(.)
85
A'(.)
85
A"(.)
85 85
MU)
Mg(A(U)) Ae9(H) A(cz) B
(X)
1
129
111,114 89 13 3
(BD)
146
BP(X)
109
(BQT) SIX)
146
237 159 F(H)
13
GUI) II
UAbnor() HL(.)
HN()
15
MCI')
59
1
M_(r)
59
62
101
19 13
U(A;T)
H(a;T)
C*(fJ)
97
md k(.,.)
K(H)
C1(H)
75
La
C2(H)
76
ElK)
Ext(.)
237
ExtS(.)
237
E(X;T)
5
E(a;a)
2
5
NFk(tI)
19
(NF(U)/#,<) (NFk(H)/#,<)
20
20
NIH)
6 6
183
60 9
231
200
N
6
Nk(H)
183 1
(L(cd)/#,<) LIII)
200
41 51
3,109 25
19
5
80
dH(.,.)
NFUI)
ker
43
D(.,.)
6
129
112(r)
H2(r;H)
108
N{A)
N(A(H))
17 62
U(T,A) (a)
59
101
r (.)
U
97
64,132
29
1
62
99
d(.)
10
m(T,A)
129
U0(.)
C'(.) C*(.)
83,159
144
me(.) min.ind(.)
104
G(A(H))
3
2
13
42
}t_1)
Nk,h(U;r,bn j Nk,h(t(;r;ø)
17
216
218 220
1
32
83,159
1.u.b.
18
L(X,Y)
1
L2(r)
in(.)
j=r
89 144
Nok(H)
85
nul
9
NU2)
89
N_U2)
89
89
ONB
30
253
PF(H)
140
r tr (.)
32
T
32 q(.)
qd(.)
T
20
140 159 146 203—204
146
Q(H)
193
E
a(.)
2
U(.)
11,84
E(.)
100
(1(H)
11,84
(S,)
20
W(.)
54
69
57 54
17,18
WN
(.) w— $
im
199
Ed
18
ae(.)
5
e C.)
106
E
x,
107 107
30,199
13
217
E (k)
14 (OT) (QT) *
140
Bk
iS'
k
R
RA(.)
51
80
ran
Rr(•t•) S(.)
4
45
'k
ni
198
135 135
9
25
9
81
106—107
3
107
20
5
25
5
20
ar(.)
25
are(.)
9 9
69
85
1
36 9
182 20
145
193
10 20
125
upper
203—204 (QD)
C.)
62
26,75
11
1
a6
42
PC.)
2
a
42
PT C.)
9
T
it
ab
'AB 10
41 50 80
210
28 10
'I'(A(H))
130
214
10
W(H)
125
s—sim
9..
199
10
S(A)
115
10
TpJrnor
101
r
10
62
$
10
s(m)
21
Tiower
124
PH
45 101
TN0r
30
2 3
136
62 254
30
+
1
136
10
a
+
2
5,14
.L
13
12
0
13,59—60
13
*
[.1
13,75 13,76
12
1
12
<.,.>
12
s
18
12
V
22
12
()*
42
75
I•l1
75 75
1.12
1.,.] a
76 84
97,100 203 203
255
