Domingo A Herrero Arizona State University
Approximation of Hilbert space. operators VOLUME I
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Domingo A Herrero Arizona State University
Approximation of Hilbert space. operators VOLUME I
Pitman Advanced Publishing Program BOSTON· LONDON· MELBOURNE
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Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto
© Domingo A Herrero 1982 First published 1982 AMS Subject Classifications: Primary 47ASS, 41A6S, 47A60; Secondary 47A15, 47AS3, 81C12 British Library Cataloguing in Publication Data Herrero, Domingo A. Approximation of Hilbert space operators. Vol. 1-(Researcb notes in mathematics; 72) 1. Hilbert space 2. Operator theory I. Title II. 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. QA329.H48 1982 515.7'24 82-10163 ISBN 0-273-08579-4 (v. 1) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechaniall, photocopying, recording and/or otherwise without the prior written permission of the publishers. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the publishers. This book is sold subject to the Standard Conditions of Sale of Net Books and may not be resold in the UK below the net price. Reproduced and printed by photolithography in Great Britain by Biddies Ltd, Guildford
To Buenos Aires, on her four-hundred first birthday
"A mi se me hace cuento que empez6 Buenos Aires, la juzgo tan eterna como el agua o el aire" (Jorge Luis Borges)
Contents
1. Stability and approximation
1
1.1 Lower estimated derived from the Riesz-Dunford functional calculus 1.2 Lower estimates for the distance to Nk(H) 1.3 Lower semicontinuity of the rank 1.4 Stability properties of semi-Fredholm operators
2 6 8 9
1.5 On invariance and closure of subsets of L(H)
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 2.1.3 The lattice (N(Ek)/i, S (T)
c
R,
where -1 .S (T) = { WTW :
W
E
G(H)}
is the simiLarity orbip of T. If K(H) denotes the ideal of all compact operators acting on Hand ~:l(H) + A(H) = L(H)/K(H) is the canonical projection of L(H) onto the (quotient) Calkin algebra, then the image ~(T) = T+K[H) ofT € l(H) in 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 A nonempty bounded open subset n of the complex plane a: is a Cauchy domain if the following conditions are satisfied: (i) n has finitely many components, the closures of any two of which are disjoint, and (ii) the boundary an of n is composed of a finite positive number of closed rectifiable Jordan curves, no two of which intersect. In this case, r = an will be assumed to be positively oriented with respect to n in the sense of complex variable theory, i.e., so that 1 2'11'i
J
d>..
r >..-~
{1, if =
o,
~ e n 1 n- = nur
if ~
(the upper bar will always denote closure with respect to the metric topology of the underlying space). Clearly, r is uniquely determined by n (and conversely). We shall say that r is a rectifiabZe contour. If all the curves of r are regular analytic Jordan curves, we shall say that r is an analytic contour (or n is an anaZytic Cauchy domain). If A is a Banach algebra with identity 1 and a e A, the spectrum of a will be denoted by a(a). The complement p(a) = 0:\o(a) of a(a) in the complex plane is the resoZvent set of a and the function >.. ~ (X-a)-l (from p(a) into A) is the resolvent of a. It is well-known that (X-a)-l is an analytic function of >.. in that domain that it satisfies the first resoZvent equation: (>..-a) -l_ (p-a) -l = (ll- X) (>..-a) -l (p-a) -l (X rll e p (a) ). Furthermore, if a, be A and>.. e p(a)np(b), then (>..-a)- 1 - (X-b)-l = (X-a)- 1 (a- b) (X-b)-l (second resolvent equation). If a is a nonempty clopen subset of a(a), then there exists an analytic Cauchy domain n such that a c nand [a(a)\a]nn-= -; in this case 1 Jr (>..-a) -1 dX E(o;a) = 2 '1fi
is an idempotent of A commuting with every b in A such that ab = ba. (E(a;a) is the Riesa 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
~ith
identity 1. Assume that the spect~um o(a) of a is the disjoint u~ ion of two compact subsets a 0 and o 1 such that o 1 is nonempty (o 1 t< ~)
a~d Zet n be a Cauchy domain such that a 1
lla-brJ< minfmc;~.-a)-~J- 1 : (where JI.Jidenotes the norm of A)
II = II a- at II
;~.
If
an},
€
= min{ 11!>.-a) -~1- 1 :
A "- an} and
= t6 < li -~~- 1
for all A E an, so that o(at)nan = fJ for all t E Thus, the idempotent
....!..-f 2w1 an
et
nn- = -·
then o(b) nO 'I ffand o(b) nan =fJ.
PROOF. Assume that II a-b II = 6 < m let at= (1-t)a+tb, 0 ~ t ~ 1: then 11!>--at> -
0 and a 0
c
(>.
-at
~0,11.
) -1 d.>.
is a well defined element of A for all t E [0,1]. Furthermore, if 0 ~ t < s ~ 1, the second resolvent equation implies that
whence it readily follows that t + et is a continuous mapping from [0,1] into A. Since o 1 = o(a)nn 'I fJ, it follows that 1
eo = 2w1
f
(;i.-a)
-1
d).
F o,
an so that II e 0 II :2; 1 (Recall that et is idempotent, 0 ity, lle 1 11 :2; 1 and therefore el
= 2!i f
(>.-b)-1 d). an This is clearly impossible, unless o(b)nn
~
~
t
1). By continu
F 0. F fJ.
0
Recall that if (X,d) is a metric space and B(Xl (Bc(X]) is the family of all nonempty bounded (closed bounded, respectively) subsets of X, then ~(A,B)
= inf{e
>
0:
B
c
AE, A
c
BE},
where AE = {x EX: dist[x,AJ s e},defines a pseudometric in B(Xl (a metric in Bc(X], resp.);dH(A,B) is the Hausdo~ff distance between A and B, A, B € Bc(X]. The qualitative form of Theorem 1.1 is the following COROLLARY 1.2. (i} Let a be an element of a Banach algebra A with identity. Assume that o(a) is the disjoint union of two compact subsets a0 and a 1 such that o 1 F fJ. and that a 1 is contained in abounded openset 0. Then there ezists a constant C = C(a,o 1 ,n) > 0 such that 3
o(b)nO t: ~foro aZZ bin A satisfying lla-bll 0, there e~ists 6 > 0 suah that a(b) a(a)E, provided lla-bll < 4, i.e., the mapping a+ a(a) from A into Be(~) (Hausdorff distanae) is upper semiaontinuous.
c
PROOF. (i) follows immediately from Theorem 1.1: Given a 1 and 0 (as above) , there exists a Cauchy domain o1 such that a 1 c n1 c n 1 - c S2 and a 0 no 1 - =~-Take c = min{lj(A-a) -1II -1 : Ac: an 1 }. (ii) Apply (i) with a0 ~and 0 =interior a(a)E. 0 COROLLARY 1. 3 . (i) If S2 is an open subset of ~. then {a c: A: c S'l} is an open subset of A. (ii) If E is a G6 subset of ~. then {a € A: a(a) c E} is a G6 subset of A. In par-tiauZar, the set {a e; A: a (a) = {0}} of aZZ quasiniZpotent eZements of A is a G6 in A. a(a)
PROOF. (i) follows from Corollary 1.2(ii) and (ii) is an immediate consequence of (i) • 0 The limit case of Theorem 1.1 yields the following COROLLARY 1.4. Let a, o 1 (o 1 7: ~) and assume that b c: A satisfies the inequality
n be
II a- b II s min{ II< A-a) -11 I -1 :
as in Theorem 1.1 and >.
c:
anl;
PROOF. Define at as in the proof of Theorem 1.1; then Theorem 1.1 implies that a(atlnan = g and o(at)nS'l t: ~for 0 s t < 1. Since lim(t _,. 1) lib- at II= 0, i t follows from Corollary 1.2(i) that o(b)nO- cannot be empty. 0 It is convenient to observe that the result of Corollary 1.4(and, a fortiori, the result of Theorem 1.1 too) is sharp. In fact, we have EXAMPLE 1.5. Let P c: L(H) be a non-zero orthogonal projection and let Q = {A: I A-ll < ~}; then II< A-P) -lll-l = ~ (A € an ) • Thus' by Corol lary 1.4, a(B)nS'l- 7: ~ for all B in L(H) such that liP- Bll s ~On the other hand, if A = ~. then II P- A II = ~ and o (A) = {~} c an. If T c: L(H), o is a clopen subset of o(T) and E(o;T) is the corresponding Riesz idempotent, then the range ran E(o;T) and the kernel ker E(o;T) of E(a;T) aresubspacesofH invariant under every B-in L(H) commutinq with T (i.e., hyperinvariant forT), and H can be written as 4
the algebraic (not necessarily orthogonal!) direct sum H = ran E(a:T)+ ker E(a:T). (Here and in what follows, subspace will always denote a closed linear manifold of a Banach space.) Furthermore, the spectrum of the restriction T)ran E(a:T) ofT to ran E(a:T) coincides with a and the spectrum of the restriction T!ker E(a:T) coincides with a(T)\a [l73,Chapter XIV]. In what follows, ran E(a:T) will be denoted by H(a:Tl. If a = {X} is a singleton, we shall simply write H(X:T) (E(X:T)) instead of H({X}:T) (E({X}:T), resp). If a = {X} and dim H(X:T) is finite, then X is called a normaL eigenvaLue ofT: in this case, H(X:T) coincides with ker(X-T)n for some n ~ 1. The set of all normal eigenvalues ofT will be denoted by a 0 (T). Clearly, a 0 (T) is contained in the point spectrum ap(T) ofT (i.e., the set of all eigenvalues ofT). The esse~ tiaL spectrum ofT, i.e., the spectrum ofT in A(H) will be denoted by ae (T).
COROLLARY 1.6. Let A, Be L(H); then (i)
Assume that a is a nonempty
~Lopen)subset
of a(A) and Let Q
~Cauchy domain) be a neighborhood of a such that
[a(A)\a]nO- =-.If X dO}. then a 1 =a (B) nO -,; ~; (ii) furthermore, dim H(a:A) =dim H(a 1 :B) (0 s dim H(a:A) s ~>. (iii) If a is a nonempty cLopen subset of a (A) and the Cauchy e domain n is a neighborhood of a such that [a e (A)\a]nO- = ~. then a e (B) nO-,;- for aZZ Bin L(H) suah that IIA-BII < min{I!CX-A)-~1-l: A e an}.
II A- B II
< min{
II< X-A) -~~-l:
PROOF. (i) and (iii) follow immediately from Theorem 1.1, applied to A= L(H) and to A= A(H), respectively. (ii) This follows from the proof of Theorem 1.1. Observe that, if At= (1-t)A+tB, then the continuity of the mapping t + E(a(At)nO:At) (0 s t s 1) implies that the idempotents E (a :A) = E (a (A 0 ) nO :A0 J and E (a 1 :B) E (a (A1 ) nO:A1 ) necessarily ha-ve the same (finite or infinite) rank.O Until now, we have only applied the arguments of functional calc~ 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 su~ 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 n of a(a). Given £ > 0, there ezists ~ > 0 such that f(b) is weLL-de5
fined for aZZ b in A satisfying
II a- bll
II f (a) - f
< ti and. mol'eovel'.
(b)
II
< e.
PROOF. Let n 1 be a Cauchy domain such that a(a) c n 1 c Ql- c Q. By Corollary 1.2(i), there exists o1 > 0 such that a(b) c n 1 for all b inA satisfying lla-bll < ti 1 • Clearly, f(b) is well-defined for all these b, by means of the integral f(b)
= 2!i
Ianl f(~)
and therefore, it is a nonempty (compact) subset of ~. Its complement ~\aire(T) = pie(T)upr~(T), where pie(T) = ~\~ie(T) and pre(T) =~\are (T), coincides with ps-F(T) ={A e ~: A-T is semi-Fredholm}, the semiFredholm domain of T.
ll
The following results are an immediate consequence of Theorem 1. and its proof (see [153,Chapter IV]).
COROLLARY 1.14. Let T E L(H); then (i) Ps-F(T) is the disjoint union of the (possibly empty) open n ® -oo sets {ps-F(T)}_oof, where denotes the inner product of H. FIHl = {Lj~l fj6gj:
fj, gj
€
H, j = 1,2, ••• ,n; n = 1,2, •.• }
is the ideal of all finite rank operators acting on H. Let {e 1 ,e 2 , ••• ,ek} be the canonical orthonormal basis (ONB) of ~k and let qk" L(~k) be the operator defined by (2 .1)
(k
= 0,1,2, .•• ;
q 0 is the 0 operator acting on the trivial space {0},
q 1 is the 0 operator acting on the one-dimensional Hilbert space ~.
~l ~
and qk admits the matrix representation 0 0 0
1 0
0 1
0
0
0 0 0
0 0 0
(k
qk
0
0
0
0
1
0
0
0
0
0
X
k)
with respect to the canonical ONB, for k =2,3, •.• ). These operators will play a very important role throughout this monograph. A £cr(a)} will denote the spectral raFinally, sp(a) =max{ IXI: dius of a £ A (a 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
£
L(~
d
) and let p(X)
m
k"·
ITj=l (X-Xj) Jrxi ~ Aj• 13
for i F j) be its minimal (monic) polynomial; then the closure of the similarity orbit of T is equal to S(T)
= {A£ L(~d):
rank q(A) s rank q(T) for all qlpl,
~here
q!p denotes a monic polynomial q dividing p. Furthermore, if L £ L(~d), then L # T if and only if rank q(L) rank q{T) for all q!p if and only if L ~ T.
COROLLARY 2.2. Let T e L(~d); then the follo~ing are equivalent: (i) S(T) is maximal ~ith respect to inclusion (equivalently, [T] is a maximal element of (L(~d)/#, 1, then T sim Tl by n1-1 n1 !.emma 2.4._If r=l, then qn 1 "' }:j=l ej6ej+l E: L(a: ) is similar to e:e 1 ee 2 +-!~! 2 1 ej6ej+.l (« ~ 0) and, lett-Ulg e: + 0, we conclude that T s!m 0 T1 • In either case, Ts!m T1 sim A, and therefore A £ S (T)-.
2.1.2 Proof of Theorem 2.1 L (Q: d )
(0 S d < oo) 1 the minimal polynomial of k· J (A.-I' A., i f i -F j) and rank q(A) s rank T is p, p(A) = nj=l (;\-A.) J ~ J q(T) for all qiP· We want to show that T simA. Clearly, we can directly assume (without loss of generality) that T and A are unitarily equivalent to their Jordan forms: let T = $j:l (Aj+Qj), where a:d = $j:l Hj and Qj is a Jordan nilpotent acting on the subspace H., 0 0
~ v ~·
It is easily seen that (I:d,s) is a poset, d = max{m 0 ,m
0} >
max{m 1 ,mil
2
•••
~
max{md,md}
0
and 2 max{mj,mj} s max{mj_ 1+mj+l'mj_ 1 +mj+ 1 } s max{mj_ 1 ,mj_ 1 }+max{mj+l'mj+l}, so that p v ~· e I:d. It is completely apparent that~ v ~· is the least upper bound (l.u.b.) of~ and u' with respect to the partial orders. Since I:d is finite, every subset of I:d has a l.u.b .. In particular, ~A~·
= l.u.b.{v e I:d:
v s
~and
v s
~·}
is the (unique) greatest lower bound of~ and~·· It readily follows that (I:d,s) is a finite lattiae with supremum (d,d-l,d-2, ... ,2,1,0) and infimum (d,O,o, ... ,o,o,o). Given~= (m0 ,m 1 , .•• ,md) e I:d' define T e N(~d) by d (Tj) ~ T~ = IBj=l qj where Tj = mj_ 1 -2mj+mj+l for j = 1,2, ... ,d-1 and Td =md_ 1 : (2.2) and (2.3) guarantee that the mapping ~ ~ [T~] (2.4) is a bijection from I:d onto N(~d)/# and, moreover, that rank T j =mj 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 bijeation from (I:d,s) onto (N(~d.)/#,s). In partiaular, (N(~d)/#,s) is a finite lattiae with supremum [qd] and infimum [0] (0 = ql (d)) .
18
2.1.4 Closures of similarity orbits of finite rank operators Let H be an infinite dimensional Hilbert space and let T, A E F(H); then T and A are algebraic operators with nul T =nul A= oo and there exists a finite dimensional subspace H(T,A) reducing both, A and T, such that TIH(T,A)L = AIH(T,A)L = 0 (H(T,A) can always be defined so that dim H (T ,A) s 2 rank T + 2 rank A) • Assume that A E S(T)-; then we can prove exactly as in the finite dimensional case that rank q(A) :S rank q(T) and, by using Proposition 1.12(ii), that nul q(A) ~ nul q(T) for all qJp, 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) sim AIH(T,A). A
fortiori, T = TIH(T,A)$0 simA= AIH(T,A)$0; hence, we have
COROLLARY 2.8. Let T E L(H) be a (necessarily algebraic) finite Pank operator with minimal polynomial p, then S (T)
= {A E L (H) : rank q (A) s rank q (T) and nul q (A) for all qlp}.
Let L
E
L(H);
then L#T if and onl-y if L
~
:?:
nul q (T)
T.
REMARK 2.9. Since H is infinite dimensional, the conditions "rank q(A) s rank q(T) and nul q(A) ~ nul q(T) for all qlp" of Corollary 2.8 still imply that cr(A) = cr(T). However, the following example shows that those two conditions cannot be replaced by "cr(A) = cr(T) and rank q(A) s rank q(T) for all qJp": Let T be an orthogonal projection of rank 2 and let A be an orthogonal projection of rank 1; then cr(A) = cr(T) = {0,1}, and rank q(A) :S rank q(T) for all qlp (p(;\) = J.(J.-1) ),but A cannot belong to s(T)~, because 1 =dim H(l;A) ~dim H(l;T) = 2. Reversing the roles of A and T, we see that conditions of Corollary 2.8 cannot be replaced by "cr(A) = 0 (T) and nul q(A) ;:, nul q(T) forall qlp: Let NF(H) = N(HinF(H) and NFk(H) Since H is infinite dimensional, NF 1 (H)
c
NF 2 (H)
c
...
c
NFk (H)
c
{T
E
NF(H):
NFk+l (11)
c
rank T
:S
k-1}.
...
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
(NF 1 (H)/*, 1/k for all E in
2).
The last result of this section says that the above conjecture is true at least for k = 2. PROPOSITION 2.19. If 2 s h
= dim
n2 = 152 = inf{IIE- Oil:
E
H s €
ro,
E(H),
then Q
€
N2 !Hl}
but this infimum cannot be attained foP any paiP (E,Q), E
~. €
E(H),Q
€
N2 (H).
PROOF. If E = 1, then i t is clear that liE- oil = 1 for every quasinilpotent Q. Let E =
Ill- oil ~ sp(l-Q)
(~ :):::nEE)L
be the matrix of E € E(H)\{1} with respect to the decomposition H ran ES(ran E)L and let
26
lc oJ
0 =
be the matrix of Q
€
L(H)
(with respect to the same decomposition). ~ 2 (H)
if and only if
= AB+BD
= CA+DC = 0.
It is immediate that Q e
= o 2 +CB II E- Oil :S ~:
A2 +BC Assume that
then Ill- All
~ and
:S
liD II
:S
~ and therefore
o(A) c {A: Jl-AI s ~}and o(D) c {A: !AI s ~}.Thus, by the spectral mapping theorem, o (A 2 ) o(-BC) c {A: :le A ~ ~}
and 0
(D 2 )
=
o(-CB)
c
{A:
Since o(-BC)\{0} = o(-CB)\{0}
I AI
~}.
:S
(see, e.g., [ll9], [153], [172])and
A is invertible (recall that Ill-All s ~ < 1 ) , it readily follows that o(-BC) = o(A 2 ) = {~} c o(-CB) = o(D 2 ) c {0,~} and o(A) = {~} c o(D) c {0,!1}. Hence, liE- oil ~
Ill-All ~ sp(l-A)
= ~-
Assume thati!E-QJJ =~.Since o(A) = {~} = oR.(A), there exists a sequence {xn}n:l of unit vectors in ran E such that JJ(A-~)xnll-+- 0 (n +
oo) • On the other hand, A2 = -BC and A invertible imply that
Ellxll
II Cx II ~
for some e: > 0 and for all x in ran E, so that JJE- Oil
:?.:
lim sup(n
-+- co)
IJ(E- Olxnll
~
lim sup(n
-+- co) {
i!(l-A)xn11 2 +11-Cxn11 2 l~
~ (~+£2)~
>
~.
a contradiction. Hence
II E - Q II
> ~
for all 0 in ~ 2 ( H) •
On the other hand, it readily follows from Proposition 2.17 that inf {II E - 0 II:
2.3 On the distance to
E
€
E (H) , Q
€
~ 2 (H) }
s
~.
0
~k(H)
2.3.1 A general upper bound LEMMA 2. 20. Suppose that T
€
II T II s 1 and II Tk II s E: for some (T*T)~ = JLO,l]A dE (speatral deaomp£ L (H),
k ~ 2 and some e:. 0 < E s 1. Let sition) and let P = E([O,Ie:J); then
11((1-P)T(l-P)]k-41 s (k-1)/e:. PROOF. Clearly,
II TP II
s le:, and 27
e:11xlr
~
11Tkxll2 =
II~k- 1 xl1 2
=
IIPIT*T)~Tk-lx+ll-P) IT*T)~Tk-lxll2
~II (1-P) (T*T)~Tk-lxll 2
=
II
I 2. 7) wheroe ~kle:) is a continuous, positive, non-decroeasing function defined on 10,1] such that lim(e: + 0) $k(e:) = 0. Moreovero, ~kle:) can be inductively defined by $ 2 le:) = l2e:)~ and $k(e:) = {e:+$k_ 1 1(k-l)/e:) 2 }~. fork= 3,4, •••• PROOF. Let P be defined as in Lemma 2.20 and let T = [E 1 T1]ran P E 2 T 2 ker P be the matrix of T with respect to the decomposition If k = 2, define T' = PT(1-P) =
H
ran P$ker P.
(~ ~l).
Since
(~ ~,.2 ) Iran P and ker P
1
=
0
ran P, it follows from Lemma 2.20 and its proof that
II T- T' II
=
1 (:~ ~J 1 {II(:~ ~JII \II(~ ~JI 2 r s
s (e:+£)
~ = ( ~.
~ 3 and (2.7) holds for j s k-1, with ~ 2 (e:) and 4>.(£) = {e:+~.-l((j-2)/£) 2 }\ for j = 3,4, •• ,k-l. Assume that k
J J Clearly, IIT 2 11
28
=
11(1-P)Til-P) II s 1 and, by Lemma 2.20, IIT 2
2 £)
= (2e:)~ k-L
11 s
(k-1)/E. Thus, by our inductive construction, we can find T2 (ran P)
f
Nk-l
such that II T 2 -T211 s lj)k-l ( (k-1) /E). Define
T' It
is easily seen that T'
€
Nk(H) and a formal repetition of our
previous argument (for the case when k = 2) shows that IIT-T'Il s lj)k(E)
{E+Ij)k-l((k-l)IE) 2 }~
(def)
4
D
k.
J ().i~ Aj if i ~ jJ be a potynomiat and tet Mp(H) = {A € L(H): p(A) = 0}. Given E > 0, there e:J:ists 6 > 0 suah that, if IITII s 1 and IIP(T) II s 6, then COROLLARY 2.22.
=
Let p().)
rrj:l (A-Aj)
dist[T,Mp(H) J < E. PROOF. I f m = 1, then the result follows from Theorem 2.21. As~
2. There exists E1 > 0 such that the m open disks D1 , o2 , ••• ,Dm of radius El centered at ). 1 , A2 , ••• , Am' respectively, are pairwise disjoint. Let E0 > 0 be such that i f IP(A) I < E0 , then A is sume that m
contained in one of these disks. Let 1 I = 2ni aD. J
PJ. We shall show that
().-T)-1 d)..
is a bounded function of IIPII, IITII, e 0
Pk
for small IIP(Tlll· Observe that p(A)- p(T) = a1 q
in
(A-T)q(A,T)
,
p(J.)
for a polynom!_
the variables A, T. Thus 1-p(T)p(A)-l = ().-T)q(A,T)p(A)-l.
Now, for A such that I A-Aj I = El we have that IIP(A) small i f IIP(T)II
-1
p(Tlll
is
is. Let E=p(A)-lp(T); then
().-T) - l = p(A) -lq().,T) (1-E) - l = where II J II s II p (A) -lq (A ,T>II ·II E
IV (1-JI E II>
p(A) -lq(A,T)+J,
is small since II p (A) -lq (A ,T)II is
bounded by a funtion of II T II and p (A) • Thus the P. 's are bounded ( j
J
2, ••• ,m) • On the other hand, k. 1 II
Tj. Then p(T') = 0 and 1 ~ j ~ m} < e:,
0 small enough.
0
Let T, k and e: be as in Theorem 2.21 and let n. 0 < n ~ 1, be the k-th root of e: (i.e., 11~11 ~ nk) and define ljlk(n) = cpk(e:), k = 2,3, ••• ~ then Theorem 2.21 implies that, if k = 2, then dist[T,N 2 (H)J s l2n!kif k = 3, then dist[T,N 3 (H) J ~ (n3+4n3 / 2 )~, •• , and dist[T,Nk(H) 1 = O(e: 2 = 0 0 on (0,1], lji(O) = 0 and dist[T,Nk(H)J s 11J(n) for all Tin L(H) such that IITII ~ 1 and IITkll ~ nk (i.e., the functions ljlk(n) 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 ~+l = 2 (m+l) >nul Lm+l=2m+l. Thus, if W (U) is an invertible (unitary, resp.) operator, we can always find a unit vector x = x(W) (= x(U), resp.) such that IILm+lxll = 1, but (WTW-l)m+lx = 0 ( (UTU*)m+lx = 0, resp.), whence it readily follows from Proposition l.lO(i) ((ii), resp.) that dist[L, S(T)] :?: 2l/(m+l)- 1 (dist[L,U(T) J :?: 1/(m+l), 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): COROLLARY 2.24. (i) Let {gn}-m h ~ 3: then the operators A qk (h) (A, B € L(~kh)) satisfy dist[B,S(A)]
1:
qh (k) and B = (2.8)
however, 21/k-1 s dist[B(ool,S(A(oo))] s 4{sin 'IT/([(k-1)/2]+1) (2.9) +sin 'IT/([(h-1)/2]+1)}
Bn(l/k+l/h).
nul B =h. Thus, dist[B,S(A)] ~ 1. (Use Proposition l.lO(i) as in our previous observations at the begi~ ning of this section.) On the other hand, A~ EA for all E > 0, so that 0 € S(A)-, whence we obtain (2.8). The lower estimate of (2.9) follows from Proposition l.lO(iii) ([A(oo)]h=O, II[B(oo)Jhll = IIB(oo) II= 1.) The upper estimate follows from the proof of Example 2.25: For suitably chosen·T = ses*(oo), A' =A and B' = B, we have IIB'-A'II s IIB'-Tit+JIT-A'II s 4{sin 'IT/([(k-1)/2]+1)
+sin 'IT/([(h-l)/2J+l)} < B'IT(l/k+l/h).
D
Example 2.26 suggests that, if H is infinite dimensional, then 1 N (H) = {Q € N(H): !lOll s 1} "looks like" IU+iQI (where !1J denotes the set of all rational numbers), in the following sense: Observe that Gl+iGI = uk:l (Qik +i!IJk) , where !Ilk = {m/n: "large" nowhere dense subset of !IJ+i 0 independent of k.
0
E
Nl(H)} ~ C/k for
An affirmative answer to this conjecture would provide some heuristic explanation to the wild structure of N1 (H) - (see Chapter V).
2.3.3 An example on approximation of normal operators by nilpotents > SO, p = [,/k/2hrJ, n
aj =
p
-1
and r = [kn/2] and let Qk E k-1 L(~k) be the operator defined by Qk = Lj=l aj ej+ll8ej with respect to the ONB {ej}j~l of ~k, where Let k
l
nn, for r (n-1) < j s rn, n = 1, 2, ••• ,p, nn, for r(2p-n) < j s r(2p-n+l), n = 1,2, .•• ,p, 0, for 2rp < j ~ k-1.
(Roughly speaking: Qk is a truncated weighted shift; the weights aj grow from n 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+l) .) . . . . J . .. . In the f~rst mod~f~cat~on, we shall "~gnore" the coord~nates 1,2, ... ,r(p-1) and r(p+l)+l,r(p+l)+2, ••• ,k and apply Corollary 2.14 to the subspace V{er(p-l)+l'er(p-l)+ 2 , ••• ,er(p+l) }. It is easily seen that we can modify Qk in order to obtain an operator Ri = Tieur+l' where U +l ~ e 10e +l+L.: 1 e.+ 10e. is a unitary operator acting on a subspace r r JJ J 1 1 1 of dimension r+l and there exists an orthonormal system {f 1 ,f 2 , ••• ,f2t r (p-1) 1 2r k . R , _ such that {ej}j=l u{fj}j=lu{ej}j=r(p+l)+l ~san ONB of a:, T1ej-Qk ej for j i 1
1
(r(p-l),r(p+l)J, Tier(p-l) =fi, _
1_
Tif~=f~+l for h = 1,2, ••• , h 1
r-2, T1 fr_ 1 - (1-n)er(p+l)+l and Ur+lfh- (wr+l) fh for h = r,r+l, ••• ,2r (-wr+l is a primitive (r+l)-th root of 1); furthermore, IIOk-Rill = s(r-1). Let T1 be the operator obtained from Ti by replacing each weight equal to 1 by 1-n and R1 = T1eur+l; then II Qk- R111 ~ s (r-1) +n. Now we can apply the same argument to R1 in order to obtain an OE erator R2 = T2e (l-n)u 2 reur+l, where u 2 r is a unitary operator acting on a subspace of dimension 2r, whose eigenvalues are equal to minusthe 2r 2r-th roots of 1, T 2ej = ~ej for all j I. (r(p-2) ,r(p+2) ], T2er(p- 2 ) = 2
2
2
2
(l-2n)f 1 , T2 fh = (l-2n)fh+l for h = 1,2, •••,r-2, T2 fr-l = (l-2n)er(p+ 2 )+l'
,£;_
{fi,f~, ••• 1 } is an orthonormal system that spans a subspace orthor u{ej}j=r(p+ k gonal to the span of the vector ({ej}j=l 2 )+1), II R1 - R2 II ~ (1-n)[s(r-l)+n]and this second modification only affects the vectors 33
· th e su b space spanned by (e.}. { r (p-1) r (p+2) 1n J J=r ( p- 2 ) +1 u{e.}. J J=r ( p+ l) +1 ), so that IIOk- R2 JJ = max{JJok- R1 JJ,JJR 1 - R2 11} s s(r-l)+n, etc. An easy inductive argument shows that after p-1 steps we finally obtain an operator L = u mre:£?-1 (l-· )U 1...,.. (k+l-(2p-l)r) k r+l J=l Jn 2r ~~1 such that JJok-~11 s s(r-ll+n < 2n/r+[lk/2lnJ < 5(n/k)~ for all k >SO. On the other hand, if 1 s k s 50, then 5(1T/k)~ > 1. Thus, we have the following PROPOSITION 2.28. (i)
FoP eaah k
~
1 thePe exists a noPmaZ
opeP~
toP~ E L((tk) suah that JJLkJI = 1 and dist[~,N(a:k)] < 5(1T/k)~. (ii) If H is infinite dimensional, thePe exists a noPmaZ opePatoP M suah that cr(M) = D-, !Uhepe D = {;\: !AI< 1} and distrM,Nk(H)J < 5(1T/k)~ fop all k = 1,2, •••. In partiauZar, ME N(H)-.
PROOF. (i) If k > 50, define ~ as above. If 1 s k s SO, take Lk = 1, Qk=O. (ii) Let {Am}m:l be an enumeration of all those points A in Osuch that both lAml and (arg Am)/1T are rational numbers (arg 0 is defined equal to 0) and let M be a diagonal normal operator with eigenvalues A1 ,A 2 , ••• ,A , ••• such that nul(A -M) =cofor all m = 1,2, ••• , i. . m (co) m e., M = (d1ag{A 1 ,A 2 , ••• ,Am, •.. }) Given k, it is easytoseethatMcanbe written as M" (EDm:l Am Lk) (co), whence it readily follows that dist[M, Nk (H) J A
fortiori, ME N(H)-.
0
The result of Proposition 2.28(i) is, in a certain sense, the best possible. Observe that if Nk E L(a:k) is normal and there exist k Qk € N(a: ) and e:k > 0 such that JJNk- QkJJ < e:k' then (by Corollary 1.6 (i)) cr(Nk) is a connected set containing the origin. If the points e:k of cr(Nk) are more or less evenly distributed in a connected neighborhood n of the origin with smooth boundary (namely, n = D), then 2 cr(Nk) will include n and therefore k1re:k ~ m2 (n), where m2 denotes e:k the planar Lebesgue measure. Hence, e:k ~ [m 2 (0)/(1Tk)J~ = O(k-~). cannot be connect (On the other hand, if e:k is too small, then cr(Nk) e:k ed, a contradiction.) CONJECTURE 2.29. There exists a constant C > 0 (independent of k) 34
such that dist[N,N(~k)J ~ Ck-~ for every normal operator N such that II Nil= 1 (k = 1,2, ••. ) .
E
L(~k)
The following result provides some extra support to this conjecture. Observe that if A E L(~k) is hermitian and 0 ~ A ~ 1, then the points of a(A) are not evenly distributed in any set of positive measure. (More precisely, m2 (a(A)£) ~ 2c+~c 2 independently of k, and 2c+ 2 .,. 0, as £ ..,. 0.) nE k
PROPOSITION 2.30. If A € l(~ ), 0 dist[A,N(~k)] > (1/2/k), k = 1,2, ••••
~A~
1, and 1
E
a(A), then
PROOF. Assume that IIA-QII ~£for some Q E N(~k), Q = H+iJ (Cartesian decomposition); then IIA-HII = IIJR.e(A-Q) II~ IIA-QII ~£and trace (H) = trace (llle Q) = Ie trace (Q) llle 0 = 0. On the other hand, it is easily seen that a(A) ~ ~O,l]{Use Carol £ lary 1.6(i)), so that trace {A) ~ l+{l-2c)+{l-4c)+ ••• +{l-2nc), where n = [l/2c](= integral part of {l/2c)). It is clear that l/2c ~ n > l/2c-l. Hence, {n+l) > l/2c and n-1 trace {A) ~ {n+l) - 2c}:j=O j = (n+l)- {n+l) nc = {n+l) {1-nc) > l/4c. Let A f A dE and H = J A dF (spectral decompositions) • If a E [0,1], c'> c and rank F{{a-c',aa)) trace {A) - kc > 1/ 4c - kc. Hence, c > 1/2/k. By a compactness argUMent (exactly as in the proof of Theorem 2.12), we conclude that dist[A,N{a:k)J = min{IIA-OII=
Qk =0,
11011 ~
2} > 1/2/k.o
2.3.4 On the distance to a similarity orbit LetT E L{a:d) be a ayalia operator with minimal polynomial p, p(A) = n.m1 {A-A·)kj {A. -1- A·, i f i -1- j); then }:.m1 k. =d and Tis sim J=
J
l.
J
J=
J
ilar to the Jordan form ej:1 {Aj+qk·). Let A E L{a:d) be an operator wfth spectrum a(A) = {a 1 ,a 2 , ••• ,an} 35
and dim H(a.;A) =h .• (Clearly, }:.n 1 h. =d.) Define JJ 1 =JJ 2 = •.• =JJk =A 1, J
l.
J=
1
l.
JJk +l=JJk +2= ... =JJk +k =:\2' ••• 'JJd-k +l=JJd-k +2= ... =JJd=A and 1 1 12 m m m 11 1 =1!2= ••• = 11 h = al' 11 h +1 =l!h +2= ••• = 11 h +h =a2, ••• ' 11 d-h +1 1 1 1 1 2 n = Sd-h +2 = Bd-h +3 = • • • = Bd = an; then A admits a representation as . n
n
an upper triangular matrix of the form
A
0
Bd (with respect to a suitable ONB of ~d). It is not difficult to conclude from Theorem 2.1 that
T sim B = 0 )Jd
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)] ~ min
max
aEE(k) whe~e
E(k)
l~j~k
I JJ. - S (.) J
0
J
1.
(2.11)
denotes the set of aZZ permutations of k elements.
Unfortunately, the estimate (2.11) is very poor, in general. Name ly, if~ and Qk have the form of Proposition 2.28(i), then qk is cyclic, qk sim Qk (by Theorem 2.1) and
dist[~,S(qk)] ~ IILk-Qkll
< 5(1T/k)!oz
+
0 (k
+co).
However, sp (Lk) = 1 (-1 E a (Lk)) and a (qk) = {0}, so that the only information that we can obtain from Corollary 2.31 is that dist[Lk' S(qk)J s 1. PROBLEM 2.32. Find a formula for 36
dist[A,S(T)] (A, T
E
l(~d)).
We shall close this section with a partial answer to this problem. n
n
hk)
(o;k)
COROLLARY 2.33. If T=EDk=l ~k and A=EDk=l qk al'e finite rank opel'atol's, rank TJ =rank AJ fol' j = 1,2, ••• ,r and rank Tr+l < rank Ar+l fol' some r ;:.: 2, then 21/(r+l) -1
$
dist[A,S(T) J s 2 s([{r-1)/2]}.
PROOF. The lower estimate follows from Proposition l.lO(i). In order to obtain the upper estimate, we can directly assume without loss of generality that A, T E L(~d) (for some d, 0 < d < ~>. Then, our hypotheses and formula (2.2) imply that T· = o;. for j = 1,2, J J ... , r-1, but 'r < o:r· Since r > 1, this means, in particular, that T and A have exactly the same number of direct summands, which is equal toT=
Lj~l
'j"
After eliminating all common direct summands, we can directly assume (without loss of generality) that T =ED n q (Tk) and A=q (o:r> n {r
:X: >r'-v-'
>
m
m blocks of length r
n" - - - - _ _ _ _ __,).,.........::.:...__:'O 0 be given. Then there exists a finite rank operator F such that II K- F II < e:/3. £ £ Moreover, by the upper semicontinuity of the spectrum (Corollary 1.2),' F can be chosen so that sp(F ) < sp(K)+e:/3. £ £ Since F e: F I H) , there exists a finite dimensional subspace H of £ £ H, dim H£ = d ~ 1, such that H£ reduces F£ and F £ IHi£ = 0. Let F = F I"H , let M be a subspace of dimension kd containing H for some k £ £ £ £ large enough to guarantee that 2n/k < e:/3 and define G e: L (M ) as above £ and G e l(H) in such a way that G IM = G and G IMi = 0. £ kd £ £ £ £ Then G e F(H), G = 0 and £ e: IlK-Gil !> IIK-Fe:II+IIF'e:-Ge:ll < e:/3-+JIF-GII < e:/3+~ sp(F)+e:/3 < J;z sp(K)+e:.
Since e: can be chosen arbitrarily small, we obtain the following upper bound: PROPOSITION 2.34. If K e K(H) (H an infinite dimensionaZ space), then the distance from K to the set of aZZ finite rank nilpotent oper~ tors cannot e~ceed ~ sp(K). In partiauZar, every compact quasinilpotent operator can be uniformZy approximated by finite rank niZpotents.
2.5 Notes and remarks The problem of characterizing the closure of a similarity orbit in simple terms was raised by D. A. Herrero in [139]. This reference contains all the basic properties of the sets S(a) (for a in a Banach algebra A), the notion of asymptotic similarity, several properties of 38
the poset (A/i, (iv) => (i) of Corollary 2.3 (in the above mentioned more general setting [139,Proposi tion 1]) • Theorem 2.1 and Corollary 2.8 are due to J. Barrra and D. A. Herrero [43,Theorem 1.1], who also proved that (F(H)nN(H)/#,~and T)k < ~+(8 log k)/k (D. A. Herrero, [149,Proposition 6.5]) and 5) ok < ~+sin 1T/([ (k-1)/2]+1) (D. A. Herrero, [150,Corollary 5.2]). P. R. Halmos and L. J. Wallen called an operata~ T in L(H) a pown• parotiaZ isometroy if Tk is a partial isometry for all k (A):
cp cp
COROLLARY 3. 2.
M8 J = {cp(A)cp(B):
E
E
M8 }.{cj>(B):
o (-r ab)
cp
E
cp " M8 }
M8 } = o(A).o(B).
0
cr (a) - cr (b) .
c
PROOF. Since La~ = ~La' it follows from Lemma 3.1 that a (La-~) c a (La) -a(~). On the other hand, a (La) = a (a) and cr(b), whence the result follows. We shall see later (Corollary 3.20) that when A sion is actually an equality.
L(H)
this inclu
3.1.2 Approximate point spectrum of a sum of commuting operators The appr>oximate point spectrum of A OTT(A)
~:
{A
E
{A
E ~:
E
L(X) is the set
A-A is not bounded below}
and
a 6 (A)
=
A-A is not onto}
is the approximate defect spectrum of A. It is completely apparent that op(A) c oTT(A) c oi(A) and a 6 (A) c or(A). Furthermore, if At denotes the Banach space adjoint of A, then it is not difficult to see that crTT(A) =·cr 6 (At) and o 6 (A)
=
crTT(At).
(If X is a Hilbert space, on(A) = oi(A) = crr(A*)* and a 6 (A) oi(A*)*, where O* = {~: A f Q} for each Q c ~.)
42
LE~~
3.3. Given any Banach space X, there is an isometric imbedding of X into a larger Banach space X', and a mapping A+ A' of L(X) int9 L(X') which is an isometric isomorphism such that every A' is an extension of A and a (A') =a (A') =a (A) • p 7r 7r PROOF. Let ~ 00 (X) be the Banach space of all bounded sequences of elements of X with the norm ll{x } : 1 11 = sup llx llx and let e (X) "' n nn n o {{xn} E ~ (X): II xn II + 0 (n + "')}. It is easily seen that e 0 (X) is a subspace of ~"'(X). We define X'= ~"'(X)/e (X) and the imbedding of X into X' by x + 0 [{xn}J (=the coset of {xn}), where xn = x for all n = 1,2, . . . • Clear ly, this mapping is an isometric isomorphism of X into X'. Similarly, given A in L(X), we define A' E L(X') by A'[{xn}J = [{Axn}]; then A+ A' defines an isometric isomorphism from L(X) into L(X') •
If A ~ a7r(A), then there exists a sequence {xn}n:l of unit vectors in X such that II. E pr(T) (R 1 (~,T)(1J-T) = 1, ~ E p 1 (T), resp.). However, Rr(.,T) (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. LetT E L(H). Given E > 0, there e~ists a right resolvent for T defined on pF (T) npr (T) e~cept for an at most denumerable set S c pr(T) which does not accumulate in pr(T), such that
s
c
cap r (T)J £ ={A
€
cr:
distn,ap r (T)J s £}.
Applying the above theorem toT*, we obtain the following dual result. COROLLARY 3.9. LetT E L(H). Given E > 0, there e~ists a left re~ oZvent forT defined on pF(T)npR.(T) e~cept for an at most denumerable setS' c pR. (T) which does not accumulate in pR.(T), such that S' is included in [, Pker(TIM>* will be a finite rank projection. Thus, if we write 45
N = {z < M.l:
Pker(TJM)*PMTz=O}
we haveN c M.l, dim (M.leN) 0 there exists a right resoLvent R of T on n1 except for an at most denumerabLe set s1 , which does not accumuZate in n1 , and satisfies s1 c ran 1 J£.
=
PROOF. By Lemmas 3.14 and 3.15 and Proposition 3.16, there exists 48
a vector y E H such that Pker{A.-T)y f 0 for all A. E n 1 \s 1 , where s 1 is an at most denumerable subset which does not accumulate in n1 and such that sl c canlJ£. (Take 0 = {A. E Ql: dist[A.,an 1 J ~ £}, in Proposition 3.16.) Fix A. 0 E n 1 ,s 1 and let z be a non-zero vector in ker(A. 0 -T). If R0 is a fixed right inverse of A0 -T, then any right inverse of A. 0 -T is of the form Rt = R +zet, where t E H. Choose t
0
= -
-1
R~y
(Since Pker (A. a-T) y f 0, it readily follows
that f 0, so that t i s well-defined.): then =
=
0
for all x in H, so that y L ran Rt. For A E nl the following identity holds: (A.-T) Rt = ( {A.-A. 0 ) Rt+1) • -1
This shows that A.~ -(A.-A. 0 ) is a mapping from n 1 into the camp~ nent of pF(Rt) which contains the point at infinity. Also, for A. E n 1 , the Fredholm index of {A-A 0 )Rt+l is zero. Suppose thatker[{A 1 -A 0 )Rt+l] f {0} for some A. 1 E n 1 ,s 1 • Then it follows from the above identitytha~ for some x f 0, (A. 1 -T)Rtx = 0. In this case, Rtx E ker(A. 1-T) and since the last space is one-dimensional, ker ( A. 1 -T) c ran Rt. This contradicts ran Rt and Pker(Al-T)y f 0. It readily follows that the operator (A.-A 0 )Rt+l is invertible for all A. in n 1 ,s 1 • The operator valued function
y
E
R(A.) = Rt((A.-Ao)Rt+l]-1 is a right resolvent forT in n 1 \S 1 • This completes the proof. D LEMMA 3.18. Let T E L{H) and let Qn be a aomponent of pr(T) suah that nul (A.-T) = n, A. E Qn. Foro any £ > 0, there exists a right resolvent F ofT on Qn exaept foro an at most denumerable set Sn, whiah does not
accumulate on nn, and satisfies sn
c
cannJ£.
PROOF. We proceed by induction on n. The result is clear if n = 0, for then nn is a component of the resolvent set p(T) ofT (and F(A) = (>.-T) -l, A. E Qn) and the case n = 1 is contained in the preceding lemma. Suppose the result has been obtained in the case n = k-1. Let Qk be a component of pr(T) such that nul(A.-T) = k, A. E nk. It follows from Pr~ position 3.16 that for any £ > 0 there exists a vector y E H for which Pker(A.-T)y f 0, for all A E Qk\S', where S' is an at most denumerable set which does not accumulate in nk and satisfies S' c cankJ£. Let MA = ker(A.-T)n{y}L, for A E nk' and let M = V{MA}AEnk· Obviously, M is invariant under T and relative to the decomposition H = MGlM.L, 49
T [TM A ) • 0
T
1
M
It is easy to establish that A-TM is onto for A " Qk and clearly (A-TM ... ) is onto for A € Qk. It follows that for A " nk \S', nul ( >..-TM) k-1 and nul(A-TM.d = 1. By Lemma 3.17 and our inductive hypothesis TM has a right resolvent R(A) on Qk\S' and TM1 has a right resolvent G(>..) on Qk\S", where S" is a (possibly empty) finite or denumerable subset of Qk which does not accumulate on ~ and satisfies S" c [oQk]c· Define F (A) = (
R( >..)
0
(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 Qk\Sk' sk is at most denumerable, Sk does not have any accumulation point in Qk and Sk c rankJc·
o
PROOF OF THEOREM 3.8. Clearly, it suffices to define a right resolvent F on each component of pr(T), except for an at most denumerable subset S with the desired properties. Let n be a component of pr(T) such that nul(A-T) =n(n) ~ 0, A" n. I f n(Q) =0, then n c p(T) and the only possible definition for F is F(A) = (A-T)-l =the resolvent of T restricted to n~This is true, in particular, for the unbounded component of pr(T).) If 1 $ n(Q) < oo, then Q is a bounded component of pr(T). If Q intersects the compact set~= {A< pr(T)\p(T): dist[A,apr(T)] ~ c}, then we define F on Q\S(Q), where S(Q) is an at most denumerable subset of n which does not accumulate in n and satisfies S(Q) c Can> £ by using Lemma 3.18. If Qn~ = ~, then we can use the same arguments as in that lemma in order to construct a right resolvent F on Q\S(Q),where S(Q) is an at most denumerable subset which does not accumulate in Q (the condition S(Q) c Q c Can) is trivially satisfied in this case). £ It is completely apparent that this defines a right resolvent for Ton pr(T)\S, where S = u{S(Q): n is a component of pr(T)\p(T)} is an at most denumerable subset of pr(T) with the desired properties.o
3.1.4 The left and the right spectra of tAB THEOREM 3.19. Let A, B" L(H); then (i) a 6 (TAB) = crr(TAB) = crr(A)- aR, (B). (iiJ cr'lf(TAB) = aR,(tAB) = aR,(A) -crr(B).
50
PROOF. (i) By Corollary 3.7 and our observations at the beginmng of Section 3.1.2, cr~(TAB) c crr(A) -cr.R.(B) and cr~(TAB) c crr(TAB). Assume that \J E crr(A)- cr.R.(B) (i.e., \J can be written as \J =.a-a, where a E crr(A) and a E ot(B)) and that TAB-\J is onto. Then, given C ~ L(H), there exists X E L(H) such that (TAB-\J) (X) =TA-a,B-a(X) = (A-a)X -X(B-a) =c. Since ran(TAB-\J) is closed, there is a constant m > 0 such that II(TAB-\J) (Tlll ~ m dist[T,ker(TAB-\J) ], for all T in L(H). In parti£ ular, X= X(C) can be chosen so that (m/2liiXII :;;; IICII. Since a E or(A) and 8 E ot(B), we can find unit vectors x, yin H such that IIllciHIIJ s e < 1 and therefore max{Janl ,JenJ} s M= (1-62)-!:1 (otherwise ll'hxn+ f3nyJI < 1) for all n sufficiently large. We have 1
Jl!'u n IJ2 = = JJun IJ2 - < (1-T*T) U n ,un >
Since 2
IJ(l-T*T)xn ll=llxn 112 -2JI!'xn II +IIT*Tx n 112 2 = (1-JtJ.'x n 112 >- 11 2 > s 1-JI!'xn II .. 0 (n .. "') and, similarly, 11