Spectral Decompositions on Banach Spaces

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

623 Ivan Erdelyi Ridgley Lange

Spectral Decompositions on Banach Spaces

Springer-Verlag Berlin Heidelberg New York 1977

Authors Ivan Erdelyi Department of Mathematics Temple University Philadelphia, PA 19122/USA Ridgley Lange Department of Mathematics University of New Orleans New Orleans, LA 70122/USA

AMS Subject Classifications (1970): 47A10, 47A15, 47A60, 4 7 A 6 5 , 47B99 ISBN 3-540-08525-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-38?-08525-4 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2140/3140-543210

FOREWORD There is a new trend developing in the spectral theory of linear operators. In contrast to the classical spectral theory of linear operators and to the Dunford-type spectral operators which depend on some algebraic and topological structures outside their domains of definition, the contemporary spectral decomposition is defined only in regard to the operators invariant subspaces.

In

this way, the spectral theory can be conceived as an axiomatic system functioning within the underlying Banach space with possible extensions to more general topological vector spaces. Our purpose in this work is to extend and unify the intrinsic axiomatic perspective on spectral decompositions,

in such extension we wish to consider the

widest feasible generalization of the notion "spectral decomposition" in order to learn more about the special cases.

In this spirit we start in Chapter II the

study of the most abstract form of spectral decomposition so that when we come to the more special theory of "decomposable operators" (Chapter IV) we find that many of the known results of the latter theory are easy consequences of the preceding material.

More importantly, however, we obtain solutions to deep

problems which have been open and vigorously studied (e.g. the dual theory). Chapter I presents various classes of invariant subspaces a given operator may have.

Special attention is devoted to the single-valued extension property

as an essential tool in the study of spectral decompositions.

Chapter II is the

foundation of our axiomatic attack in the general problem of spectral decomposition.

We show that the single-valued extension property is an intrinsic element

of the spectral decomposition.

The more recent theories of "asymptotic spectral

decompositions" are treated in Chapter III.

Chapter IV brings the full power

of the spectral decomposition to bear in the theory of duality. The Appendix is aimed to supplement a few topics in the context of spectral decompositions.

The example given in section A.I offers the opportunity to

develop some spectral features of the multiplication operator in an elementary way.

Section A.2 provides an additional tool, the set-spectrum, for some proving

techniques.

Section A.3 gives a link between the two major properties which are

present in every aspect of the spectral decomposition problem, namely the singlevalued extension property and the approximate point spectrum.

Finally,section A.4

lists some open problems of the theory to open the way to further exploration. The main prerequisite for reading these Lecture Notes is the reader's interest in the spectral decomposition problem. The reader interested in this topic will be familiar with some classical properties of linear operators as the open mapping, the closed graph and the Hahn-Banach theorems.

Except for a few

other theorems known for at least twenty years with references given from Dunford

IV

and Schwartz's

"Linear Operators"

the material presented herein is self-contained.

As a final word, we wish to express our appreciation

to ~ s .

Geraldine S.

Ballard for the neat and careful typing of the manuscript. The a u t h o r s .

NOTATIONS

C,

the complex field (plane). For a set U:

U0,

tile i n t e r i o r

U,

the closure

Uc,

t h e complement ( i n a g i v e n t o t a i

aU,

the boundary

d(X,U)

the distance

set)

from a p o i n t X t o U.

For a l i n e a r o p e r a t o r T on a Banach s p a c e X: DT,

t h e domain

Ker T,

the kernel

T*,

t h e d u a l ( c o n j u g a t e ) o p e r a t o r on X*

T = T**,

t h e second dual on X = X**

o(T),

the spectrum

Ca(T),

the approximate point spectrum

Op(T),

the point spectrum

(null manifold)

~(x,T),

the local spectrum

0(T),

the resolvent

p(x,T),

the local resolvent

R(-;T),

the resolvent

~cT,

(~ i f T i s u n d e r s t o o d ) ,

XT(H),

definition

Inv(T),

the family of invariant

set set

operator t h e maximal a n a l y t i c

extension of R(.;T)x,

x c X

p. 5 subspaces under T

Inv(T,F), definition p. 26 AI(T),

the family of analytically invariant subspaces under T, Definition 2.7,

SM(T),

the family of spectral maximal spaces of T, Definition 3.1, p. 26.

p. 16

For a subspace

(closed linear manifold) Y and a linear operator T with

DTCY: TIY,

the restriction of T to Y

TY ,

the coinduced operator on the quotient space X/Y,

Y~,

the annihilator of Y

^ X,

a coset (vector) of the quotient space X/Y.

VI

Other notations:

I,

the identity operator

B(x),

the Banach algebra

of bounded linear operators defined on a Banach

space X

D(X),

the class of decomposable operators defined on a Banach space X, Definition ii.i, p. 73.

F,

the family of closed subsets of C

K,

the family of compact subsets of C

s(x),

the family of subspaces of a Banach space X

AT ,

the algebra of analytic functions on an

open neighborhood of ~(T),

T e B(X)

~r(K),

the algebra of analytic

C[a,b],

t h e a l g e b r a o f c o n t i n u o u s c o m p l e x - v a l u e d f u n c t i o n s d e f i n e d on a c l o s e d

of operators

interval

f u n c t i o n s on a compact K v a l u e d i n t h e s u b c l a s s

from B(X) which commute w i t h T E B(X)

[a,b]

supp f ,

the support of f ~ C[a,b]

E,

spectral

supp E,

s u p p o r t o f E, p.

capacity,

Definition

8 . 1 , p.

60

61.

Abbreviations:

h-T,

for kI-T

k-TIY,

for ~IIY- TIY

SDP (2-SDP),

(2-) spectral decomposition property

SVEP,

the single-valued extension property

det,

determinant

V,

span

c.l.m.

the smallest closed linear manifold spanned by a family of vectors

[]

end of proof. The arrow has two uses:

(a) x

n

+ x

indicates that the sequence x n tends to the limit x (in a given topology),

(b) it expresses a mapping between two sets which can be a linear operator, a function, a relation between families of sets, etc. Except for the standard notations of real intervals and references, the round and square brackets are indiscriminately used for the convenience of a clearer separation of various terms.

CONTENTS

Introduction 3

Chapter I: INVARIANT SUBSPACES

3

§

i.

Invariant subspaces and the single-valued extension property

§

2.

Analytically invariant subspaces

14

§

3.

Spectral maximal spaces

26

Chapter II: THE GENERAL SPECTRAL DECOMPOSITION

36

§

4.

Operators with spectral decomposition properties

§

S.

Operator-valued

36

functions with spectral decomposition

properties

45

Chapter III: ASYMPTOTIC SPECTRAL DECOMPOSITIONS

50

§

6.

~lalytically decomposable operators

SO 55

§

7.

Weakly decomposable operators

§

8.

Spectral capacities

60

§

9.

Decomposable spectrum

64

Quasidecomposable

69

§ i0.

operators

Chapter IV: DECOMPOSABLE OPERATORS

73

§ ii.

Properties and characterizations

§ 12.

The duality theory of spectral decompositions

of decomposable operators

80

§ 13.

Spectral decompositions

97

of unbounded operators

APPENDIX A.I.

A.2. A. 3.

A.4.

73

106 An example of an analytically invariant subspace which is not absorbent

106

The set-spectra of decomposable operators

109

The approximate point spectrum and the single-valued extension property

112

Some open problems

114

BIBLIOGRAPHY

116

Author index

120

Subject index

121

INTRODUCTION "Concerning general non-normal transformations, it. is quite easy to describe the s t a t e of o ~ knowledge; i t is non-existent. No even u n s ~ f a ~ t o r y generalization exists for the t ~ n g ~ a r form or for the Jordan cano~cal form . . . " P.R. Halmos from "Finite-Dimensional Vector Spaces" D. Van Nestrand Co., Princeton, 2nd Edition 1958, Appendix, p.192.

The spectral theory of linear operators on some organized topological vector space has undergone a prodigious development from the time Halmos wrote the comments quoted above from his popular book. A unified treatment for various classes of linear operators which perform a spectral decomposition of the underlying space and give rise to some functional calculi, impels for an axiomatic formulation of the problem.

We shall confine th?

presentation of this problem to bounded and some closed linear operators acting on an abstract Banach space X. The basic requirement imposed on operators by any spectral theory is the existence of proper invariant subspaces.

A proper invariant subspace Y under a

given linear operator T may be an element of the spectral decomposition.

Also Y

produces the restriction TIY of T as well as the coinduced operator T Y on the quotient space X/Y, operators which may inherit some basic properties of T. For a linear operator T, the functional calculus is based on the isomorphic mapping f ÷ f(T) from the algebra A T of analytic functions defined on an open neighborhood of o(T) into the Banach algebra B(X), for which 1 ÷ I and ~ ÷ T, where 1 and ~ denote the constant function f(~) ~ 1 and the identity function f(~) ~ X, respectively.

An indication that an intimate relation exists between T

and f(T) is given by the spectral mapping theorem which asserts that o(f(T)) It is known set

that

for

every

compact

= f(o(T)). K CC

and

open

D C

K there

exists

an open

G s u c h that (i)

K CGCG

CD;

(ii)

G has at most a finite number of components

(iii)

every component

rectifiable Jordan curves (iv)

G.i

n ~Gi}l;

has a boundary formed by a finite number of simple

ri~;

K N r.. ij = ~, for all i,j.

We denote by

r = ~)

rij

and f o r e a c h f e AT, we p u t I f(X)d% = ~ f f(X)d% . r i,j r . . xj We call F, endowed with the above properties, an a c ~ s £ b l e

contou~ which surrounds

K and is contained in D. For T e B(X) and K = o(T), Dunford's formula 1 f(T) = ~-

establishes the isomorphic mapping

f f(X)R(X;T)d% r

f ÷ f(T)

of the functional calculus.

If c(T) is disconnected then 1 E = 2 - ~ f R(),;T)dX F is a projection whenever the admissible contour F disconnects the spectrum. The range EX of E is a subspaee of X invariant under both T and f(T).

CHAPTER I INVARIANT SUBSPACES § I.

Invariant subspac~ and the single-valued extension property. The concept of single-valued extension property

is a major unifying theme

for a wide variety of linear operators in the spectral decomposition problem.

I.I.

Defi~on.

A closed ~@~zr operator r : DT ( C_ X) ÷ X ~

have the single-valued e ~ e n s ~ n property

(abbrev.

SVEP)

said to

i f for every function

÷ DT analytic on Df, the condition

f : Df ( C C )

(% - T ) f ( ~ )

=

0

on Df

/mp//es f = 0. Equivalently, for every x ~ DT any two analytic extensions f,g of R(X;T)x agree on Df N Dg.

When this property holds, the union of the sets Df as f varies

over all analytic extensions of R(%;T)x is called the local resolvent set and is denoted by p (x,T).

The SVEP implies the existence of a maximal analytic

~

extension x(-) of R(.;T)x to p(x,T).

This function identically verifies the

equation (i.i)

(%

- T)X(~)

=

x

on

O(x,T).

The local spectrum o(x,T), defined as the complement in C of p(x,T) is the set ~

of the singularities of x. Not every bounded or unbounded linear operator enjoys the SVEP as it is seen from the following useful property due to Finch [i].

I. 2. Proposition. A Hosed l i n e ~ operator T which ~ s u r j e c t i v e but not i n j e c t i v e does not have the SVEP. Proof.

Given T as stated by the Proposition we can exhibit a function f that

violates Definition I.I. II x 0 II = I.

Since T is not injective we can choose x 0 ~ Ker T with

T being closed and surjective, by the open mapping theorem, there

exists k > 0 such that for every y ~ X there is an x ~ DT satisfying conditions Tx = y, II xll Choose x

< k II Y~I

inductively so that n

TXn = Xn-l' II xrill < k II xn_lll

, n = 1,2 ....

Define

f(X) =

(1.23

Xnxn •

[

n=O II x n II ~ k n , t h e s e r i e s

Since

(1.2)

converges

for

It[

< k -1 and t h e n f i s a n a l y t i c

on D

0 such that II ( X - T ) x The range of I -T I-T

*

~i > k

i~x* II , for all

being closed, I-T* is injective.

X ~

~ X .

But then the kernel of

is the zero point and consequently the range of I-T is the entire space X.

This, however, contradicts the hypothesis. hence i e Oa(T ).

Thus X-T

is not bounded below and

The second part of the Corollary follows by a similar proof. []

Relation (1.3) will be shown (Theorem 4.5) to be an intrinsic property of operators which have a general spectral decomposition. Some immediate implications of the SVEP are expressed by the following

1.5.

Propos~on.

If m ¢ B(X) ha~ the SVEP £ h e n t h e foLtowd~g ~ s ~ w t i o ~

hold: (i) (ii) (iii) (iv) (v) Proof.

O(x+y,T) ~ o ( x , T ) U e ( y , T ) ,

x,y s X;

ax(X) + by(X) = ( a x + b y ) ( X ) ,

a , b e C, x , y ¢ X, ~ ¢ p ( x , T ) N p ( y , T ) ;

O'(x,T) = ~

iff

X = O;

~ ( S x , T ) C_ c~(x,T), for e v ~ y S ¢ B(X) which c o y o t e s wddch T; o[~¢(X),T] = o ' ( x , T ) , x e X,

% ~: O ( x , T ) .

Since properties (i) - (iv) are well-known (see e.g. Dunford and Schwartz

[i, XVI., 2.1 and 2.2]) we shall prove (v). For every I ~ p(x,T) there is an X-valued function ~

analytic verifying

equation (1.4)

( p - T ) ~ x ( l i ) = x(X) on

p [x(X) , T ] .

Apply (h-T) to both sides of (1.4), use (i.i) OJ-T) ( X - T ) ~ ( ! a ) note that (I-T)$ X is analytic on

= (X-T)x(.),) = x,

p[x(X),T] and conclude that U ~ p(x,T).

Thus

o(x,T) C o'[(xCX),T]. To obtain the opposite inclusion define the analytic function gX:p(x,T) ÷ X x(l~)-x(t) (1.5)

if ~ ~ X,

gx(~) = -

For

by

~ # X

x'(k),

if

~

+

x

=

k.

we obtain (v-T) gx (V') =

and this extends by ~ ÷ X.

x V.-X + x(X)

= 7¢(~,),

Consequently,

~[~(~),T]C_. ~(x,T). [] Given T ~ B(X), for every set H c C , (1.6)

XT(H) = {x g X : o ( x , r ) CZH}

is a linear manifold in X (Proposition 1.5, (i) and (ii)).

For K compact in C,

denote by AT(K) the set of functions analytic on a neighborhood of K and valued in the class of operators from B(X) which commute with T. to an algebra as the one mentioned in the Introduction.

AT(K) can be extended For every f e AT(K ),

consider the mapping f:XT(K ) ÷ X defined by (1.7)

f[TI~(K) ]

=

~

1

I f(l)x(l)di F

,

where r is an admissible contour surrounding o(x,T) and contained in

p(x,T).

A functional calculus which parallels the Riesz-Dunford functional calculus can be developed in terms of the local resolvent x by (1.7). Actually, that mapping is homomorphic with I + TIXT(K ) and 1 ÷ IIXT(K), (Apostol [6]). This functional calculus gives rise to certain "localization" theorems extensively developed by Bartle [1,2], Battle and Kariotis [i] and applied to some special operators by Stampfli [1,2].

In the next theorem which generalizes the spectral

mapping theorem, we follow the proof of Bartle and Kariotis [i].

1.6.

Theorem.

hood D of ~(T).

Given T ~ B(X), l e t f:D

(1.8)

Proof.

÷

C be a n a l y t i c on an

neighbor-

f[o(x,T)] = o[x,f(T)].

If f is constant (1.8) is trivially satisfied, assume therefore that f is

nonconstant.

First, we show that f(X0) ~ p[x,f(T)] implies that k 0 ~ p(x,T).

10 E D with f(k 0) ~ p[x,f(T)] and let G be an open neighborhood of f(G) C p [x,f(T) ].

Then

(1.9)

[f(l)-f(T)]x[f(l)]

I (1.10)

gx (~)

is analytic in both X and p.

k0 such that

÷ C d e f i n e d by

f(X)-f(P) for p # - p '

=

f'(X), for

and (I.i0).

Let

= x, f o r all X ~ G.

S i n c e f i s n o n c o n s t a n t on G, t h e f u n c t i o n g:G x g

(1.11)

open

I f both T and f(T) have t h e SVEP then for every x ~ X,

~ = X

Note the analogy of the functions defined by (i.5)

The functional calculus applied to (I.i0) produces

f ( X ) - f ( T ) = (X-T)gx(T).

With the help of (1.11), equation (1.9) becomes [f(1)-f(T)]x[f(l)] = (;~-T)gl(T)x[f(l)] = x and since gl(T)x[f(1)] is analytic on G(3 I0 ), we have ~0 g ~(x,T) implies that f(X0)c o[x,f(T)] and hence

f b ( x , T ) ] ~ ~[x,f(T)].

l 0 ~ p(x,T). Then

In order to obtain the opposite inclusion, let

~0 ~ f[o(x,T)]. We separate

90 from f[o(x,T)] by disjoint neighborhoods V and W of v0 and f[q(x,T)], respectively. Let H C D

be a neighborhood of

an admissible contour surrounding and we have

9 -f(k) # 0

o(x,T) such that

f(H) C W ,

and let F be

o(x,T) and contained in H. Then f(P) C

for all

~ e V and ~ e H. Denote

W

C = {~ :Ill= liT U+ I} .

The functional calculus in AT[q(x,T)] , with the help of (i.ii) gives

I

[~-f(T)]2-~{ f [v-f(k)]-IxT(k)d% = F 1 f = 2~iifFXT(l)d% +2-~ F i f R(X;T)xd% 2~i C

[f(l)_f(T)][9_f(%)]-l~T(%)d%

i f [~_f(X)]-igx(T)xd~ = x, + ~-~ r

the last integral being analytic in the region bounded by P. Since i 2~i f [v-f (%) ]-ixT (k)dX F that

is analytic on the complement of

f[o(x,T)], it follows

v 0 e p[x,f(T)]. Thus we have obtained o[x,f(T)] C f[o(x,T].

1.7. Corollary. Given T c B(X),let

[]

f:D + C be a n a l y t i c on an open neighbor-

hood D of ~(T) and nonconstant on every component of D. I f both T and f(T) have t h e SVEP then f o r every F ¢ F, (1.12)

Proof.

Xf(T)(F ) = XT[f-I(F)]. By Theorem 1.6, for every

x e Xf(T)(F),

f[a(x,r)] = q[x,f(T)] C F, and consequently

~(x,T) C f-l(F).

Thus

F e F

x e XT[f-I(F)]

and hence

Xf(T)(F)C XT[f-I(F)]. Conversely, for

x ~ XT[f-I(F)], we have

o(x,T) C f-l(F) and then Theorem

1.6 implies q[x,f(T)] = f[~(x,T)] C f[f-l(F)] = F. Thus XT[f-I(F)] C Xf(T)(F), and property (1.12) follows.[]

We shall s e e later (Corollary 2.21) that in Corollary 1.7, the hypotheses that both T and f(T) have the SVEP are redundant. In the search for invariant subspaces the SVEP may be very helpful. it provides us with h y p e A J 2 t v c ~

Actually,

subspaces, i.e. with subspaces that are in-

variant under every operator which commutes with the given one.

1.8.

Proposi/sion.

L e t T g B(X) have the SVEP.

For every subset H of C,

the subspac66 XT(H ) a n d XT(H)& a r e hyperinvariant under T and T * , respectively. Proof.

First let x ¢ ~ - ~ .

There is a sequence {xn} C ~(H) which converges

(in the norm topology) to x.

If S ¢ B(X) commutes with T then Proposition l.S

(iv) implies that g(Sx,T)

Cg(x,T)C

H, for a l l

n.

Thus, for every n, s ~ n ~ XT(H) and the continuity

Next, let y

of S implies

c XT(H)~ .

that

Sx ~ XT(H).

By the first part of the proof, for every x ¢ XT(H) < Sx,y

Consequently,

Hence, SXT(H ) C ~(H).

> = 0.

we h a v e 0 = < Sx,y

and hence S'y* s ~(H) ~

> = < x,S y

>, f o r

all

x e XT(H)

The proof is complete. []

The relationship between the spectrum and the local spectra is expressed by the following

I. 9.

Theorem.

I f T ~ B(X) h a s t h e SVEP t h e n co(T) =

(,_j O ( x , T ) . x ¢ X

Proof. The inclusion

o(T) D

~j

o(x,T)

x c X

follows directly from the definition of the local spectrum. (1.13)

X0 ¢ o(T) -

U o(x,T). x e X

Let

The operator

%0-T is surjective because for every x ~ X, we have ~

(Xo-T)x(~,O) = x. Then Corollary 1.3 implies that

k0 ~ 0 (T) but this contradicts

(1.13). []

The SVEP is inherited by the restrictions of the given operator.

I. 10.

Proposition.

Le~t T ¢ B(X) have the SVEP and let Y ~ Inv(T).

Then

TIY h ~ the swP ~ d o(y,T) C o ( y , T I Y ) , Proof. every

for every y e Y.

The first assertion of the Proposition follows at once.

Let

y ~ Y.

For

k ¢ p(y,T[Y), we have (X-T)y(X)

= (x-rlv)~-(x)

= y

and h e n c e O(y,T[Y) C p ( y , T ) .

[]

The SVEP i s s t a b l e u n d e r u n i f o r m c o n v e r g e n c e .

1.11.

Theorem.

Given T ¢ B(X), l@~ {Tn}

be a sequence i n B(X) s a t i s f y i n g

(i)

Each T commies w ~ h T;

(ii}

Each T has t h e SVEP;

(iii)

The sequence { T } c o n v ~ g e s t o T i n t h e uniform operator topology.

n

n

Then T has t h e SVEP. Proof.

Let f:D ÷ X be analytic and verify equation

(1.14)

(~.-T)f(~.)

=

0 on D.

Let ~ ¢ D be arbitrary and let K i = {v ¢ C:[~-~ I _< r i } C D ,

i = 1,2 and r 2 < r I.

By the uniform convergence, for every 6 > 0 there is an n such that the operator = Tn-T has the norm ;] ~

II < 8.

Take B = min (r2,rl-r2) and denote

% For ~ ¢ K6c' ~-~ ¢ P ( ~ )

--

_
II T ]I

Fix

10 in V and put x 0 = g(10).

we have R(I;TIYI)X 0 = R(I;T)x 0 s Y.

Since V lies in the unbounded component of p(TIYI), by analytic continuation, we have R(X;TIYI)X 0 g Y, for all k ~ V.

For I = lO, R(X0;TIY1)x0 = R(x0;TIY1)z(I 0) = f l ( 1 0 ) ¢ Y. The p o i n t X0 b e i n g a r b i t r a r y

in V, we have f l ( X ) ¢ Y GY2 on V. f(t)

Hence

e Y2 on V

and on a l l o f D by a n a l y t i c c o n t i n u a t i o n . Next, we can devise another spectral decomposition of X with respect to a couple of covering open half-planes G I, G 2 such that H 2 N G 2 = ~ and D-G 1 J ~. There are subspaces Z I, Z 2 s Inv(T) which perform the spectral decomposition X = Z 1 + Z 2,

o(TIZi) C G i ~

o(T), i : 1,2.

Then we have

(4.15)

~(TIY 2) n :(TIZ 2) = ~ •

44

By repeating the above procedure, we find that

f(X) ~ Z2 on D. Thus both invariant subspaces Y2 and Z 2 satisfy hypothesis

(4.5) of Lemma 4.7.

Hence if f ~ 0, then Lemma 4.7 and relation (4.15) imply that

D C e p ( T J Y 2 ) ~ ap(TIZ2) = ¢ . Thus f = 0 and t h e p r o o f i s concluded. [ ] Now we have a decomposition of the operator with the underlying space. I~at about the spectrum?

4.10.

Theor~r,.

I f T has t h e SDP then f o r any open cover {Gi} I of o ( T ) ,

t h e r e i s a s p e c t r a l decomposition (4.1) such t h ~ n

e(W) Proof.

=

U °(r[Yi)i=l

In view of Lemma 4.6, there is a spectral decomposition (4.1) with n

U

~(T[Yi) C ~(T)

i=l

To prove the opposite inclusion, let x g X be arbitrary but fixed.

We have a

representation n

x =

~ Yi' with Yi ~ Yi' i = 1 , 2 , . . . , n . i=l

Proposition 1.5 (i) implies rl

o(x,~') c U

n

~(yi,'r) c U

i=l

a(~tYi)-

i=l

Now, with the help of Theorem 1.9, we obtain n

X

~

X

=

Corollary' 1.3 applied to the dual operator T* implies that if T* has the SVEP then

X-T* is not surjective as long as X E ~(T).

As an application, we

prove that if the original T has the SDP then a restriction of

X-T* is at least

injective.

4.11.

Corolla~y.

I f T has t h e SDP then for every F ~ F and X ~ F c,

(x-T*) ]XT(FC)X Z6 inje~/LLve.

45

Proof.

Let ~ ~ F c and let G be open such that F C G and X e G c.

open cover of C.

Then {FC,G} is

By the SDP, there are subspaces YI' Y2 c Inv(T) which perform

the spectral decomposition

X = Y1 + Y2

o(TIYt) C F c, oiTIY2) C G . Let y* ¢ ~ ( F C ) ~

verify equation (~-T*)y* = 0,

and let x g X be arbitrary.

By the spectral

decomposition,

there is a representa-

tion x = x I + x 2 with x i s Yi' i = 1,2. Since Y1 C

~r(FC), we have < xl,Y* > = 0.

As ~ e p(TIY2),

there is a unique Y2 E Y2 such that (~-T)y 2 = x 2.

Then < x2,Y* > = < (~-T)Y2,y* > = < y2,(l-T*)y*

> = 0.

Thus, we have < x,y* > = < Xl,y* > + < x2,Y* > = 0. Since x is arbitrary in X, it follows that y* = 0. The surjectivity

of (X-T*)I~(FC) ~

holds under some more restrictive

conditions as it will be seen later (Proposition

5.

12.6).

Operator-valued functions w ~ h s p e c t r a l decomposigion p r o p e ~ i e s . in this section we shall examine the stability of the SDP under the functional

calculus.

5. I.

Lemma. Let T have the SDP and l e t f:D + C be a n a l y t i c on an open

connected n~ighborhood D of Proof.

o(T). Then f(T) has t h e SDP.

For f constant the assertion of the Lemma is obvious.

forth consider non-constant

functions.

Let {Gi} 1

By the spectral mapping theorem, we have n

(S.l)

f[~(T)] = ~[f(T)] C U i=l

Gi

We shall hence-

be an open cover of ~ If(T)~.

48 and consequently n

n

o(T) C f -I (~) G i) : [.J f-l(Gi). i=l i=l So {f-l(Gi)} ~

is an open cover of c(T).

By Lemma 4.4 there is a spectral

de-

composition n

(5.2)

X =

(5.3)

[ Y. l i=l

a(TIYi) c f-l(Gi ) ~ ~(T), i = 1,2 ..... n. In view of Proposition 1.15 the spectral inclusion property expressed by

(5.3} implies that ever)" Yi is invariant under the resolvent and hence under f(T) by the functional calculus.

With the help of (5.S) we obtain successively:

o[f(T) IYi] = o[f(TIYi)] = f[o(TIYi)] E f[f-l(Gi)] C G i, i ~ i f n. Now (5.1) and (5.2) complete the spectral decomposition of f(T). [] The SDP is stable under finite direct sums. 5.2. property. Proof.

Lemma.

If T i s B(Xi), (i = i,2] have the SDP H e n

Ti ~

T 2 ha~ that

Let {Gi} ~ be an open cover of

(5.4)

o(T I (~) T 2 ) =

O(Tl)~J

o(T2).

Let {Yi}~ C InV(Tl) and {Zi} ~ C Inv(T 2) be corresponding systems of invariant subspaces pertinent to the spectral decompositions n

n

XI = i!l Y i'

X2 = i!l Zi;

o(TIIY i) C G i, Then Yi Q T1 ~

T2

o(T21Z i) C G i, i = 1,2 .....n.

Zi (i _< i < n) are invariant subspaces of X 1 Q

X2

under

and n

El Q xz= ~ (Yi Q zi)" i:l

Furthermore, in view of (5.4) we have

~[(T i Q 5.3.

z2) l(Yi G

zi)]

: o(TiIYi) U ~(T21Zi) c o i, i = 1,2 .....n . D

Lemma. Let T have t h e SDP and l e t T be a s p e c t r a l s e t .

corresponding s p e c t r a l p r o j e c t i o n then TIE(z)X h~as t h e SDP.

I f E(T) /S t h e

47

Proof.

Let {Gi} 1

be an open cover of

Without l o s s o f g e n e r a l i t y from T' = ~ ( T ) - x .

we may asswne t h a t

each G. l i e s 1

at a positive

Let H be an open n e i g h b o r h o o d o f x' d i s j o i n t

distance

from e v e r y G.. 1

Then {Gi U H}~ c o v e r s ~(T) and t h e SDP o f T i m p l i e s t h e e x i s t e n c e o f {Zi} ~ C I n v ( T )

satisfying

the spectral

decomposition n

(5.53

x =

Z

z.

i=l (5.6)

o(T]Zi)~GiU

Let E = E(T) and Y. = EX t% Z.. I i

1

H, i =

1,2

.....

n.

Since E commutes with T it follows from (5.5)

that n

EX =

(5.73 Moreover,

EX e SM(T),

[

i=l

Y.. 1

(Example 3.14) so

o(TtYi3 C ~(TIEX) = ~. Now use (5.6) and apply the functional calculus to T[Z i, to find invariant subspaces V i, W.I such that Zi = V i Q

W i,

o(TIV i ) C Gi , cs(TiWi) C H ,

i = 1,2 . . . . . n.

Then, for every i, we have

(5.8)

Yi = ( Y i n Vi) Q

(YiN Wi3

so that

(5.9)

~(TIY i A Wi) ~ o ( T [ Y i) ~ o(TiW i ) ~ - r ~ H = ~ ,

by the choice of H.

From (5.9) it follows that Yi ~ Wi = {0}.

Thus (5.8) implies that (5.10)

~(TiYi) c o ( T I V i )

~ G i, i = 1,2 ..... n.

Hence (5.73 and (5.10) prove that TiEX has the SDP. p~ The foregoing proof of Lemma 5.5 was prepared to be suitable for the 2-SDP case.

48

5.4.

Theorem.

Let T have t h e SDP.

I f f:D ÷ C iS a n ~ y t i c

on an open

neighborhood D of o(T) then f(T) has t ~ e SDP. Proof.

Let D1, D 2 , . . . ,D m be t h e components o f D which i n t e r s e c t

• j = Dj N o ( T ) ,

(1 < j < m).

The s u b s p a c e s E(Tj)X b e i n g i n v a r i a n t

o(T) and p u t under both

T and f ( T ) , we have m X = @ E(Tj)X, j =1

m T = @ TIE(~j)X, j =1

(5.11) m

f(T) = @ f(T) I E ( z j ) X . j=l Since by Lemma 5.3 every T]E(Tj)X has the SDP, Lemma S.I implies that every f [ T ] E ( ~ j ) X ] = f(T) IE(Tj)X has t h e SDP.

Finally,

Lemna 5.2 a p p l i e d t o t h e d i r e c t

sum (5.11) p r o v e s t h a t

f(T) h a s t h e SDP.[-] In order to carry over the SDP from f(T) to T we need some additional conditions on f.

5.5.

Lemma. Given T e B(X), l e t

open connected n~ghborhood D of ~ ( T ) .

f:D + C be anagytic and i n j e c t i v e on an

Then T has t h e SBP i f

f(T) has t h a t

pro-

perry.

Proof.

Let {Gi} 1

be an open cover of o(T). Then, we have n n o[f(T) l = f[o(T)] C f ( U G i) = U f(G i) i=l i=l n

and hence {f(Gi)} 1

is an open cover of

In view of Lemma 4 . 6 , t h e r e system

{Yi}l~Inv[f(T)]

o[f(T)].

is a spectral

decomposition in terms of a

, as f o l l o w s n

X = (5.12) Hence every

o[f(T)[Yi] C f ( G i ) N Yi

~ Yi' i=l o[f(T)], i = 1,2 ..... n.

is a v-space for f(T) and then Theorem 2.4 implies that the Yi

are invariant under T.

Then, with the help of (5.12) we obtain successively:

f[o(T[Yi)] = a[f(TIYi)] = o[f(T)[Yi ] C f ( G i )

49

and hence we have

o(TLY i) c f-1[f(Gi)] = G i, i = 1,2 ..... n. []

5.6.

Theorem.

Given T s B(X), £ e t

an open neighborhood D of ~(T). Proof.

f:D ÷ C be ~ a l y t i c

and i n j e c t i v e on

Then T kas t h e SDP i f f(T) has t h a t p ~ p ~ y .

For any spectral set ~ of ~[f(T)] with the corresponding spectral pro-

jection E(T), f(T) IE(T)X inherits the SDP from f(T), by Lemma 5.3.

Then the

assertion of the Theorem follows via Lemmas 5.5 and 5.2 in similar lines with that of Theorem 5.4.[-]

NOTES AND COMMENTS. Lemma 4.8 with several applications in spectral decompositions was proved by Foias [3] . For an interpretation of Theorem 4.5 see Appendix A.3. A partial isometry T satisfying the hypotheses of Proposition 4.3 turns out to be a spectral operator in Dunford's sense.

The sum (4.2) represents the

canonical decomposition of the spectral operator T (Dunford [3] , Dunford and Schwartz [i, )0/.4.6]).

Such a decomposition of a partial isometry T was obtained

by Erdelyi and Miller [I] under the asymptotic condition lim n÷~

II T*T n - TnT*III/n

=

0.

CHAPTER III ASYMPTOTIC SPECTRAL DECOMPOSITIONS How much remains true of the spectral theory if we drop the fundamental condition of linear sum decomposition?

A spectral theory can be built for an

operator which avails itself of finite systems of invariant subspaces with the linear sum dense in the underlying space.

Thus the sum representation for

the vectors in the given space can be weakened by the n o ~ vectors from the invariant suhspaces.

limit of sums of

We shall refer to this type of spectral

theory as asymptotic spectral decomposition. The study of this spectral theory is rewarding.

Many basic properties of

the spectral decomposition can be extended to the asymptotic case.

Thus, for

instance the spectrum of an operator which satisfies an asymptotic spectral decomposition is equal to the approximate point spectrum.

Indeed, the asser-

tion of Lemma 4.4 remains valid if we replace the linear sum of the invariant subspaces by its closure and then Theorem 4.5 follows directly.

§ 6.

Analytically decomposable operator. 6. I.

Definition.

T ~ B(X) i s said to be analytically decomposable i f

for every open cover {Gi} I

of ~(T) t h e e is a system of anaZytically invariant

subspae~ {Yi}l performing the following ~ y m p t o t i e s p e c t r ~ decomposi~on n

x=

Yi' i=l

(6.1) o(TIYi) C G i (or o(TIYi) C F i ) ,

I < i < n.

We shall refer to (6.13 as analytic spectral decomposition.

6.2. Proof.

Theorem.

Let

Every analytically decomposab£e o p e r ~ r

has the SVEP.

T be analytically decomposable and let f:D ÷ X be analytic and

verify equation (%-r)f(k) = 0 on an open D C C . If f ¢ 0 then D ~ o(T) ~ @ .

In this case, Lemma 4.4 applied to analytically

decomposable operators, implies the existence of an analytically invariant subspace Y with

o(TiY) C D. On t h e o t h e r hand, Lemma 2.19 i m p l i e s t h a t

51

D C o(TIY), but this is impossible because D # ~ cannot be open and compact at the same time. This contradiction

implies that f = 0 on D. []

The extension of analytic spectral decomposition

from T to f(T) follows

easily.

6.3.

Theorem.

Given T a n a l y t i c a l l y decomposable, l e t f:D + C be analytic

on an open n6ighborhood D of Proof.

n

Let {Gi} 1

o(T).

Then f(T) /S analytic~61y decomposable.

be an open cover of o[f(T)] = f[~(T)].

covers o(T), there is a system {Yi}l C A I ( T )

n

Since {f-l(Gi)} 1

which performs the following

analytic spectral decomposition

X=

n [ Yi ' i=l

~(TIYi) c f - l ( G i ), i = 1,2 ..... n. By Theorem 2.20, each Yi is analytically invariant under f(T) and ~oreover, we have ~[f(T) IYi] = o[f(TIYi)]

6.4.

P~position.

= f[~(T]Yi) ] C f[f-l(Gi)] C G i, i = 1,2 ..... n. []

Let T be a n a l y t i c a l l y decomposable.

I f E i s a projection

c o m m ~ g with T then TIEX i s a n a l y t i c a l l y decomposable. P~of.

Denote S = TIEX and let {Gi} 1

be an open cover of o(S).

G O = p(S), {Gi} 0 forms an open cover of o(T).

Putting

There is a system {Yi}o C A I ( T )

such that X =

n I Y" i=0 i

and

o(TIYi) C G i, i = 0,i ..... n.

By Example 2.14, EX e AI(T) and hence the subspaces Z. = Y. A EX, i = 0,1,...,n 1 1 are analytically invariant under S, by Proposition 2.17 (i).

Since

o(TIZ O) c o ( T [ Y O) ~ a(T]EX) C GO f] a(S) = fl , Z0 =

{0} .

E being a projection,

we have EX =

n ~ Z.. x i=l

52

Again, by Proposition 2.17 (i), Z i is analytically invariant under TIY i and hence we have

a(S]Z i ) = o(T]Z i ) C o(T[Yi) C G i , 6.5.

Theorem.

i = 1,2 . . . . . n. [ ]

Let Tj ¢ B(Xj), (j = 1,2) and let T = T 1 (~ T 2.

analytically decomposable i f f each T. is a n a l y t i c ~ y 3 Proof.

Then T ,66

decomposable.

The "only if" part follows from Proposition 6.4. For the "if" part, let

n be an open cover of {G i}l a(T) = o(TI)[.j o(T2). There are analytically invariant subspaces Y.. under T. such that •J ] n

xj = a ~

Yij and o(Tj[Yij ) ~ G i ,

i = 1,2 .....n; j = 1,2.

By Proposition 2.18, for every i, Yi = Yil G is analytically invariant under T.

Yi2

Since

n

X =

[ Y. and a(TlYi) C G i, i = 1,2 ..... n i=l i

T is analytically decomposable. [] The property expressed by the foregoing Theorem 6.5 is valid for any finite direct sum of continuous linear operators.

6.6. Proof. (ii),

Proposition.

I f T is a n ~ y t i e a l l y decomposable then so is T[T'X.

Write S = T I T X a n d let {Gi}~ be an open cover of a(S). TX

By Example 2.16

(as a special case) is analytically invariant under T and hence a(S) C a ( T ) .

Without loss of generality we may assume that {G i} covers a(T). analytically decomposable, there is a system {Yi}~ CAI(T)

T being

such that

n

X = i=l [ Y.i and

The Z i = TY i

a(TIYi) c G i, i = 1,2 ..... n.

form a requisite system of analytically invariant subspaces under

S, for Z i ~ AI(T), by Exan~le 2.16 (ii) and then Z i ¢ AI(S), by Proposition 2.17 (i). Moreover, Z i c AI(T[Yi) and hence we have the following inclusions

53

~(S[Z i) c ~ ( T I Z i) C (~(TIYi) ~ G i, i = 1,2 ..... n. Finally, n

TX =

~ Z. i=l i

follows from the continuity of T. [] Analytic spectral decompositions

6.7. space

XI ,

Theorem.

are stable under similarity transformations.

Let T be a n a l y t i c a l l y decomposable on

S ~ B ( X I ) i s s i m i l a r to T then S i s ~ a ~ y t i c ~ l y

X.

I f for a Banach

decomposable on x I .

PJwo~. Let P:X + X 1 be a bounded invertible linear operator which performs the similarity transformation between S and T, i.e.

PT = SP.

First, we show that if Y ~ AI(T) then PY ¢ AI(S). invariant under S.

Let

Y E AI(T).

Then PY is

Let f:D + X 1 be analytic and satisfy (X-S)f(X) ~ PY on D.

Then P-I(x-S)f(X)

e Y on D,

or --(X-T)p-If(x) e Y on D. Since P-If(h) is analytic on D, by the hypothesis on Y we have P-if(h) E Y, and hence f(k) s PY on D. Thus PY c AI(S). Now let {Gi} ~ be an open cover of o(T) = g(S) and let { Y i } ~ c A I ( T ) perform the analytic spectral decomposition n

x=

~ Yi'

~(rIYi) cGi, i= 1,= ..... n .

i=l Since SIPY. 1

is similar to TIY i under the invertible restriction PIYi, we have a(SIPYi) = o(TIYi) ~ G i ,

Also

i = 1,2 ..... n.

54 n

n

Z Ph

p

:

i=l

i=l

Yi

is dense in X 1 because P is a continuous surjection. [] Analytic Dunford-type A scalar

spectral scalar

operator

decompositions

operator

S has an integral

(6.2)

s =

over the complex plane

are stable

(Dunford

[2.3]

under a perturbation

, Dunford and Schwartz

by a [1,

Part

III]).

representation f C

XdE(x)

C in terms of a resolution

of the identity

E.

Each T ¢ B(X)

w h i c h c o m m u t e s w i t h S, commutes w i t h e v e r y e l e m e n t o f E.

6.8.

Theorem.

Let T be a n a l y t i c a l l y decomposable and S a s c a l a r operator.

I f T commutes with S then TS a n d T + S a r e a n a l y t i c a l l y decomposable. Proof.

Let E denote the resolution of the identity for S and let 6 > 0.

Then for

a suitable partition {bj} of a(S) by pairwise disjoint Borel sets and for l j ¢ bj, we have II S-YjXjE(Dj)_ II

~

6 II T II - 1 ,

or (6.3)

IJ TS-~j~jTE(bj)II < 6.

Since each Ej = E(bj) commutes with T, every component Tj = TE D = T IEjX is analytically decomposable by Proposition 6.4. Then the direct sum operator P = ~DljTD = @ j

(6.4)

is analytically decomposable by Theorem 6.5.

~.T. ] ] By the integral representation (6.2)

of S a~d in view of (6.3), there is a sequence {Pn ] of operators of the type P as given by (6.4) which converges to TS in the uniform operator topology. Now let Y ¢ AI(T).

Then, by Proposition 2.17 (i), Y. = Y N E.X is analytiJ ] cally invariant under T. and by Proposition 2.18 J Y = ~)j Yj is analytically invariant under P. 2.13, Y ¢ AI(TS).

Thus, for every n, Y ¢ AI(Pn) and by Corollary

55

If Z ¢ AI(P) then every Z.3 = E.Z] a AI(Tj) and Z = __Oj Z=j is analytically invariant under T = Q

j Tj, by Proposition 2.18. Hence it follows from what we

have shown that Z ~ AI(TS). Let {G i} be a finite open cover of o(TS). For n sufficiently large {G i} also covers P

n

(e.g. Dunford and Schwartz [i, VII.6.3]). Since every P

n

is analytically

decomposable, there is a system {Yi } C AI(Pn) which satisfies (6.5)

X = ~i Yi'

Since every Y. ¢

°(PnlYi) C Gi'

for every i and all large n.

AI(TS) and for n sufficiently large we can have (rf.above cited)

i

~(TSIYi) c Gi,

for every i,

we conclude that TS is analytically decomposable. Finally, the identity T + S = (X + S ) [ ( T reduces

t h e sum T+S t o a p r o d u c t

the scalar

§ 7.

operators

R(X;-S)

- X)R(X;-S)

+ I],

for X ¢ p(-S)

between the analytically

d e c o m p o s a b l e T-X a n d

a n d X+S. [ ]

Weakly decomposable o p e r a t o r . 7. I.

Definition.

T ~ B(X) / S called weakly decomposable i f

cove~ {Gi} 1 of o(T) there i s a s y s t ~

for every open

of speoO~al ma~Omal spaces (Yi}l which

performs the following ~ y m p t o t i c s p e c t r a l decomposition n

X=

(?.i)

~Yi'

i=l ~(rlY i) c c i, (or ~(T]Y i) C G i ) ,

i = i,2 ..... n .

We shall refer to (7.1) as a weak spectral decomposition.

7.2.

Proposition.

then there e x i t s

Let T be weakly decomposable. I f G C C i s open such t h a t

a nonzero s p e c t r a l m a ~ , a l space Y of T with t h e proper~y o(TIY) C G.

Proof.

The a s s e r t i o n

t o a weak s p e c t r a l

of the Proposition

decomposition.

[]

follows

directly

f r o m Lemma 4 . 4 a p p l i e d

56 7.3.

Lemma. Given T s B(X), let Y g SM(T) and let f:D ÷ X

be a nonze~to

function a n a l y t i c and v e r i f y i n g equation (7.2)

(X-T)f(~) = 0 on an open D C C.

Then e i t h e r

D N cr(TlY) Proof.

D C C~p(T[Y).

or

: ~

Suppose that D (] o(TIY ) ~ ~ and let 10 c D (] o(TIY ).

By differentiating

(7.2) n times, we obtain (7.3)

Tf(n)(x) = xf(n)(x) + nf(n-l)(x),

where f(-l)(X) = 0.

n = 0,i .... ; ~ s D

The linear manifold Xn = V { f ( x 0 ) ,

f'(x O) ..... f(n)(10) )

being of finite dimension is a subspace of X.

X n is invariant under T since for

every x =

n ~ ~k f(k)(X0) E Xn, (~k ~ C) k=0

with the help of (7.3), we obtain Tx =

n n [ ~kTf(k)(XO ) = [ ~k[lof(k)(lO ] + kf(k-l)(iO)] k=O k=O

It is easily seen that we have a triangular matrix representation l-X

-I

0

... 0

]

i....... ITI..... :i:::i I 0

0

0

with the determinant det [(X-T) IXn] = (Â-x0)n+lThus, R(X;TIXn) exists for any X # l 0 o(T[X n) C The hypothesis on

X0

and therefore {XO}, n = 0,i ....

implies

g(T[ Xn) C a(TIY), and since Y s SM(T), we have X C Y, n = 0,i,... n

X-XOJ

s Xn.

for (X-T)IXn:

57

Thus it follows that f(n)(l 0) g Y, n = 0,i .... and since f is analytic, there is an open U C C such that {f(~):~ ~ U} C Y. Then Len~na 4.7 implies that D C ~p(TIY).

7.4. Proof.

Theorem.

[]

Every weakly decomposable op6~tor has the SVEP.

Let T be weakly decomposable and let f:D ÷ X be analytic and verify

equation (~-r)f(~) = 0 on an open D C C . We may assume that DCo(T)

and is connected.

By Proposition 7.2, there is a

nonzero spectral maximal space Y such that

o(TIY) c D. If f # 0 on D, by Lemma 7.3,

D C o(TIY). S i n c e D i s open and n o n v o i d , t h i s

7.5. Corollary. decomposable. Proof.

is impossible.

Thus f = 0.

Every weakly decomposable operator i s analytically

Since every weakly decomposable operator has the SVEP, by Theorem 7.4,

Theorem 3.9 implies that the spectral maximal spaces of T are analytically invariant under T. [] The stability of the weak spectral decomposition under the functional calculus is subject to a restrictive condition on f.

For T weakly decomposable, Theorem

6.3, Corollary 7.5 imply that f(T) is only analytically decomposable.

7.6.

i~nma.

Given T c B(X), let f:D + C be analytic and injective on an

open neighborhood D of o ( T ) . A subspace Y of x i s spectral maximal for T i f f i t i s s p e c t 2 ~ maximal for f(T). Proof.

First, we prove the "if" part of the assertion.

Let

Y ¢ SM[f(T)] .

Then Y is a v-space for both f(T) and T, by Corollary 3.5 and Theorem 2.4, i.e.

(7.4) If Z ¢ Inv(T) satisfies

~(T[Y) C o ( T ) . condition ~(T[Z) C o ( T [ Y ) ,

58

then by (7.4) and Theorem 2.4, Z is a v-space for both T and f(T). o[f(T)]Z] = o[f(T[Z)] = f[o(T[Z)] ~

f[o(T[Y)]

Hence we have

=

= o[f(TIY)] = o[f(T)]Y] and since Y s SM[f(T)] Conversely,

let

, we find that Z C Y .

Thus, Y e SM(T).

Y e SM(T) and let Z ~ Inv[f(T)] satisfy G[f(T) IZ] C G[f(T) IY]-

Now Y is a ~-space for both f(T) and T, and so is Z. f[o(TIZ)] = o[f(T]Z)] =

o[f(T) IZ] co[f(T)[Y]

Then, we have

= G[f(T[Y)] = f[o(T[Y)],

and hence o(TIZ) C o(TIY). Since Y ~ SM(T), it follows that Z C Y

7.7.

Theorem.

Given

open neighborhood D of

T a B(X),

and hence Y a SM[f(T)]. [] let

f:D

÷ C be analytic and i a j e c £ i v e on an

Then T i s weakly decomposable i f f

o(T).

f(T)

is

weakly

decomposable. Proof.

Let f(T) be weakly decomposable and let {Gi} 1 be an open cover of

Since

~(T) G D ,

o(T).

Then

the sets

~(T).

H. = G. f~ D (I < i < n) also form an open cover of 1 1

n

{f(Hi)} 1 is an open cover of

o[f(T)]

and we can find spectral

maximal spaces Y. of f(T) such that 1 n

X=

(7.5)

[ Yi ' i=l

o[f(T)]Yi ] C f(Hi), i = 1,2 ..... n.

By Len~ma 7.6, Yi e SM(T) and from (7.5) we obtain o(TIY i) C H i C Thus, T is weakly decomposable. Weak spectral decompositions Nevertheless,

perturbations

The "only i~' part of the proof is similar. [] are highly perishable under perturbations.

of weakly decomposable operators by spectral operators

result in analytically decomposable

7.8.

Theorem.

G i, i = 1,2 .... ,n.

operators.

I f T is weakly decomposable and Q is q ~ i n l l p o t e ~

with T then T + Q i s a n a l y t i c a l l y decomposable.

commuting

59

Proof.

Let

Y s SM(T).

Then Y is invariant under T + Q.

T being weakly decom-

posable, it has the SVEP, hence Y e AI(T) by Theorem 3.9 and T Y has the SVEP by Theorem 2.11.

QY being quasinilpetent commuting with T Y, T Y + QY = (T + Q)Y has

the SVEP by Corollary 1.12 and then Theorem 2.11 implies that Y is analytically invariant under T + Q.

Since o[(T+Q) IY]

= o(TIY),

we see that every weak spectral decomposition for T is an analytic spectral decomposition for T + Q. []

7.9.

Theorem.

which c o m m ~ Proof.

I f T i6 weakly decomposable and A ,is a spe~£ral operator

with T then T + A i s a n a l y t i c a l l y decomposable.

L e t A = S + Q be t h e c a n o n i c a l

and Q i s t h e q u a s i n i l p o t e n t

part

d e c o m p o s i t i o n o f A, w h e r e S i s t k e s c a l a r

o f A. T h e n , T commutes w i t h b o t h S and Q, and

T + A = (T + Q) + S. Since T + Q is analytically

d e c o m p o s a b l e by Theorem 7 . 8 ,

so i s T + A, b y T h e o r e m

6.8. ~] 7. I O.

Theorem.

t o r comm~ging with Proof.

Let T be weakly decomposable and l e t A be a s p e c t r a l operaT.

Then TA / S a n a l y t i c a l l y decomposable.

Let

A=S+Q be the canonical decomposition of A, where the scalar part commutes with the quasinilpotent Q.

We have TA = TS + TQ

where TS is analytically decomposable by Theorem 6.8 and TQ is quasinilpotent. Also TS commutes with TQ.

Let E be the resolution of the identity for A.

Any spectral maximal space Y of T has the following properties: (a)

Y e AI(T), (Theorem 3.9);

(b)

Y e AI(TS), (part of the proof for Theorem 6.8);

(c)

Y c AI(TA), ((b.) and Corollary 2.12);

(d)

For any Borel set b,

Y (3 E(b)X is analytically invariant under

TIE(b)X , TSIE(b)X and TAIE(b)X ((a), (b), (c), Example 2.14, and Proposition

2.17 (i)); (e)

If for Borel sets bj, we put Ej = E(bj), (j, finite), P =

(~)jjjX.TE.

and Y = v(~)j Y fl E.Xj

60

then Y ~ AI(P), ((d) and Proposition 2.18). Now, let {G i} be a finite open cover of

E = (~j as in (6.4) with each

Xj ~ 0.

o(TA) =

o(TS) and let

X.TE

J J Since for every j, XjT is weakly decomposable,

there are spectral maximal spaces Y.. satisfying the weak spectral decomposition x] X = ~i Yij' °(T]Yij)C Xj-iGi , for every i. Then by (d), for every i, the

y3[ ..~ z = y 1j

E.X j

are analytically invariant under PIEjX, TSIEjX and TAIEjX. =

i

=

By putting

YJ

we have o(P[Y i) = Uj[Xjo(TEjlY~)] C

UjXj(xilG i) = Gi, for all i.

By (e), (b) and (c) the Y.l are ar~alytically invariant under P, TS and TA.

Finally,

we choose the approximation P of TS such that ofTSIY i ] ., - C G i, whenever o(PlYi%., - C Gi, for every i.[-]

§ 8.

Spectrc~ c a p a c i t i e s . One of the most important tools in the classical spectral theory of self-

adjoint operators in Hilbert spaces is the set of orthenormal projections which is extended to the resolution of the identity in Dunford's theory of spectral operators.

There is an analogue of these concepts for more general operators

which possess some kind of spectral theory and this is the spectral capacity.

g.I.

Definition.

Given a Banach space X over C, a s p e ~

capacity i s a

mapping E:F + S(X), (S(×) denotes the family of subspaces of X) w~ich possesses t h e following p r o p ~ e s : (i)

E(~) = (o}, E(C)

(ii)

E( N Fn) =

(iii)

X = ~jE(G~),j for every f i n i t e open cover {G~} of C.

n

= x;

~ E(Fn), for any sequence {Fn} C F; n

J

We call E a weak s p e c t ~

capacity if condition (iii) is weakened by

(iii') X = ~jE(~),j for every finite open cover {Gj} of C.

61

E is said to be a 2-spectral capacity if the original condition

(iii) is

replaced by (iii")

X = E(~I) + E(G2) , for every couple of open sets G I, G 2 which cover

C. We define the Support of the (weak) spectral capacity E to be supp E = (-~ {F ¢ F:E(F) = X}.

8.2. E

Definition.

T ¢ B(X)

iS said to possess a (weak) s p e c t r a l capacity

i~ (iv)

E(F) e I n v ( T ) ,

(v)

[TIE(F)] C F ,

I t f o l l o w s from ( i i ) The intersection property

for a l l

F ¢ F

for each F e F. that

E(F1) C E(F2) whenever F 1 C F 2, f o r F l ,

F2 ¢ F .

(ii) likewise holds for arbitrary families of closed

sets as it can be shown with the help of Lindelof's covering theorem. For brevity, we shall refer to the defining properties of the spectral capacity given in Definitions 8.1,8.2 and in some pertinent remarks as (i), (ii), (iii), (iii'),

8.5.

(iii"),

(iv), and (v) throughout this section.

Proposition.

Let T possess a weak spect)u~ capa~bty E.

open s e t such t h a t G N supp Proof.

I f G i~ an

E # ~ then ECG) ~ {0}.

There e x i s t s a s e c o n d open s e t H such t h a t G and H c o v e r C but -

supp E ~ @.

Then, by (iii') X = E(~) + E(H). If E(~) = {0}, then E(H) = X and consequently X = E(supp E N ~). But the last equality contradicts the definition of supp E.

8.4.

Co~l~y.

Thus E(~) # {0}.

l e t T possess a weak s p e c t 2 ~ capacity E.

Then

X = E[o(T)].

Proof.

Let K = supp E.

a t ¢ K-~(T). and d i s j o i n t

Since

We p r o p o s e t o show t h a t K C ~(T).

Suppose t h a t t h e r e i s

~(T) i s compact t h e r e i s a c l o s e d d i s k F w i t h c e n t e r a t

from a ( T ) .

By P r o p o s i t i o n 8 . 3 , E(F) ~ {0} and t h e n by ( v ) , o [ T [ E ( F ) ] ~ ~(T) C F ~ o(T) =

62

but this is clearly impossible.

Hence K C o(T), and it follows that

X = E(K) C E[o(T)] C X.

8.5.

Corollary.

[]

Let T possess a weak s p e c t r a l capacity E.

Then

a[TIE(F)] C o ( T ) , for a l l F ¢ F, i.e.

E(F) i s a v-space for T.

Proof.

Corollary 8 . 4 ,

(ii) and (v) imply

o [ T I E ( F ) ] = o[TIECF ) n ECoCT))] = o[T[ECF n oCT))] C F I"1 o ( T ) c o ( T ) .

8.6.

Corollary.

Let T ~ BCX) p o s s e s a (weak) s p e c t r a l capacity E and l e t

{Gi} 1 be an open cover of ~(T).

n ~7 E ( F i ) ,

x =

Then t h e r e i s a system { F i } I C F such t h a t (resp.

X =

i=l

There is an open set

o[TIECFi)] C Gi' i = 1,2 ..... n.

H with the properties

n

n

C = [U

Gi ] UH = U

i=l

n Y ECFi)); i=l

and

GicFi, Proo f.

E]

(Gi U H) and

n o(T) = 9.

i=l

If we put F i = Si U H, Ciii), (resp. (iii') implies n

X =

n

~ E(Fi), (resp. X = ~ ECFi) ). i=l i=l

Furthermore, (v) with the help of Corollary 8.5 implies

a[TIECGiU ~)] c:: CG-iU G) n oCT) 8.7.

Theorem.

= Gi

n °CT) c:::Gi' i

=

1,2 . . . . .

n. []

I f T ~ B(X) possesses a s p e c t r a l capacity then T h ~ t h e

SVEP.

Proof.

In view of Theorem 4.9, it suffices to show that T has the SDP.

If E is

the spectral capacity possessed by T then Corollary 8.6 exhibits a spectral decomposition for T.[]

8.8.

Theorem.

I f T ~ B(X) p o s s e s s ~ a s p e c t r a l eapacdty E then for every

F g F, E(F) e A I ( T ) .

Proof.

Let f:D ÷ X be analytic on the open connected domain D such that (X-T)f(X) ~ E(F), for all X ¢ D.

63

First assume that

DOFC~. S i n c e by ( v ) ,

o[TIE(F)] there is an open disk G C D

c

N p[T[E(F)].

F,

For X e G put

g(k) = (l-T)f(k] and n o t e t h a t h(k) = R[k;T]E(F)]g(k)

¢ E(F),

f o r k e G.

It follows that

g(X) = ( k - T ) h ( k ) ,

~ e G

and by Theorem 8.7,

f(k)

= h(k) e E(F) on G.

By analytic continuation

f(X) a E(F) on D. Next, assume that

D C F. Let G be an open disk with G C D and put K = G c.

Since D and K 0 cover

by (iii), we have

x = ECD) + E(K). By Lemma 4.8, there is an open disk

V CG

and analytic functions

fl:V + E(D), f2:V ÷ E(K)

s u c h that f(l)

= fl(X)

+ f2(X) on V.

A l s o f o r I e V, (X-T)f2(),)

= (~.-T)[f()~)-fl(k)]

E E(F)N

E(-ff) CE(F),

and hence (k-T)f2(~) c E(K) N E(F) = E(K (] F), Since

V fl (K N F) = @, i.e.

v ~(KNF) c~ by the first part of the proof

@,

k e V.

C ,

64 f2(x) e E(K N F), for

X ¢ V.

Hence for X ~ V, f(~) = fl(X) + f2(K) ~ E(D) +

E ( K A F) C E(F),

and by analytic continuation f(X) ¢ E(F), for all

8.9.

Corollary.

{Gi} I

cover

of

Let T possess a s p e c ~ x ~ capacity.

~(T), t h e r e i s a system

following s p e c t ~

X ~ D. []

Then f o r every open

{Yi}~C AI(T) which performs t h e

decomposition n

x=

Y Yi' i=l

o(T]Yi) C G i, i = 1,2,...,n. Proo f.

The assertion of the Corollary follows from Corollary 8.6 and Theorem

8.8.[]

§ 9.

Decomposable spectrum.

9. I. D e f i n d ~ o n . T iS ~aid to have decomposable spectrum i f f o r e v ~ y open cover {Gi} ~ of o(T), t h e r e d~ an asymptotic s p e c t ~ decompositAon n

X = ~ Yi' i-1

o(TtY i) C G i, i = 1,2 . . . . . n

~/th {Yi}l C Inv(T), such t h a t n

(9.1)

o(T) = U (~(T[Yi)" i=1 By Corollary 4.10, every operator with the SDP has decomposable spectrum.

It is not yet known whether every operator with an asymptotic spectral decomposition has decomposable spectrum.

The decomposable spectrum, however, may be

a helpful spectral property.

9.2. Theorem. are equivalent: (i)

T

(ii)

If

Let T be weakly decomposable.

Then t h e following statements

has decomposable speGG~um;

F C~(T) iS closed and G D F i s open then t h e r e exists Y ¢ SM(T)

such t h ~

F Co(TIY ) c G .

65 (iii)

Every system {Yi}~ C SM(T) s a t ~ f i e s

(9.1) whenever

n (9.2)

X=

Proof.

7. Yi" i=l

Since the implication (iii) => (i) is obvious, we prove only (i) => (ii)

and (ii) => (iii). (i) => (ii): Let F C o(T) be closed and G ~ F be open.

Then G and Fc

cover o(T) and hence there are Y, Z E SM(T) satisfying conditions

oCTIY) c G, o(TIZ ) N F = ~, o(T) = oCT]Y) O o(T[Z).

Consequently, F Co(TIY ). n

(ii) => (iii):

Let {Yi}l be an arbitrary system of spectral maximal spaces

of T satisfying (9.2).

If n

F = U o(TIY i) ~ o(T), i=l then there is a Z ¢ Sk~(T) such t h a t

(9.3]

F C o(T[Z) ~ g(T).

Then n

n

o(T[ ~ Yi) C U i=l

o(TiYi) C ~ ( T [ Z )

i=l

and since Z is spectral maximal, it follows that

n X= but this contradicts (9.3).

9.3.

Corollary.

~ Y. C Z i=l l

Therefore, F =

o(T). []

L ~ T be weakly decomposable witch decomposable s p e ~ m .

I f Y, Z ~ Inv(T) are such t h a t o(TIY ) and o(Tiz) are d i s j o i n t and both contained i n o(T), then Y + Z i s a d ~ r e ~ s~7. Proof.

First, assume that Y,Z ~ SM(T).

Since o(TIY ) and o(TIZ ) are compact,

there exists decreasing sequences {Gn} and {Hn} of open sets with o(T[Y) = Q n

Gn

and

o(T]Z)=

~'~ Hn. n

By Theorem 9.2 (ii), for each n there is a spectral maximal space W °(TIY) U ~(T[Z) ~o(TlWn) C Gn U Hn"

n

such that

66

Then, for every n, Y + Z C W n, and hence Y+ZCW=

~ W n

n

.

Furthermore,

a(TIW) C A

a(TlWn) C A (Gn U Hn)

n

=

o(TIY) U °(TlZ)"

n

Thus o(T[W) is the disjoint union of two spectral sets. E[o(T[Z)] be the corresponding projections in W.

Let E[o(T[Y)] and

We have

o(T1E[o(T[Y )]W) C o ( T I Y ) , o(TIE[o(T[Z)]W) C o ( T t Z ) , and s i n c e

Y, Z ¢ SM(T), E[~(T[Y)]W C,Y, E[o(T[Z)]W C Z .

Consequently, W=Y

Q

Z.

Now let Y and Z be arbitrary invariant subspaces with disjoint and contained in o(T).

~(TIY ) and

o(T]Z)

Let G and H be disjoint open neighborhoods of

c(TIY ) and ~(TIZ), respectively.

By Theorem 9.2 (ii) there are spectral maximal

spaces X 1 and X 2 of T with the properties o(TIY) Co(T[XI) c G ,

o ( T I Z ) C o(TIX2) C H .

By the first part of the proof, X 1 + X 2 is a direct sum.

Since Y C X 1 and

Z C X 2, it follows that Y + Z is a direct sum.[]

9.4.

Theorem.

Let T be a weakly decomposable operator with decomposable

spect~tam. Then T p o s s ~ s ~ F ~ F, E(F) ¢ SM(T).

Proof.

a weak sped~ral capacity E such t h a t f o r e v ~ y

For closed F C a(T), define

(9.4)

E(F)

= (~

{Y:Y ~

SM(T), o(TIY)~

F}

and for arbitrary F ~ F, let

E(F) =

E[F 0 o(T)].

We propose to prove t h a t E i s a weak s p e c t r a l c a p a c i t y possessed by T. Obvious ly, E(~)

= {0},

E(C)

: x

and for every F s F, E(F) e SM(T), by Proposition 3.6. with

(9.53

o(T I Y ) ~ F,

o[T[E(F)] Cc~(T[Y)-

Moreover, for Y g SM(T)

67

To prove (iii') of Definition 8.1, let {G.} be a finite open cover of i There is a system {Yi}CSM(T) performing the weak spectral decomposition X = ~i Yi'

c(TIYi) C G i

C.

for all i.

Then, for every i,

Yi C

(-]{z:z ¢ SM(T), o(TIZ)D [i n oCT)} = E[[ i N o(T)l = E(g i)

and hence (iii') of Definition 8.1 follows. Next, we show that

(9.6)

o[TIE(F)] C

Let k e Fc be arbitrary and let

G D F be open with X ¢ G c.

there exists

F. By Theorem 9.2,

Y ¢ SM(T) satisfying F C~(TIY ) CG.

In view of (9.5), ~ e o[TIE(F)] and then (9.6) follows. Now let {Fn} C

F

and put F=~F

n

n

.

Then E ( F ) C M E(Fn) n and since for a l l n, o[TIE(Fn)] C Fn, we have

~[Tll-]E(Fn)] n

C

~

o[TIE(Fn)]CF.

n

Thus

for every

Y ¢ SM(T) with

E(Fn) C2Y, n o(TIY)'~ F. Then i t follows from (9.4) t h a t ~

E(Fn) C E(F). n

and hence E satisfies (ii) of Definition 8.1. With this the proof is complete. []

9.5. Coasllary. Let T ~ B(X) be weakly decomposable with decomposable spectrum. T h e e i~ a weak spectral capacity E possessed by T such that e v ~ y Y c SM(T] has the represent~tion y = E[o(TIY)].

PtLoof.

Let

Y ~ SM(T) be arbitrary.

Then for every weak spectral capacity E

q(T[E[o(TIY) ]) co(TIY)

68

in, lies that E[g (T1Y) ] C Y . By Theorem 9.4, there is a weak spectral capacity E of the type (9.4), so that for F = o(TIY), we have Y C ~{Z:Z

¢ SM(T), o(T]Z) D a(TiY)}

= E[o(TIY)].[~

9.6. Corollary. Let T be weakly decomposable with decomposable spectrum. Then t h e r e ~ a weak spectral capacity E possessed by T with supp f = o(T). Proof.

By Theorem 9.4 and Corollary 9.5, there exists a weak spectral capacity

E possessed by T such that every Y ¢ SM(T) has the representation Y = E[a(T]Y)]. In view of Corollary 8.4, supp E C a ( T ) . Suppose t h a t supp E ~ a(T), and denote G = (supp E) c.

Then

G Q o(T) # ~ and by Proposition 7.2, there exists

a nonzero Y ¢ SM(T) such that a(T[Y) C G. We h a v e E[o(TIY)] ~ X = E[o(TIY)] N E(supp E) = E[(~(TIY) (] supp E] = {0}. But this contradicts {0} ~ Y = E[o(T]Y)]. []

9.7. Theorem. Let T possess a weak s p e c t ¢ ~ capacity E. Then in each of the following cases: (i) Every Y ~ SM(T) has a r e p r e s e ~ i o n Y -- E [ a ( T I Y ) ] ; (ii) T is

supp E = ~(W);

weakly decomposable with decomposable spectrum.

Proof.

In view of Corollary 8.6, T is weakly decomposable.

cover of o(T) and assume to the contrary that n F = U °[TIE(Gi)] ~ a(T). i=l Let G = F c.

Then

c N o(r) ~ 9.

Let {Gi} ~ be an open

69

In case (i), there exists a nonzero Y ~ SM(T) with 7.2.

~(TIY) C G ,

by Proposition

By denoting K = ~(TIY), in view of the representation of Y, we have

(9.7)

E(K) # (0} with

~[TIE(K)] C G N ~(T).

In case (ii), there is a closed K c G 8.3.

Ao(T) with E(K) ~ {0}, by Proposition

Thus, (9.7) summarizes both cases (i) and (ii). We may asstuae that K C G j ,

for some j.

Then E(K) C E(Gj), for some j and

hence

(9.8)

~[TIE(K)] c ~ [ T I E ( F ) ].

Combining (9.7) and (9.8), we obtain o[T[E(K)] C C ~o[T[E(F)] C G ~F = ~, but then

§ 10.

E(K) = {0} is a contradiction. []

Q~ideaomposable o p ~ z ~ o ~ . A weakly decomposable operator acquires some interesting properties if it

has decomposable spectrum.

A subclass of weakly decomposable operators endowed

with this property contains the quasidecomposable operators.

These operators

are characterized by a basic property (Definition i0.i, below) of the (nonasymptotic) decomposable operators (Chapter IV). There is now a possible doubt that the reader might have.

By our trend to

acquire more properties for the asymptotic spectral decompositions we may end up in the class of decomposable

operators.

More to the point the question is

whether the class of decomposable operators is or is not a proper subclass of the quasidecomposable operators.

The answer to this question was given by an

ingenious example constructed by Albrecht [i] of a quasidecomposable operator which is not decomposable.

This justifies

our further interest in this last

type of asymptotic spectral decomposition.

10. I.

Definition.

A weakly decomposable operator T iS said to be quasi-

decomposable i f XT(F) /S closed whenever F c

C i s closed.

An example of an operator T with the SVEP for which XT(F ) is not closed for F closed is given in Colojoar~ and Foias [3, 1.3.9].

Another example of

this type was given by Radjabalipour [I] .

10.2. Proposition. repr~ ent~on

(lO.1)

Let T be q~idecomposable.

Y = ~r[a(T]y)].

Every Y ~ SM(T) has the

70

Proof.

By Theorem 3 . 1 1 ,

a[TIXT(a(T]Y))] c

~(T]Y),

and t h e h y p o t h e s i s on Y i m p l i e s t h a t

XT[O(TIY) ] C Y. To ascertain the opposite inclusion, let y e Y.

It follows from Corollary

3.12, that c(y,T) and h e n c e

10.3.

= ~ ( y , TIY ) C ~(TIY),

y ¢ XT[a(T[Y)].[-]

Theorem.

Every quasidecomposable operator has decomposable spectrum.

Proof.

Let T be q u a s i d e c o m p o s a b l e ,

{Yi}C

SM(T) perform the asymptotic spectral decomposition

{Gi ) a f i n i t e

open c o v e r o f o(T) and I e t

X = [i Yi" o(TIYi) C G i, for all i. If

F = ~i

~(T]Yi)

is proper in a(T), then ~(F) is a proper subspace in X and contains each Yi" Then

x c XT(F), which is impossible.

Hence T h a s decomposable s p e c t r u m . [ ]

10.4. (i)

T iS quasidecomposable;

(ii)

T iS weakly decomposable with decomposable spectrum such t h a t

(10.2)

o(x,T) = ~

(iii)

(:0.3) Proof.

Theorem.

Given T ¢ B(X), the following statemen~ are equivalent"

{o(TIY ) : x c Y, Y ¢ SM(T)}, f o r e v e r y x e X;

T has t h e SVEP and possesses a weak spectra£ e a p a ~ y ~(x,T) = ~

(i) => (ii):

E such t h a t

{F ~ F : x ~ E ( F ) } , f o r e v e r y x ¢ X.

In view of Theorem 10.3, it suffices to prove the contain-

ment o(X,T)D ~{~(TIY):x

This follows easily.

¢ Y, Y ¢ SM(T)} = S x.

Z = XT[~(x,T)] b e i n g a s p e c t r a l

X~

S X C ~(TIZ ) C o ( x , T ) .

maximal s p a c e which c o n t a i n s

71

(ii) => (iii):

T has the SVEP (Theorem 7.43 and possesses a weak spectral

capacity E such that E(F) is spectral maximal for all F a F

(Theorem 9.4).

Moreover, by Corollary 9.S, every Y ~ SM(T) has the representation Y = E[~(TIY)].

So, for every x ~ X, there is an F ¢ F which gives rise to the spectral maximal space Y = E(F)

with

x ¢ E(F)

and conversely, to every Y ¢ SM(T) there corresponds a closed F = o(T]Y) with Y = E(F).

Hence we have (~{P

e F:x ¢ E(F)} = ~ { o ( T I Y ) :

x ¢ Y, Y s SM(T)}.

Then (10.3) follows from (10.2). (iii) => (i):

In view of Corollary 8.6, for every finite open cover {G i} of

o(T), there corresponds a system { Y i } C l n v ( T )

which performs the asymptotic

spectral decomposition

x=XiYi, We p r o p o s e t o show t h a t ~ ( F )

o(tlYi) cGi foralli.

i s c l o s e d on F.

E(F) C XT(F) f o r a l l

Clearly, F ~

F.

On t h e o t h e r hand, i f x e XT(F) t h e n ~ (x,T) c F and by ( 1 0 . 3 ) ,

X s E[o(x,T)] CE(F). Thus XT(F) c E ( F ) and c o n s e q u e n t l y XT(F) i s c l o s e d f o r e v e r y F a F.

Now, the subspaces Xi = ~ [ o ( T I Y i ) ] ~ Y

i

form a s y s t e m o f s p e c t r a l maximal s p a c e s o f T w i t h t h e p r o p e r t i e s X = Xi Xi •

°(TIXi) c ~ ( T I Y i ) ~ G i "

for all i.

NOTES AND COMMENTS. The notion of analytically decomposable operator was introduced by Lange [2] .

and

studied

The study of weakly decomposable operators was suggested by

Colojoara and Foias [3], and Jafarian was the first to treat them (Jafarian [i]).

72

The concept of spectral capacity was introduced by Apostol [6] and the weak form by Lange [2]. Theorems 7.9 and 7.10 were proved by

Lange [2]. The

2-spectral capacity is contained in an extension of the spectral capacity concept by Albrecht and Vasilescu [i]. The concepts of quasidecomposable operator and decomposable spectrum were introduced by Jafarian [I] who proved Corollary 9.3 (in a restricted form) and Theorem 10.3.

CHAPTER IV DECOMPOSABLE OPERATORS A class of operators with a well-developed spectral theory was introduced by Foias [2]under the name of decomposable operators.

The decomposable operators

have a satisfactory duality theory and functional calculus.

They are closely

related to the operators studied in Chapters II and IiI by uniting the properties and filling some gaps of the previous theories.

§ 11.

Properties and characterizations of decomposable operator.

11. I. Definition. T ~ B(X) / s called decomposable i f for every open cover {Gi} ~ of G(T) there i s a sys£em {Yi}~ CSM(T) performing the following spectral decomposition n

x= (11.1)

IY i=l

i

u(T[Y i) C Gi (or o(TlYi) C ~ i ) , i = 1,2 . . . . . n.

We denote t h e c l a s s o f decomposable o p e r a t o r s on X by D(X). from the d e f i n i t i o n

t h a t T ¢ B(X) i s decomposable i f f

o f s p e c t r a l maximal s p a c e s .

Thus, T ¢ D(X) i n h e r i t s

It is clear

i t has the SDP in terms from the g e n e r a l s p e c t r a l

decomposition the following properties: (ll.a)

o(T) = Oa(W );

(ll.b)

T has the SVEP;

(ll.c)

T has decomposable spectrum.

We remark that property (ll.c) has a stronger meaning for decomposable operators, in the sense that every spectral decomposition (ii.i) entails the decomposable spectrum property. As a link to the quasidecomposable operators we have the following

11.2.

Proof. H CC

Theorem.

Given T ¢ D(X), for every F ~ F, XT(F) /~ closed.

Given F ~ F, let

G D F be an arbitrary open set.

Pick another open

satisfying

o(T) C G U H

andF~=

~

There are YG' YH ~ SM(T) performing the spectral decomposition (ii.2)

X = YG + YH'

a(TIY G) C G,

~(TIYH) c

H.

74

L e t x a XT(F ) b e a r b i t r a r y .

By ( 1 1 . 2 1 ,

x has a representation

x = YG + ~ t w i t h YG

E YG' YH ~ YH"

For X E F c A p(TIYH), we have (11.5)

(X-T) [~(X)-R(X;TIYH)yH]

and s i n c e

= x - y H = YG

the function

(11.4)

yG(X) = x(X) - R(X;TIYH)y H

is analytic

on F c ~ p(TIYH), we h a v e X ¢ ~ ( y c , T ) .

Consequently,

~(YG,T) C F U ~(TIY H) C F U H. For an admissible contour r surrounding F and contained in F c ~ ~-c, (11.3) and (11.4) imply (ll.S)

1 --2~i / r

1 YG ( x ) d x = 2 ~ -

/ r

~(X)dX-

~

1

/r R(X;TtYH)YHdX "

By Corollary 3 . 1 0 , i

~

2~i" and s i n c e

~cCO(T[YH)

' 1

24i

Furthermore,

1

f r

-__

x(X)dX = ~

C = (X ~ C: [X I = I I T II + i } .

(11.6)

L

f R(X;T)xdX = C

Thus ( 1 1 . 5 )

implies

X,

that

x ¢ YG a n d h e n c e

XT(F) C YG"

Since

(11.6)

holds

f o r e v e r y YG E SbI(T) a s s o c i a t e d

(11.71 Now

f R(X;TIYH)YHdX = O. F

we have 2~i

where

/ YG ( x ) d x ¢ YG" F

let

~r(F) C y ¢ Y.

Then f o r e v e r y a(y,T)

(-3 GDF

YG = Y"

g D F, = a(y,T[YG)

c~(rtYG) C G,

and h e n c e G=F.

~ (y,T) C GDF Thus y ~ XT(F ) and h e n c e

w i t h an open G ~ F, we h a v e

75

YCxT(F). The l a t t e r

i n c l u s i o n t o g e t h e r with (11.7) gives XT(F ) = Y,

proving that XT(F ) is closed.~]

11.3.

Corollary.

Ev~y decomposable o p ~ o r

is q~idecomposable.

T e D(X) inherits the following properties from the quasidecomposable operators: (ll.d) For every F g F, ~(F)

g SM(T) and

o[T[XT(F)] C F Do(T), by Theorem 3.11; (ll.e)

Every s p e c t r a l maximal space Y of T has the r e p r e s e n t a t i o n Y = ~[o(T]Y)]

by Proposition 10.2. The interesting case when (ll.d) holds with equality is presented in Appendix A.2.

11.4. Corol£ary T ¢ B(X) iS decomposable i f f T has the SDP and XT(F) iS ~osed for every F ~ F. Proof.

Let {Gi} ~ be an open cover o f ~(T).

There i s a system {Yi}~ C Inv(T)

satisfying the general spectral decomposition n

X =

[ Yi' ~(TIYi) c G i ' i=l

i = 1,2,...,n.

Since YiC

~[o(T[Yi)]

= Z i,

and o(T[Zi) C o(T[Yi) c G i , T is

i = 1,2 ..... n

decomposable.~]

11.5.

Tkeorem.

T ~ B(X) /S decomposable i f f

o(T) there is a system {Yi}~CAI(T)

for e v ~ y open cover {Gi} ~ of

performing the spectral decompos~on

n

X =

~ Yi' °(T]Yi) c G i ' i=l

i = 1,2 ..... n.

76

Proof. the S ~ P

The "only if" part of the assertion follows from the fact that for T with every spectral maximal space is analytically invariant

(Theorem 3.9).

In view of Corollary 11.4, the converse property needs the proof that ~ ( F ) is closed for every F ¢ F. 11.2 with "spectral

This however, has the same proof as that of Theorem

maximal" replaced by "analytically

invariant" and the

reference to Corollary 3.10 replaced by the reference to Corollary 2.10. [-]

11.6.

Corollary.

T : T 1 (D T 2 ~ D ( X 1 ®

Proof.

Let {Gi} 1

If Tj e D(Xj) for j = 1,2 then x2).

be an open cover of"

~(T) T1

and

T2

being

y

n { i2}i C, SM(T2)

decomposable,

performing

= o(Tz) U a(T2)"

there

exist

the spectral

systems

{Yil}IC

SM(TI)

and

decompositions

n

xj =

¥ij' J = 1,2;

[

o(TIYij) C G i, i = 1,2 ..... n; j = 1,2. By Lemma 5.2 and its proof, T has the SDP in terms of the invariant subspaees Yi = Yil (~ under T.

The Yi's are analytically

Yi2' i = 1,2,...,n

invariant under T (Proposition 2.18) and

then Theorem 11.5 concludes the proof.[]

11.7. Co~lla~y. Given Z s D(X), l e t • be a spectral s ~ of o(T). If E(T) is the ~rresponding spectral projection then TIE(~)X is decomposable on E(T)X. Proof.

By Lemma 5.3, S = TIE(z)X has the SDP.

In view of Corollary 11.4 it

suffices to show that for every closed F C z, Ys(F) is closed, where Y = E(z)X. Let x ¢ Ys(F).

Since E(¢)X is spectral maximal for T (Example 3.14), Pro-

position 3.13 implies that x £ XT(F ) and hence

(11.8)

Ys(F) C ~r(F).

On the other hand, therefore (11.9)

~(S)

= T (e.g.

Dunford

and Schwartz

[i, VII.3.20])

we have Y = E(~)X = XT[a(S)]

So, if x c XT(F) then o(x,T) C F C ~

= XT('O.

and by (11.9), x c XT(~) = Y.

Proposition 3.13, x ~ XT(F ) N Y = Y (F) and hence S

Now by

and

77

XT(F) c YS(F). T h i s coupIed w i t h ( 1 1 . 8 ) g i v e s Ys(F) = XT(F ) . S i n c e T e D(X), XT(F ) i s c l o s e d and t h e n so i s Y s ( F ) . F 7

11.8. (i)

Theorem. T e g(x) /S decomposable i f f the following conditions hold: T has the svEP;

(ii)

For any s y s t e m

n

{Fi}~ C F

with

~(T) C k_) F0i "

the XT(F i ) a r e M o s e d

i=l

x Proof.

=

n [ i=l

~r(Fi)-

If T e D(X) then (i) holds and Theorem 11.2 with property (ll.e) prove

(ii). Conversely, assume that conditions (i) and (ii) are satisfied and let {Gi}~ be an open cover of o(T).

We can find a system {Fi}~ of closed sets such

that n e(T) C U F~ i=l i

and

Fi c

Gi ,

i = 1,2 . . . . . n .

Then by Theorem 3 . 1 1 , t h e s u b s p a c e s Yi = Y~ (Fi) are spectral maximal for T.

Thus we obtain the following spectral decomposition

n

X =

11.9. capacity. Proof.

Theorem.

~ Yi' i=l

o(TIY i) C F i C G i, i = 1,2 ..... n. []

T ~ B(X) /S decomposable i f f i t possesses a spectral

If T ~ D(X) then for every F ¢ F,

(11.10) is a spectral

E(F) = XT(F ) c a p a c i t y p o s s e s s e d by T.

as wei1 as ( i v ) o f D e f i n i t i o n

Indeed,

8.2 are clearly

( i ) and ( i i )

satisfied

of Definition

by ( 1 1 . i 0 ) .

8.1,

Furthermore,

(iii) of Definition 8.1 and (v) of Definition 8.2 follow from Theorem 11.8. Conversely, if T possesses a spectral capacity g then Corollary 8.9 and Theorem ii.5 imply that T is decomposable. []

78

11.10.

Proposition.

Given T ¢ D(X), for every Y c SM(T) we have ~(T Y) = ~(T) - o(TIY)

Proof.

I n view o f P r o p o s i t i o n

1.14 ( i ) ,

we o n l y have t o p r o v e t h e i n c l u s i o n

a(T Y) C o(T) - ~(TIY). Suppose t h e r e i s a ~ ¢ ~(T Y) - ~(T) - o(T]Y). Then t h e r e i s an open c o v e r {61,6 2 } of o(T) such that o(T)

- 6(TIY)C

GI,

~ e G c1

and

G 2 63

. o(T[Y) = ~.

There correspond YI,Y2 e SM(T) which perform the spectral decomposition (ii. Ii)

X = Y1 + Y2'

o(TIY i) C 6 i,

i = 1,2.

Since

g(T[Y 2) C 6 2 n o(T) c o ( T ] Y ) , we have

(11.12) Let

Y2 C Y.

y ¢ Y have

a representation

Y = Yl + Y2'

with

Yi e Yi'

i = 1,2.

On the quotient space X/Y, for y with y ¢ y, we have y = Yl because (11.12) implies that Y2 = 0. Since X c p(TIYI) , there is an x c Y1 verifying equation (X-T) x = YI" The corresponding equation on X/Y

(x-TY)x = Yl = y shows that ~-T Y is surjective. In view of Theorems 3.9 and 2.11, T Y has the SVEP and then Corollary 1.3 implies that X ~ p(TY). But this contradicts the hypotilesis on ~. []

11.11. Corollary.

Given T ~ D(X), for every open G C C with G A ~ ( T ) ¢ :~ and

there ~ (11.13) Proof.

a proper s p e c t r a l m a ~ a l

cF(T)(~ G,

space Y of T with t h e following p ~ p e r t i e s :

~(TIY) C G

and

We s h a l l u s e t h e c o n c e p t o f s e t - s p e c t r u m

o(T Y) N 6 = 9. (Appendix A . 2 ) .

Put

79

and denote by F I the interior of F in the topology of o(T). by Corollary A.2.4 (Appendix A.2), F is a set-spectrum of T.

We have F I = F and Then for Y = ~(F),

we have aCT[Y) = F

c~.

With the help of Proposition 11.10, we obtain successively: oCTY) = g(T)-F = q(T)-[G (]oCT)] = gCT)-G-CoCT) - G cG c. [ ] 1 I. 12.

Application.

Let T have t h e SDP. I f ~ (T) has ~mpty i n t e r i o r then

i s decomposable. P~u~of.

In view of Corollary 11.4, it suffices to ascertain that ~ ( F )

is closed

for every closed F Co(T). Let Y e Inv(T) be such that o(TIY ) C o(T) and let y e Y. V be a component of the local resolvent set

p (y,T).

cannot be contained in oCT), there is a disk D C V

Furthermore, let

Since by hypothesis V

• p(T).

By Proposition 1.15,

R(I;T)y E Y on D and by analytic continuation to all of V, the range of y(l) lies in Y, i.e.

(11.14)

{y(1): i ~ p(y,T)} CY.

To prove that XT(F) is closed for F closed, apply the proof of Theorem 11.2 with the following modifications: a)

consider YG and YH just invariant subspaces instead of spectral maximal

spaces ; b)

the reference to Corollary 3.10 be replaced by the reference to

property (ii. 14) above. [] Some basic results of the functional calculus on operators with the SDP have a straightforward application to the class of decomposable operators.

11.15.

Theorem.

n~ighborhood D of

Proof.

Given T ~ D(X), l e t

o(T).

f:D ÷ C be a n a l y t i c on an open

Then f(T) ~ D(X).

For f constant, f(T) is obviously decomposable.

We assume therefore that

f is a nonconstant function By Theorem 5.4, f(T) has the SDP.

Moreover, by Corollary 1.7, for every

F e F we have

(11.15)

Xf(T)(F) = XT[f-I(F ) ] .

Since f - l ( F ) is closed and T is decomposable, i t follows from (11.15) that Xf(T)(F) is closed.

The proof is now concluded by Corollary 11.4. [ ]

80

11.14.

Theorem.

Given T ~ B(X), l@.~ f:D + C be ~ y t i c

an open neighborhood D of o(T).

Proof.

and i n j e c t i v e on

Then T IS decomposable i f so i s f ( T ) .

By Theorem 5.6, T has the SDP and then for every F ~ F, Corollary 1.7

implies that XT(F ) = Xf(T)[f(T)]. Thus ~ ( F )

is closed and then Corollary 11.4 concludes the proof. []

The injectivity condition on f is strongly restrictive. that the implication f(T) g D(X) => condition on f.

Apostol

[6] showed

T s D(X) holds under a different restrictive

That is, if f is locally nonconstant on ~(T),

(i.e. if the zeros

of f' have no accumulation point in ~(T)) then T c D(X) if f(T) ~ D(X).

§ 12.

The d u ~ y

£heory of s p e c t r a l decompositions.

For the theory we shall develop we need both a strengthening and a weakening of the decomposable operator concept.

We shall reach the best result of this

theory when we conclude that both modifications insubstantial.

12. I.

of the basic Definition ii.i are

We begin with the presentation of the stronger concept.

Definition.

T ~ B(X) is c a l l e d s t r o n g l y decomposable i f

f o r any

open cover {Gi} 1 of ~(T) and f o r every Y ~ SM(T), t h e r e i s a system y n { i } l C SM(T) which g i v ~ r i s e to t h e following s p e c t r a l decomposition n Y=

I i=l

Y~Yi'

(12.1) o(TIY i) C

G i, i = 1,2 .... n.

We shall refer to (12.1) as a strong spectral decomposition.

12.2.

Theorem.

T ~ B(X) /S s t r o n g l y decomposable i f f

~or any Y e SM(T), TIY

decomposable.

Proof. { G i}ln covers

First assume that T is strongly decomposable. be an open cover of o(TIY ) o(T)

and

G O ~ o(TIY) = ~.

Choose an open set y

n Let { i}0 C S M ( T )

Let Y ~ SM(T) and let GO

such that {Gi} O

which performs the

following strong spectral decomposition: Y =

n ~ Y~ i=0

Yi' ° ( T I Y i ) c G i '

i = 0,1 ..... n.

81

The subspaces Z i = Y A Yi (0 f i ~ n) are spectral maximal for TIY [Proposition 3.6 and Theorem 3.15 (i)) and verify properties: (12.2)

o[(TIYIIZ0] C~(TIY01 ~ ~(TIy ) C G 0 ~ a(TIY I = ~,

(12.3)

~[(TIY) IZi] c o(TIYi) C G i, i = 1,2 ..... n.

Inclusions (12.2) imply that Z 0 = {0}. Furthermore, the strong decomposability of T i m p l i e s (12.41

Y :

The d e c o m p o s i t i o n (12.4)

n n n l Y ~ Yi = I Z i = [ Z i" i=0 i~=0 i: 1

coupled w i t h (12.3) prove t h a t TIY e D(Y).

T[Y i s s t r o n g l y decomposable i t s e l f . Theorem 3.15 ( i i ) .

In f a c t ,

i f Z e SM(TIY) t h e n Z e SM(T) by

T b e i n g s t r o n g l y decomposable and s i n c e n

Actually,

Z0 = {0} , we have

n

[ Z ~ Zi = [ (Z ~ Y) ~ Y. = Z N Y i=l i=l i

= Z.

Next, assume that for any spectral maximal space Y of T, TIY is decomposable. In particular, for Y = X, T is decomposable and hence it has the SVEP. be an open cover of a(T1 and let { F i } ~ c F FicG i Putting

and

Let {Gi} ~

have the following properties n ~ F 0i ~ o(T). i=l

F = ~(TIY ) and Yi = ~(Fi)' we have Y = XT(F ) and with the help of

Proposition 3.13, we obtain n

n

n

i=l[ Y N Y.I = i =[l Y ~ ( F i )

(12.51

= 111"=YTIy(Fi) = Y"

The YTIy(Fi) being spectral maximal spaces for (12.6)

a(TIYi) C F i C

T]Y

and hence for T, we have

Gi, i = 1,2 ..... n.

Relations (12.5) and (12.61 prove that T is strongly decomposable.~

12.3.

Lemma. Let T be s ~ o n g l y decomposable and l e t Y ~ SM(T).

SM(TY) then Z = {x ~ X: x = x + Y ~

spec~ Proof. (12.7)

mazimal for T.

The assertion of the Lemma will follow from

z = xTbcrlz)].

If

82 Since Z ¢ Inv(T), we have

o(TlZ) c ~(TIXT[~(TIZ)]) and since XT[~(T]Z))]

¢ SM(T), it follows that

z

c XT[a(T]Z)].

For convenience, let us write

w

XT[a(TIZ)].

:

Having Y C Z C W, P r o p o s i t i o n 11.10 a p p l i e d to TIW g i v e s (12.8)

a[(TIW) Y] = a(TIW) - a(TIY ) C ~(TI'Z) - a ( T l ~ .

Since Y ~ SM(TIZ ) by Theorem 3.15 ( i ) ,

P r o p o s i t i o n 2.2

applied to T[Z with

restriction (T[Z) IY = TIY and coinduced (TJZ) Y gives o(TIZ ) = a(T[Y) U s[(rIz)Y] • Then (12.8) becomes a[(r]w) y] co[(TIZ) Y] , or, equivalently

~(TY]w) C o(TYIz). Since

~ SM(TY), it follows that W C Z, and hence

XT[OCTIZ)]

= WEE.[]

12.4. Thzorem. If T iS Strongly decomposable then for every Y ~ SM(T), T Y iS strongly decomposable. P~of.

Let {Gi} l_ be an open cover of o(T Y) and let G O be open such that

{Gi} 0 covers o(T) and G O N c(T Y) = ~.

Let {Yi}oCSM(T)

perform the following

strong spectral decomposition n

Y =

~ Y N Yi' i=0

~(TIYi) C G i ,

i = 0,i ..... n.

Put T i = o(TIY i) U ~(TIY) and Z i = ~(Ti), Then the Zi = Zi/Y ¢ SM(T Y) by Theorem 3.15 (iii). TI Z i gives

i = 0,i ..... n.

Proposition ii.i0 applied to

83 (12.9)

~(TYIzi ) = o[(TlZi )Y] = o(TlZ i) - o ( T ] Y ) C ~(TIYi) C

G i, i = 1,2 ..... n.

As for i = 0, with the help of Proposition 1.14 (i) we obtain ~(TtY0) C o ( T ) A

G O = [o(TIY) U °(TY)] N

and since Y e SM(T), it follows that Y0 C Y.

Consequentl)J, we have

Z 0 = ~ ( ~ 0 ) = ~[e(TIY)] and hence Z0 = {0}.

= Y

Furthermore, for every i,

Z i = ~ [ ~ ( T I Y i) U °(TIY0)] ~ Let ~ E SM(TY).

GOCO(TIY),

XT[°(TIYi)]

= Yi"

Then by Lemma 12.3, ^

Z = {x e X: X = x + Y e Z} e

SM(T)

and since T is strongly decomposable, we have n n

z:

X zOYiC ~ z n z . .

i=O

i=O

i

This, on the quotient space X/Y corresponds to n

~: ~ ~ N ~ i i=l

and with (12.9) proves that T Y is strongly decomposable. [] Now we shall weaken the concept of decomposable operator.

12.5. Definig6on. T e B(X) / s csY~ed 2-decomposable i f for eveay couple of open sets GI,G2 which cover C, there are spectral maxima~ spaces Yi,Y2 p~forming the s p e ~

decomposig6on X = Y1 + Y2

(12.10) c(TIYi) C G i , In Definition 12.5 we can also use a

i = 1,2. 2-member open cover {GI,G 2} of a(T).

Indeed, we can choose an open set H such that C = (G I U H) U (G2 U H) and H N o(T) = ~. Then, there are YI,Y2 ¢ SM(T) such that X = Y1 + Y2 and o(TIY i) C (Gi U

H) N o(T) C G i, i = 1,2.

84 As a straightforward

consequence of Definition 12.5, every 2-decomposable

operator T has the following properties: (12.a)

T has the SVEP (by the 2-SDP);

(12,b)

For every F ~ F, XT(F ) ¢ SM(T) with

o[Tt~r(P)] c P N ~(r) and every Y s SH(T) has the representation

y (note

that

the proof

of Theorem

=

11.2

~r[O(TIY)] is based

on a spectral

decomposition

of type

(12.10)); (12.c)

T is 2-decomposable

We can now proceed

12.6.

toward

Proposition.

iff T possesses the dual

theory

a 2-spectral of spectral

L~t r be a 2-decomposable

capacity.

decompositions.

operator.

Then for ev~g

FEF, (12.1i)

P~of.

u[T*IXT(FCp ] C F. Let X g F c"

by Corollary 4.11. open cover of

C.

We show that (~-T*)Ixq,(FC) i Let

G D F be open such that ~ ~ G c.

Note t h a t Y1 G XT(FC) and Y2 C % ( g ) .

(12.12)

It is injective

]~en {FC,G} is an

There are YI,Y2 ~ SM(T) which decompose X:

X = Y1 + Y2' °(TIY1 ) C F c '

arbitrary

is bijective.

°(TIY2)CG.

Let u e XT(FC) m be a r b i t r a r y .

Fix an

x i n X and c o n s i d e r the r e p r e s e n t a t i o n x = Yl + Y2" with Yi e Yi' i = 1,2.

Define the linear functional v by < x,v > = < R(I;TIY2)Y2,

u

>.

This definition does not depend on a particular representation of x. let x be another representation

=

yl + y½ with Yi'

of x.

£

Yi'

i

=

1,2

We have

Y2 - Y2 = Yl - Yl e Y1 ~ Y2" Since Y1 N Y2 e SM(T), i t i s i n v a r i a n t under R(k;TIY2), R(X;TIY2)(y ~ - y2 ) E Y1 A Y 2 ~ X T (Fc)

In fact,

85

and t h e n < R ( X ; T i Y 2 ) ( y ~ - y 2 ) , u > = O. Thus, v is well defined :


Next, we show that v is bounded. Yl @

:




The linear map Y2 + Yl + Y2

being continuous and surjective, it follows from the open mapping theorem that there is a constant K such that [I yl[[

+ [1 Y2

< K

L[ Yl + Y2 I]

= K

]] X il

Then we h a v e I < x,v >l = [ < R(X;TIY2)Y2 ,u >l = < R(X;T[Y2)Y2,U > = 0, and h e n c e v ¢ XT(FC).a.Now we can show that v verifies equation (12.1s)

(X-T*)v = u.

Let again x be arbitrary in X and have a representation (12.12).

Then (X-T)x has

the representation (X-T)x = (X-T)y I + (k-T)y 2, with (X-T)y i ¢ Yi' i = 1,2. Then, by the definition of v, we have

8@

< X,(X-T*)v > = < (X-T)x,v > = < R(%;TIY2)(%-T)Y2,U >

:

= < Y2 'u > = < Yl + Y2 'u > = < x,u > . Since x is arbitrary in X, we obtain (12.13). With this proof of the surjectivity it follows that l-T* is bijective on ~(FC) ~

and hence X e p [ T * I ~ ( F C ) ~ ]

implies (12.11).

We remark that we could

also choose to refer to Corollary 1.3 instead of referring to Corollary 4.11, at the beginning of the proof.[3

Lemma. Let T be strongly decomposable and l e t

12.7.

open set~.

{Gi}~

be a system of

Then n

n

°i!lX (ci). Proo f.

In view of Proposition 1.5 (i), it suffices to prove n

n

n

Let x ~ X_(~ JGi). Ti~'=l decomposable.

Then Y = ~ [ o ( x , T ]

~ SM(T) and by Theorem 12.2, TIY is

Since n

o(TIY) C ~(x,T) C ~ G i, i=l {G i}ln is an open cover of ~(TIY ).

There is a system {Yi}ln C SM(TIY ) such that

n

Y =

~ Yi and o ( T I Y i ) C i=l

G i, i = 1,2 ..... n.

We have Yi ~

~(Gi),

i = 1,2 ..... n

and since x e Y, it follows that n

n

x ~ XYi C i=l

X xT(Gi).

[]

i=l

The next theorem is not a necessary link in the succession of properties which lead us to the ultimate dual theory.

It is, however, interesting at this

stage to see some connections between various concepts developed so far.

12.8. (12.14)

~S a 2 - s p e ~

Theorem.

I f T is strongly decomposable then E(F) = XT(FC) a-, F ~ F

capacity possessed by

T*.

87 Proof.

We shall refer to the defining properties of a 2-spectral capacity

(Definitions 8.1 and 8.2) ~

(i), (ii), (iii"), (iv) and (v).

E as defined by (12.14), clearly satisfies (i); by Proposition 1.8, verifies (iv) and by Proposition 12.6 condition (v) holds.

We divide the

remainder of the proof in parts (A) and (B). (A).

In order to see that E verifies (ii), let {Fn} C F F=AF. n=l

n

By Proposition 1.5, for every positive integer N, we have N N

~ x~(~) ~ x ~ ( ~ )

~ ~(~c).

n=l

Thus, when N ÷ ~ we obtain oo

02.15)

I ~CF~)C XT(Fc). n=l

On t h e o t h e r h a n d ,

f o r e v e r y x e XT(FC),

oCx,T) C F c= U n=l

Fc n"

oCx, T) being compact, for N sufficiently large, N ~ (x,T) C n=l

Fc n'

and b y Lemma 1 2 . 7 , we o b t a i n N

N.

c

o~

x~ x~(~l ~:) = ! x~(F n) ~ x n

=

n=l

I

~c~).

Thus (12.15) and the latter inclusions inrply co

KT(Fc) c ~ XT(F~) c KTcFc). n=l

Therefore oo

(12.16)

ao

c.l.m. C U %(Fn~)): n=l

Applying the annihilator to (12.16), we obtain successively

~¢.l.m.CU ~(Fn))~" = ~(FC) ~ n=l

n=l

,

and put

88

E(Fn) = E(F) = E(("~ Fn). n=l (B>.

To prove

(iii"),

n=l

let {GI~G 2} be an open cover of C.

The sets

F i = G ci , i = 1,2 are closed and disjoint.

By Theorem

Y = XT(F I U

5.16,

F2) = XT(F I) ( ~

XT(F2)'

and hence each x c Y has a unique r e p r e s e n t a t i o n x = x I + x 2 with x i c ~ ( F i ) , Let y* c X* be arbitrary

and define

(12.17)

< x,y I > = < x2,Y*

It clearly

follows

from

(12.17)

theorem Yl extends

Hahn-Banach

the functional

Yl on Y by

>, x c Y.

that Yl is linear continuously

i = 1,2.

and bounded

to y~ ~ X*.

on Y.

By the

We have y ~ l ~ ( F l ) = 0

so that

Put = Y*

~(F2),

Then for x 2

-

YI"

w i t h the h e l p o f (12.17) we o b t a i n

< x2,Y ~ > = < x2,Y* Hence y~ e XT(F2)-~ C E ( G 2 )

> - < x2,Y I > = < x,y 1 > - < Xl+X2,Y 1 > = O. and since y* = Yl ÷

is arbitrary,

we have

X* = E(G1) + E(~2). As an immediate if T is strongly it follows

[]

consequence of Theorem 12.8, we note that T* is 2-decomposable

decomposable.

In this case,

for every F c F, E(F) = X~,(F)

at once that

(12.18)

XT(FC)a. = X~,(F).

Given T ~ B(X), consider the following eases: (a)

T / 6 quasidecomposable and i t s dual T* s a ~ f i e s

(12.19) (b)

a [T* t XT(FC) a'] C F, for e v e r y F c F T ~

Z-decomposable.

property

and

89

12.9.

Le~ma.

In each of cas¢s (a) and (b) above, T* h~6 t h e 2-SDP.

Proof. Let {GI,G 2} be an open cover of ~(T*). In view of the remarks following that {GI,G 2} covers C. The hypotheses on T allow

Definition 12.5, we can assume

us to apply part (B) of the proof of Theorem 12.8.

This gives us a decomposition

of X* into the sum x*:

Since XT(GiC ) ~

÷

is invariant under T* (Proposition 1.8), (12.19) applied to G1

and G 2 , o [ T * ] ~ ( G i C ) ± ] C G i, i = 1,2 c o n c l u d e s t h e proof.~]

12.10.

Lemma.

Given T ¢ B(X), / n e a e ~ of caSeS ( a ) ,

(b) above and for

every F ~ F, XT(FC)J- i s a s p e c t r a l maximal space of T*.

Proof. Let Y* ~ Inv(T*) satisfy ~(T*IY*) c o [ T * [ X T ( F C ) J'] C F. L e t y* e Y*.

By Lemma 12.9 a n d Theorem 4 . 9 ,

T* h a s t h e SVEP.

~(y*,T*) C o(T*[Y*) C So f o r e v e r y

x ~ XT(FC), < x , y * > = 0

Then we have

F.

and h e n c e

y* c XT(FC) ~ . Thus

Y* c ~rCFc) ~ . [ ] Lemmas 12.9 and 12.10 prove the following

12.11. T*/S

Theorem.

Given T E B(X), / n each of t h e cases (a) and (b) above,

2-decomposable on X*. 12.12.

Theorem.

Ev¢ry 2-decomposable op6%atoris q ~ i d e c o m p o s a b £ e .

Proof. Let {Gi} be an open cover of C.

c i = 1,2, Putting F i = Gi,

• ~ ~n~

we h a v e

n

(12.2o3

( % F i = 2. i=l

By Theorem 12.11, T* is 2-decomposable and in particular it has the SVEP. for F e F, X*T,(F) e SM(T*).

Then,

90 By (12.20), n

X*T,(i__~1 Fi)

X*T,(~ )

=

=

{0}.

On the other hand, n

n

X*T*(i-~-I Fi)

= i=I~X*T*(Fi3

and by (12.181 we obtain n

X =

n

[~-~

X*T.(Fi)]a"

=

i=l The ~ ( ~ i )

~(Gi)C

I i=l

~(Gi 3"

are spectral maximal for T and

c~[TIXT(Gi)] CG-i, i 12.13. Proof.

n

~ i=i

Theorem.

=

1,2 . . . . . n. []

T* /6 2-decomposable on X* i f f T i s 2-decomposable on X.

The "iiTM part was proved by Theorem 12.11.

Then T = T** is 2-decomposable on X** = X.

Let T* be 2-decomposable.

Let

J:X ÷ be the canonical embedding.

T may be identified with TIJX,

Therefore, T has the

SVEP (Proposition i.I0) and the spectral manifolds XT(F ) are defined on we show that Kr(F) is closed on F .

(12.213

We have

J[XT(F) ] C

K?(F),

and hence

J[~(F3 ] = J [ ~ ( F ) ] c ~ ( F ) . Th en

o[T[~(F)] C ~[TI4(F)] C F implies the inclusion

XT(F3 C XT(F) and

hence

~(F)

is

closed

on F,

Our next objective is to show that (12.223 By

~(F) = j - l i b ( F ) ) .

(12.21), XT(F) C

J-I[~(F)],

F.

Now

91

and it follows from

Ttj-I(~(F))] : j-ItT~(F)] C j-lt~(F)] , that J-I[%(F)] arbitrary.

is invariant under T.

Since I e p[TI%(F)]

Let I c Fc and let x e J-I[%(F)]

be

, there is a y e %(F) verifying the equation

(x-~)y = Jx. Then x = J-l(l-T)y = ( I - T ) j - I y

and hence X-T i s s u r j e c t i v e on J - I [ ~ ( F ) ]

.

By C o r o l l a r y 1.3,

X e o ( T 1 j - I [ x ~ ( F ) ] ) and hence

~(TIj-I[~(F)I)cF. This implies

J-I[~(F)]

C XT(F)

thus p r o v i n g (12.22). F i n a l l y , l e t {G1,G2} be an open cover o f ~(T) = ~(T).

Since T i s 2-decompos-

a b l e , t h e r e are YI' Y2 e SM(T) s a t i s f y i n g = Y1 + Y2' and o(TiYi) C Gi , i = 1,2. For i = 1,2, denote Fi = o(TI ~vi). (12.23)

We have

X : %(FI) + %(F2), with F i C

G i, i = 1,2.

The application of j-I to (12.23), with the help of (12.22) gives us the sought decomposition of X:

x = XT(F 1) + XT(F2), ~[TI~(Fi)]CFiCG

12.14. (ii)

E

Corollary. Given T e B(X), the following statements are equivalent: T ~ 2-decomposable; T ~ q~ideaomposable and

o[T*l~r(FC)'] C F , Proof.

i, i = 1,2.

F c F.

( i ) => ( i i ) :

f o l l o w s from P r o p o s i t i o n 12.6 and Theorem 12.12.

( i i ) => ( i ) :

f o l l o w s from Theorem 12.11 and Theorem 12.13. [ ]

Now we take a c l o s e r look at the 2-decomposable o p e r a t o r s .

92

12.15.

Theorem.

T i s 2-decomposable i f f

f o r every open s e t G C C, t h e r e i s

an invariaY~ subspace Y such t h a t {12.24) ~(TIY) C G- and a(T Y) N G -- ~. Proof. The "only if" part follows from Corollary ii.ii. Let {GI,G 2} be an open cover of u(r). We assume that u(r) t% G 1 • G 2 # ~, because otherwise G 1 and G 2 disconnect o(T) and the functional calculus provides the 2-decomposability. The cover can be chosen such that neither G. contains any bounded component of p(T). By hypoi thesis and Proposition 1.17, we can find Y ~ Inv(T) with the properties ~(TIY) C G I N G 2,

o(T Y) A G I Q G 2 = ~,

g(TIY) C o(T).

By Proposition 1.15, the latter inclusion implies ~(T Y) C a(T). Hence we have

~(T Y) C ~ ( T )

- (CI ~ g2 ) = b(T) - G I] U [~(T) - C2]

That is, o(T Y) is the disjoint union of two closed sets. Apply the functional calculus to find two subspaces Z 1 and Z 2 of X/Y invariant under T Y with

O2.2s)

x/Y-- z I Q

~(TYIZi ) C

(12.26)

L e t d : X ÷ X/Y b e t h e c a n o n i c a l

(12.27)

e(T)

z z,

- Gi,

surjection.

X = J-I(x/Y) = J - l ( Z l @

i = 1,2. Then J-1Z.

1

s Inv(T)

and

Z2) = J-l(Zl) + J-I(z2).

Next, we prove the inclusion (12.2s)

e[TIj-i(zi ) ] C ° ( T Y i z i ) U ~(TIY), i = i,2.

Let k s 0{TYIzi ) ~ o(TIY ) and let x e J-Iz i satisfy equation (x-TY)Jx = 0

with

(X-T)x = 0~ Then

Jx g Z.. i

Thus Jx = 0, x s Y and hence x = 0, by the choice of X, Now let x g J-Iz. be 1

arbitrary. Then Jx ~ Z. and i

(I-TY)Jy = Jx, for some y e X and Jy s Z.. Thus, there is u s J-Iz. satisfying (X-T)u = x. Hence i

1

X-T is bijective on J-Iz. and inclusion (12.28) follows. For i # j, (12.26) and i

(12.28) imply the following inclusions o(TIj-Iz i) C

[~(T)- G i] U G 1 {] G 2 C % ,

and in view of (12.27), T has the 2-SDP. Next, we prove that T Y has the SVEP. Let f : D + X be analytic and satisfy (X-T) f(1) s Y

on D.

We may suppose that D is connected. If D (] G c# ~ such that H C D

~G

then there is an open disk H

c. Since H C p ( T I Y ) by hypothesis, we have

93

f(X) = R(X;TIY)(X-T)f(X ] e Y and on all of D by analytic continuation. D C G Cp(TY).

on H,

In case that D C G, D being open we have

Then (x-TY)Jf(X) = 0 on D and hence Jr(X) = 0. This imples that

f(X) s Y on D and consequently Y e AI(T). By Theorem 2.11, T Y has the SVEP. The subspaces Z I and Z 2 of the direct sum (12.25) are spectral maximal for T Y as ranges of spectral projections. By Theorem 3.9, ZI,Z 2 s AI(TY). In order to see that the J-l(zi) s AI(T), let g : D + X be analytic and satisfy conditions (X-T)g(X) ~ J-l(zi) We have (x-TY)Jg(X) e Z i and since Z i e

on D, i = 1,2.

AI(TY), it follows that Jg(k) ~ Zi, or

g(l) s J-l(zi)

on D,

i = 1,2.

Thus, T admits a spectral decomposition in terms of two analytically invariant subspaces. Then Theorem ii.5, confined to n = 2, proves that T is 2-decomposable. []

12.16.

Lemma. Let T be 2-decomposable and l e t F C C be closed. Then f o r

each p a ~ o~ open s ~

which covers

XT(F) E_. XT(GI)

(12.29) Proof.

{GI,G 2}

F, we have + XT(~2).

We can avoid the trivial cases by assuming that F, GI, G 2 intersect o(T)

and o(T) ~ G I U imply that

G 2. Put K = G1 ~ [2 and let Y = XT(K ). Then

Theorems 3.9 and 2.11

T Y has the SVEP and by Corollary ii.ii, o(T Y) ~ K ° = ~. For notational

convenience we put X = X/Y and T = T Y. Let x e

XT(F ) and let f : F c ÷ X be

analytic and verify the identity (X-T)f(X) = x

on F c.

Then there is a function f : F c ÷ X analytic on F c such that for every X e F c, ^

(12.30) Since o(T) C

(x-r)}(x) =

x.

(G I N G2)C, f has an analytic extension (with the same notation) to ^

Fc U (G I N G2) which verifies (12.30). Then x

^

e ~ ( L I U L2) where L I = F - G 2 and

L 2 = F - G I. Since LI,L 2 are two disjoint closed sets, we may apply the local functional calculus (1.7) to obtain the decomposition (12.31)

x = x I + x2, with

x i g ~(Li),

i = 1,2.

Next, we prove that x i may be lifted so that (12.32)

xi e ~(eiu

K).

Then (12.32) follows from (12.31) since we show that (12.33)

X~(Lu

K) C

XT(L U K)/Y,

for any closed L C C.

94

To prove (12.32) let y e ~ ( L U

K) and let g : (L U K) c ÷

y = (X-T)g(~), Now let ~ ~ (L ~ K) c he fixed and let D (L U

X be its local resolvent:

~ ~ (LUK] c.

be a neighborhood of ~ contained in

K) c. By a part of the proof of Theorem 2.11, there is a disk D with

e D CD

and a function h : D ÷ X analytic with h(~) = g ( A )

Then for k E D,

f(k) -- (l-T)h(l) - y s Y

on

D.

and hence

y = (l-T)[h(k) - R(k;TIY)f(k) ]

on D

and on (L U K) c by analytic continuation. This proves that c(y,T) C L U K and inclusion (12.33) is immediate, llence (12.32) holds and we can write (12.34)

x = x I + (x 2 + z),

for some z c Y.

Now (12.29) follows from (12.34). []

12.17. L ~ a .

Let T be 2-decomposable, l ~

a f i x e d p a i r of open s e ~ which c o v ~ t h a t f o r every pair (HI,H 2} of open s ~ F C H 1 [J H 2

(12.35)

F c C be ~ o s e d and l e t (GI,G 2] be

F. Then t h e e

exists a co.rant

M > 0 such

with and

Hi C Gi'

i = 1,2

and f o r e a c h x e XT(F ) , t h e ~ e a r e v e c t o ~ x I , x 2 s a t i s f y i n g p r o p ~ : (12.36) Proof.

X = x I + x2,

~(xi,T) C Hi

(i:i,2),

llXlII+IIx2]l < MII xll .

Form the direct sum w =

®

and consider the continuous linear mapping k : W ÷ X defined by k(x I ~

x2) = x I + x 2,

By Lemma 12.16, the range of k contains ~ ( F ) .

o(xi,T) C Gi'

Hence

i = 1,2.

W 0 = k-l[~(F)]

subspace of W and the restriction k 0 = kIW 0 is surjective on ~ ( F ) .

is a closed

The closed

graph theorem gives an M > 0 such that (12.36) holds for H. = G.. i i Now let {HI,H 2} be an arbitrary pair of open sets satisfying (12.35).

Then

is a closed subspace of W. Let k I = k[W I. Since kllW 0 = k 0 (because the range of k I also contains XT(F ) by Lemma 12.16) the conclusions

{H1,H2).

(12.36) also apply to

95

12.18. Theorem. If TIY ~ strongly decomposable. P/u~of.

2-decomposable for ev~y Y ~ SM(T)then T

In view of Theorem 12.2, it suffices to prove that T is decomposable. Then

that proof may be applied to T[Y. Now Lemma 12.7 applies so that for any open cover {Gi} nl of

o(T), we have n

x=

n

n

($1Gi) : i!l (Gi) i!l ( i)cx

[]

12.19. Theorem. Let T be 2-decomposable. If w ~ SM(T*) then T*[W ~ posable. Proof.

2-decom-

By Theorem 12.13 T* is 2-decomposable, hence we can use for W the represen-

tation W = X*T,(F),

Let {G1,G 2} be an a r b i t r a r y

where

F = o(T*[W).

b u t f i x e d p a i r o f open s e t s c o v e r i n g F. We s h a l l p r o v e

t h a t t h e r e a r e two c l o s e d s e t s F1, F 2 s u c h t h a t (12.37)

Fi~Gi,

i = 1,2

and

W = X*T,(FI) + X*T,(F2)

from which it will follow that T*]W is 2-decomposable. Let u e W be an arbitrary unit vector. Let A be the family of all pairs of open sets {HI,H 2} satisfying (12.35). Then A forms a directed set under inclusion. For notational purposes index

A by ~ ~ A. Now let M > 0 be the constant determined

in Lemma 12.17. By Lemma 12.17 applied to T*, for each ~ e A there are vectors u. 1

with the following properties:

(i°1,2) and

u = u I + u2,

Hence {u~ : ~ £ A}, i=1,2 are two bounded nets in X* and by Alaoglu's theorem, each has a subnet (denoted without changing index) converging to u I, u2, respectively. For fixed ~ e A it is clear that if ~ > B then

~(u~,

(12.38)

T*) c

H-i, B ,

i =

1,2.

By Lemma 12.10, every spectral maximal space of T* is the annihilator of a linear manifold in X, hence every subspace X*T.(Hi,~) is weak*-closed. Thus u i e X*T.(Hi,~),

i = 1,2;

e E A,

and moreover, ui ~ ~ X*T.(Hi,~) " i = 1,2. seA Let

Fi = ~

H.

i = 1,2. Clearly, F I U F2 = F

and

u = u I + u 2. Hence

96

u ¢ X*T,(FI) + X*T,(F2) C W and since u was arbitrary in W, we obtain (12.37). []

12.20. Proof.

Corollary.

L ~ T be 2-decomposable. Then T . ~ strongly decomposable.

By Theorem 12.18 it suffices to prove that TIY is 2-decomposable

Y s SM(T). Let S = T**. Since T* is 2-decomposable from Theorem 12.19 that SIW is 2-decomposable

for every

by Theorem 12.13, it follows

for every W E SM(S). Now let F be

closed and put V = X**. For any open cover {GI,G 2} of F we have by Theorem 12.19

(12.39)

VS(F ) = Vs(FI) + Vs(F2)

f o r c l o s e d s e t s F 1 , F 2 C F. As in the p r o o f o f Theorem 12.13, w i t h

k : X * V as

the c a n o n i c a l embedding, we can w r i t e with t h e h e l p o f (12.39)

r(F)

= k-I[Vs(F)]

= k-I[Vs(FI) ] + k-I[Vs(F2)]

=

= XT(F1) + ~ , ( F 2 ) .

This completes t h e p r o o f . [ ] We now summarize t h e s e r e s u l t s

in the f o l l o w i n g

12.21.

Theorem.

(i)

T i s decomposable.

(zi)

T i s 2-deeom~sable.

(2d.i.}

T i s Strongly decomposable.

(iv)

TIY i s decomposable for Y s SM(T).

(v)

TY i s decomposable for Y ~ SM(T).

(vi)

TIY .66 decomposable for Y = XT(G), G open.

(vii)

T* Zs decomposable.

For T e B(X), the following statements ~ e equiv~ent:

Proof. The equivalonce of (i), (ii), (iii), (iv), (v) and (vii) follows from the foregoing properties.

The implication

(vi) => (i) follows by taking G = C. We prove

(v) => (vi): Let G be open, Y = ~ ( G )

and put Z = Y~. Then Z s SM(T*) by Lemma

12.10 and (T*) Z is decomposable by (v). But (T*) Z can be identified with (TIY)* and hence (vi) follows from the equivalence

(i) (vii). []

97

As a by-product, we obtain an example of an analytically invariant subspace which is not necessarily spectral maximal (see Appendix A.I.).

12.22. Proof.

Corollary.

For T ~ D(X) and every open G C C, XT(G) ¢ AI(T).

By Theorem 1 2 . 2 I , TY i s decomposable whence Y = ~ ( G ) .

Then TY has the

SVEP and hence by Theorem 2.11, Y E A I ( T ) . [ ]

§ 13.

S p e c t r a l decompositions of unbounded operators. If we restrict the invariant subspaces to the domain DT of a given closed

linear operator T then the extension of analytically invariant and spectral maximal spaces to the unbounded case is straightforward.

Consequently,the concept of

weakly decomposable operator extends to the unbounded case. We shall write F and K for the families of closed and compact sets in C, respectively.

13. I.

Definition.

A strong s p e c t r a l capacity i n X ~

an a p p l i c a t i o n

E : F + S(X)

tha~ s a t ~ f i ~

t h e following c o n d i t i o n :

(I1

E(~)

(II)

E(An=I Fn) = n=l~T~E ( F n ) '

(IIIl

For evo_2uj F ~ F and evecuj open cover {Gi} nl of F,

= (0},

E(C) = x;

f o r e v ~ y sequence

n ~. E(F i~ g i ) ; i=l

E(F) =

(IV)

(F n} C F;

For e v ~ y F s F, t h e l i n e a ~ manifold E0(F ) = {x ~ E(K) : K ~ K and K C F )

i4 dense i n E(F). For F = C, ( I l l ) becomes n

(zzI,) where

{Gi}

x

= ~ E(~i), i=l

nl

is an open cover of C. Conditions (I), (II) and (III') define the

original concept of spectral capacity as given by Definition 8.1.

98 In the special case, F = C, condition

(IV')

(IV) asserts that

E0(C) = {x ¢ E(K): K E K}

i s dense i n X. Condition (IV")

(IV) is equivalent to

For every F E F there exists a nondecreasing sequence {K n} C K, such

that co

E(F) = ~ E(Kn)n=l

J3.2.

Definition.

A ~osed linear ephor

T : D T ( C X) ÷ X i S said t o

possess a strong s p e c t r a l capacity E i f i t has nonvoid r e s o l v e n t s e t and s a t X s f i e s conditions

:

(v)

E(K) C DT, for adz K e K;

(vI)

T[E(F) N DT] C E ( F ) ,

(vii)

The r e s t r i c t i o n WF = TIE(F ) V~ DT has t h e spectrum

for a l l F ¢ F;

o(T F) C F, for each F ~ F.

Retook.

It follows from (IV') and (V) that every T with a strong spectral

capacity is densely defined in X.

13.3.

Theorem.

I f T possesses a strong s p e c t r a l capacity E then for e v ~ y

K ~ K, t h e r ~ t r i ~ n

TK = TIE(K ) d~ a bounded decomposab£e e p h o r

on E(K)

p o s s ~ s i n g t h e s p e c t r a l capacity EK defined by (IS.l)

Proof.

EK(F ) = E(K N F ) ,

for a£~ F ~ F.

T b e i n g c l o s e d , T K i s c l o s e d and d e f i n e d e v e r y w h e r e on Y = E(K).

closed graph theorem,

TK i s bounded.

By t h e

The a p p l i c a t i o n

EK:F ÷ S(Y), defined by (13.11 gives

(13.2)

EK(F ) = E ( K ) [ ] E(F) = Y N E ( F ) ,

The p r o o f p r o c e e d s by and (v) o f D e f i n i t i o n s sequences of (13.2),

showing t h a t EK s a t i s f i e s 8.1 and 8 . 2 . ( I ) and ( I I ) .

Conditions

for all

F c F.

conditions ( i ) and ( i i )

(i),

(ii),

(iii),

(iv)

a r e immediate con-

Let {6i) 1 b e an open c o v e r o f K.

For F = K,

(III) becomes n

y =

~ i=l

n

E(KNGi)

= ~

EK(Gi )

i=l

and hence it follows that E K is a spectral capacity on Y. of (V) and (VI) we obtain

~breover, with the k~ip

99

TK[[K(F) ] = T[[(K) N [(F]] = T[E(K N F)] C [ ( K N

F) = EK(F),

F ¢ F.

Consequently, [K verifies condition (iv) of Definition 8.2. Finally, (V) and (VII) lead us to (v) of Definition 8.2 as follows: O[TKIEK(F)]

= ~[T[E(K n F)] c

Now Theorem 1 1 . 9 c o n c l u d e s

13.4.

Proof.

Theorem.

K n F C F,

F c F.

the proof. []

Every T with a strong specY~ral capac2Y~y has £he SVEP.

Let f : G + DT be analytic and verify equation

(13.3)

(~-T)f(k) = 0

on an open

G C C.

There is no loss of generality in assuming that G is relatively compact. Let {GI,G 2} he an open cover of C with G I ( D G

) relatively compact and G2 A G = ~.

The strong spectral capacity E of T provides the following decomposition of X: x=

"

In view of Lentma 4.8, for every I c G there is a neighborhood H ( C there are analytic functions

fi : H + E(Gi), i = 1,2 such that

(13.4)

f(~) = fl(~) + f2(~),

G) of k and

for all ~ e H.

Since the ranges of both f and fl are contained in DT, we have f2(H) C E(G2) A D T. Equation (13.3) written as (z-T) [fl(~) + f 2 ( l a ) ]

= 0

gives rise to (la-T) f l ( t O

= (T-!a)f2(l~)

= g(>) ¢ E(G'I) ~'1E(G2) on H.

With the help of (VII),

an analytic function

h : H + E(GI) n E(G2) h(~)

=

[TI~(~l

is defined as follows:

n G-2) - ~]-lg(l 0 e E(G'I) n E(G"2) on H.

We h a v e h(~) - f2(~) c E(G-2) f~ DT, for all ~ ~ H. The definitions of the functions h and g produce the following identity on H: (13.5)

[TIE(G2) n D T - ~][h(~) - f2(~)] = 0.

100

By (VII)

o[T[E(~2) n DT] C G2 ~ ¢fl C p[TIE(~2) ~ DT], Consequently,

and therefore

(13.5) implies

f2(~) = h(~) ~ E(GI) and then (13.4) implies that f(~) c E(GI) on H. Since by Theorem 13.3, T IE(G1) is decomposable,

it has the SVEP and hence (13.3) implies that f(~) = 0

on H,

and on all of G, by analytic continuation.

15.5

Theorem.

[]

Given T w i t h t h e strong s p e c t r a l capacity E, for every

K c K, E(K) i s a s p e c t r a l mazimalspace of T. Proof.

(A).

Let Y( C D T) be a subspace of X invariant under T such that

o(TIY) c o[TIE(K)], and let

× ¢ Y be a r b i t r a r y .

showing that

x c E(K).

The p r o o f

wilt

be brought

to its

conclusion

by

By ( V I I ) ,

~[TIE(K)] C K. (B).

Thus the hypotheses

(13.6)

are

: Y(~DT)

is

invariant

u n d e r T, x ¢ Y a n d

~(TIY) C K.

Let {GI,G 2} be an open cover of C with K C G l, G 1 relatively compact and

~2 ~ K = ~. By ( I I l ) , x = xI + x2 Note that since

with

x i e E(Gi),

i = 1,2.

x c Y C D T and x I e E(GI) C DT, it follows that x 2 = x-x I e D T.

In view of (13.6), x is defined on K c and verifies equation (13.7)

(k-T)x(k) = x,

Furthermore,

for all

k c K c.

since by (VII)

(i3.8)

~[TIE(~z) n DT] C

G2'

there is a function ~

~z: G-2c + eCc2 ~ n Dv analytic and verifying equation ~

(13.9) Combining

(X-T)xi(k) (13.7) and (13.9), we obtain

= x2

on G--2c.

101

(X-T) [X(X)-X2(X ) ] = and x - x 2 is analytic on G.

X-X 2

= x I, for all k c G = ---c G 2 N Kc '

Since by Theorem 13.4, T has the SVEP, x I analytic

and verifying equation ~

(13.10)

(k-T)Xl(k) = x I on G,

is uniquely determined by

~I(X) = ~(x)-~2(x), for all X ~ G.

(13.11)

If F is an admissible contour surrounding K and contained in G, in view of (13.8) we have (13.12)

f x2(~)ak = 0. F

With the help of (13.6), (13.11) and (13.12) we obtain

1 x = ~#f

S R(X;TIY) xdX =

1

r

1 f ~(X)d~ = 2--~ r

~l(X)dX " ~

Therefore, to prove that x e E(GI), it suffices to show that Xl(k) e E(G I) on F. Apply Lemma 4.8 to the function Xl:G + D T analytic on G.

For each i e G, there

is a neighborhood H C G of I and there are analytic functions

(13.13)

such

gi:H ÷ E(Gi), i = 1,2

that

(13.14)

Xl(P ) = gl(p) + g2(u), for all p c

Since Xl(P) e D T and gl(p) ~

E(G't) CDT,

H.

it follows from (13.14) that

g2(p) ~ E(~2) /~ DT, for all p e H. Relations (13.10) and (13.14) imply that (~-T) [gl(V) + g2(p)] = x I on H and hence

Xl-(~/-T)gl(p) = (~-T)g2(~/). So f(~A) = (P-T) g2(P) g E ( ~ )

~ E(-G2) :

E(~ 1 ~ -G2) , on fl.

The analytic function

h(v) = [~/-TIE(GI~G2)]-If(P)

g E(G 1 ~ G--2), on H

gives rise to (p-T) [h(p)-g2(p)]

= 0 on H.

102

By the SVEP of T,

g2(!a) = h(!a) ~ E(G"I n G-2) C E(G I)

on H

and hence

Xl(~)

= gl(~)

+ g2(B) ~ E(G1).

Since ~ is arbitrary on G and hence on F, we have Xl(X ) ¢ E(GI)

on F.

Thus, x ¢ E(~ I) for every relatively compact open G 1 D sequence of the n

-I

K. So, if {Gln} is the

-neighborhoods of K, then

X C I'~ E(~ln) = E ( A n=l

G--ln) = E(K), [ ]

n=l

A slightly different version of the foregoing proof can show that if G C C is open such that E(G) c DT~ then E(G) is an analytically invariant subspace under T (Erdelyi

13.6.

Proof.

[3]).

Theorem.

I f T possesses a strong s p e c t r a l capacity E then E i s unique.

Suppose that T has a second strong spectral capacity E'. Let K ~ K be

arbitrary and let x ¢ E~(K). By (VII),

o[TIE' (K)] C K. Denote Y = E'(K) and repeat part (B) of the proof of Theorem 13.5 to obtain x e E(K). Thus, E'(K) C (13.15)

E(K). By symmetry, E(K) C E'(K) and consequently E(K) = E'(K),

Now, let F ¢ F be arbitrary.

for all K e K.

By (IV") there exists a nondecreasing

sequence

(Kn} C K such that for every strong spectral capacity E,

E(F) = U E(Kn)" n=l Then, with the help of (13.15), we have

oo

oo

E'(F) = i d E'(Kn) = ~ E(Kn) = E(F). [ ] n=l n=l

13.7.

Theorem.

I f T possesses a st~cng s p e c t r a l capacity and

then T i s weakly decomposable.

e

p (T),

103 Let {Gi } ln be an open c o v e r o f o(T).

Proof.

there is a relatively

compact open neighborhood H o f o(T). H.J. = H N G .l ,

form a r e l a t i v e l y {Hi};

The s e t s

i = 1,2,...,n

compact open c o v e r o f

c o v e r s C and g0 ~ o(T) = 0.

p o s s e s s e d by T.

S i n c e , by h y p o t h e s i s o(T) i s compact,

a(T).

Let H0 be an open s e t such t h a t

Let E be the s t r o n g s p e c t r a l c a p a c i t y

By (VII) and with the h e l p o f C o r o l l a r y 8.5 a p p l i c a b l e to

E ,

we have ~[TIE(H'o) ~ DT] C H'O ~ o(T) : and hence (15.16)

E(H%) ~ DT = {0}.

By ( I I I ' )

we have n

(13. i7) i=0 Relations (13.16) (13.17) imply DT~

n I E(Hi)' i=l

and since DT is dense in X, we have n X=

Z

i=l Furthermore, (VII) implies °[TJE(

i)J C

C

i : 1,2 ..... n.

By Theorem 13.5, for every i, E(Hi) is spectral maximal for T. [] 13.8.

Theorem.

x ~ ×, t h e r e e ~ t ~ analytic

L e t T have t h e s t r o n g s p e c t r a l

capa~g~y E.

a n o n v o i d open s e t G and a s e q u e n c e

on G w i t h v a l u e s i n DT s u c h t h ~

{f}

For e v e r y

of functions

(X-T)fn(X) c o n v e r g e s t o x u n i f o r m l y

on e v e r y compact s u b s e t o f G.

Proof.

Let x ~ X be arbitrary but fixed.

F ~ C.

By (IV") there exists a nondecreasing sequence {Km} C K such that

Choose F ~ F such that x ~ E(F) and

E(F) = 0 E(Km)" m=l

104

m•=l

E(Km) such that x n ÷ x.

There is a sequence { X n } C

For every n, xn ¢ E(Km)

=

(hence

x n ~ DT),

for

Note that

some m.

co

~-~ K c ~ m= 1 m

F c = G.

For every n, there exists a function % : G + D T analytic on G such that (~,-T)£n(X)

=

x,

for

all

~, ~ G.

Thus, on every compact subset of G, we have lim n

+

(X-T) fn (~) =

oo

lim n

+

xn = x. [] oo

NOTES AND COMMENTS. The concept of decomposable operator was introduced by Foias [2]. The J

original paper contains the proofs of Theorem 11.2 and of properties (ll.a), (ll.b), (ll.d), (ll.e).

A forerunner of some basic concepts developed for

decomposable operators (Foia~ [i]) set the foundations of the "generalized scalar operators", which together with the "generalized spectral operators" (Colojoar~ [I], Maeda [i]) are defined by means of a spectral distribution (i.e. a L. Schwartz-type multiplicative vector distribution). generalized spectral operators, [3]

The theory of the

systematically presented in Colojoara and Foia~

offers to the reader an important application of decomposable operators.

Chapter 2 of the last reference is dedicated to the general theory of decomposable operators.

The theoretical foundations of decomposable operators as well as of

the generalized spectral operators form the subject of Colojoara [2]. Some extensive studies on the functional calculus of decomposable operators including direct proofs of Corollaries 11.6, 11.7 and of Theorems 11.13, 11.14 form the topic of Colo3oara and Foias [2].

As indicated in some remarks following

Theorem 11.14, some deep results on this functional calculus were obtained by Apostol [5,6]. Some of the herein contained characterizations of decomposable operators have the following sources: Theorem 11.9, Foias [3].

Theorem 11.5, Lange [2]; Theorem 11.8, Apostol [5];

Proposition Ii.i0 was proved by Apostol [3]. Corollary

ii. Ii was given in a restricted formulation by Jafarian and Vasilescu [i] . The duality theory of spectral decompositions has been developed as the result of a relatively small nun~er of papers.

Definition 12.1, Theorem 12.2,

Lemma 12.3 and Theorem 12.4 are due to Apostol

[3]; Definition 12.5, Lemma 12.7

and Theorem 12.18 appeared in Plafker [I]; Proposition 12.6, parts of Lemmas 12.9, 12.10 and Theorem 12.11 as well as Theorem 12.12

were proved by Frunza [i].

105

Theorem 12.13 and Corollary 12.14 were proved by Lange [4] as well as Theorem 12.15, Theorem 12.19, Corollary 12.20 and Theorem 12.21 are among the results obtained by Lange [3]. An important result expressed by Lemma 12.16 is due to Radjabalipour private communication.

[2] and

By using the concept of "almost localized spectrum" intro-

duced by Vasilescu [4] and some related results of Frunz~ [3,4], Radjabalipour obtained an independent proof of the equivalence (i) (ii) of Theorem 12.21. Lemma 12.16 was instrumental for our proof of Theorem 12.21. The first part of the proof of Lemma 12.17 is due to Vasilescu [4]. Section 13 on unbounded operators is based on Erdelyi [1,3]. This chapter obviously does not cover all known properties of decomposable operators. Just to mention a few interesting properties which are beyond the scope of this work we refer to three additional papers. An illuminating characterization of spectral maximal spaces when they belong to decomposable operators was given by Foias [4]. He proved that these subspaces coincide with both strong and weak spectral manifolds introduced by Bishop [i]. As an interesting analogy with the Dunford-type spectral operators, Foias [5] introduced the scalar part of decomposable operators. Finally, it is worth mentioning that Apostol

[4] developed a

topology in which the class of decomposable operators is closed.

APPENDIX A.I.

An example of an a n a l y t i c a l l y i n v ~ a n t s u b s p a c e which i s not absorbent. A.I.1.

Preliminaries.

Let X = C [ a , b ] be t h e Banach s p a c e o f c o m p l e x - v a l u e d

c o n t i n u o u s f u n c t i o n s on [ a , b ] endowed w i t h t h e norm II xll

=

sup

[x(t)[,

x ~ X.

t ~ [a,b] Define the multiplication

operator T ~ B(X) as follows

Tx(t) = tx(t), t ~ [a,b]. For every complex ~ ~ [a,b], R(k;T) R(k;r)x(t) Hence,

o(T) C[a,b].

is defined by

= (k-t)-ix(t),

Conversely,

for all x ~ X.

for k ~ p(T),

x = R(k;T)u,

let

where u(t) ~ i.

Then (k-t)x(t) and hence

k ~

[a,b] thus implying that

(A. I. I)

c(T) =

Now, let

= 1

[a,b]C

~(T).

Consequently,

[a,b].

F C [a,b] be closed and define YF = {x ~ X: supp x C F}.

Clearly,

YF ¢ Inv(T).

We will show that

(A.1.2)

o(T[Y F) C F.

This will follow from the fact that for k e F c, k-T[Y F is bijective.

Let k e F c

and let x ~ YF satisfy (h-T) x = 0. Then (k-t)x(t)

= 0, for all t e [a,b].

Therefore, x(t) = 0 for all t ~ F and hence x = O. define y ~ C[a,b]

For the proof of the surjectivity, by

let x ~ YF be arbitrary and

107

I y(t]

(k-t)-ix(t), if t a F;

=

0, if Since supp x C F ,

we have y

YF"

¢

t c F c.

Equation (k-T)y = x

proves that

k-T is surjective on YF"

Thus I-T is hijective on YF and property

(A.I.2) follows. During our next step, we show that YF e SM(T).

o(TIY F) c F,

q(TIZ) C and let x ¢ Z.

Fix k ~ [a,b] • F c.

Let Z e Inv(T) satisfy

Since k e p(TIZ), X-TIZ is surjective and

hence there exists a function y e Z such that (k-T)y = x, i.e.

(k-t)y(t) = x(t), for all t ¢ [a,b] .

In particular for t = X , we have x(1) = 0 which proves that supp x C F and hence Z

CY

F-

We proceed by proving that T e D(X)

•

n Let {G i}l be an open cover of ~(T).

Let {~i}l be a continuous partition of the unity subordinate to the cover {Gi}, i.e. ~i e C[a,b], supp s i c

G i, i = 1,2 ..... n

n

si(t ) = i, for all t ¢ [a,b]. i=l Denote F i = supp ~i' 1 < _ i < _ n.

Then for every i, YF. l

SM(T) and for every

x e X, we have n

x =

six, with i=l

six e YF." z

Thus, we have X =

n ~ YF.' i=l i

and by (h.l.2), a(TIYF.) C supp s i c 1

Gi, i = 1,2 ..... n.

108

Finally, we show that for every x ~ X, (A.I.3)

o(x,T) = supp x.

By (A.I.2) we have o(x,T)~o(TIYsupp

x) C

supp x.

Conversely, [(%-T)x(),)] For ~ ~ 0(x,T)

(t)

/% [ a , b ]

= x(t),

for every

and t e

[a,b]

implies

that

.

, we h a v e x(~)

and hence x vanishes

k e p(x,T)

on p ( x , T )

(% [ a , b ]

= 0,

.

This,

however,

supp x C o(x,T),

thus proving (A. i. 3) •

A.I.2. operator

The Example.

T defined

Let X = [-i,i]

and consider the multiplication

by Tx(t)

= tx(t),

t

¢ [-1,1].

Let

g = (-2,0)U

By Corollary 12.22, Y = XT(G ) ~ AI(T).

(0,2).

In view of (A.I.3),

XT(G ) = (x E X: supp x c G } . The sequence {xn} C XT(G), defined for every n by

I t - t + i---! for-1-< t