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
=
We h a v e
< k-l}.
N (l-T)
LnXn =
[
IN+I
xN
n=O and
t[ tN+lxNIi
~
k N III N+I ÷ 0
when N ÷ =.
Since X-T is closed, N
f(k)
is in the domain of
=
lim
~
N ÷ ~
n:O
tnx n
I-T and
(t-T)f(t)
= 0 on D.
Obviously f, as defined by (1.2) is not the zero function. []
1.3. Corollary. If the closed linear operator i f f X-T iS not surjective. ~
T
has the SVEp then i ~ o(T)
interesting effect of the SVEP on the dual operator T
regards the struc-
ture of the spectrum.
1.4.
Corollary.
If the closed, densely defined linear o p ~ o r
T has the
SVEP then Oa(T* ) = c~(T*).
Simil~y,
i f T has the SVEP then
(1.3)
~a(r)
=
o(w).
Proof.
Given T as stated by the Corollary, let X ¢ g(T). Then X-T is not sur, jective. Suppose that I-T is bounded below, i.e. there exists k > 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 ( ~ )
--
_<
and (1.14) becomes successively:
10
(~-Tn)f(k) (L15)
= (g-X-Qn)f(k),
(~-Tn)R(~-k;Qn)f(k) c it ÷ R(~-X;Qn)f(k ) is analytic on KB,
Since (1.16)
o[f(t),Tn]
= f(k).
follows
that
C K~.
In view of (i.15), by integration along the boundary of KI, we obtain
(~_~n)
1 f
a(~-~;Qn)f(~) ,J-k
dv = ~ ]
f(')) I ~)-k ~K]
3K 1
d~
=
f(x) "
The function
1
a(~-V;Qn) f(~)
~ + 2-7~i f
~K 1
is clearly
analytic
dv
u-X
0 and therefore on K2,
(1.17)
o[f(t),Tn]C
(K~)CcK c .
It follows from (1.16) and ~1.17) that
o [ f ( k ) ,Tn] = and then Proposition 1.5 (iii) implies that f(k) = 0.
Since
k
is arbitrary in
D, we conclude that f = 0. []
The property expressed by the foregoing theorem holds under weaker conditions. Condition (i) can be skipped under a slightly different topology.
1.12.
Corollary.
If T s B(X) has £he SVEP and q ~
quasini~potent co~mu~ing
with T then T+Q has the SVEP.
Proof.
Let f:D ÷ X be analytic on an open D C C
and verify equation
(k-T-Q)f(k) = 0 on D. For ~ # k, write (~-T)f(k) = (~-k+Q)f(k) and then follow the proof of Theorem i.ii from (i.15) to the end by interpreting
Qn = -Q' Tn = T and Ka = {k}. [ ] The SVEP is stable under finite direct sums.
1.15. T 1 and T2
Theorem.
LeZ T i a B ( X i ) , i = 1 , 2 .
have that p r o p ~ y .
Moreover,
Tt @
T2
h a s t h e SVEP i f f
both
°(X 1 ( ~
Proof.
First,
f = fl @
X2, T 1 (~) T2) = ° ( K I ' T l) U ° ( x 2 ' T 2 ) "
assume t h a t T I and T 2 have t h e SVEP and l e t
f2 :D ~ Xl @ X 2
(i = 1,2) analytic on D.
be analytic on an open D C C ,
with fi:D ÷ Xi,
The condition [X-(T 1 @
T2 ) ] f ( x )
= 0 on D
implies (i-Ti)fi(l)
= 0 on D, i = 1,2.
By the SVEP o f TI and T2, we have f l = 0 and f2 = 0. Next, assume t h a t T 1 @
T2
Thus f = 0.
has t h e SVEP and l e t f i : D ÷ Xi be a n a l y t i c and
verify e q u a t i o n s (X-Ti)fi(%) = 0 on D, i = 1,2. Then 0 = (X-T1)fl(X)
@
(X-T2)f2(X) = [X-(T I @
and by t h e SVEP o f T1 @
T2
T2)][fl(X ) @
f2(X)] on D
we o b t a i n
fl(x) @ f2(x) = O o n D . Thus, f l = 0 and f2 = O. Now l e t t e O(x 1 @
x 2, T1 @
analytic function f = fl @ (X-T1)fl(X)
@
T2].
f2 :D ÷ Xl @
There i s a neighborhood D o f X and an X2 (with fl,f2 analytic) on D such that
(X-T2)f2 (x) : [ X - ( V l @
T2)]f(X) = x 1 @
x 2.
Then (k-Ti]fi(X) = x i, i = 1,2 and hence X e P(Xl,Tl) [7 P(x2,T2).
Thus, we have
~(Xl'T1) U ° ( x 2 ' T 2 ) C_ o'(x 1 @
x2,T 1 @
T2).
The o p p o s i t e i n c l u s i o n has a s i m i l a r p r o o f . ~ Given T ~ B(X), a subspace Y e tnv(T) produces two r e l a t e d the r e s t r i c t i o n X/Y.
T[Y and t h e c c i n d u c e d TY, the l a t t e r
In g e n e r a l ,
the t h r e e s p e c t r a
linear operators:
a c t i n g on t h e q u o t i e n t space
o(T), o(T[Y) and o(T Y) have t h e p r o p e r t y t h a t
the union o f any two o f them c o n t a i n s t h e t h i r d .
12
1.14.
P~oposZtton.
Given T E B(X), for eveAy Y e I n v ( T ) we have
(i)
~(T) C - ~ ( T I Y )
(ii)
q(TIY) C
(iii)
~(T Y) C
?AOO{.
(i):
U ~(TY); o(TY);
q(T) O ~(T) O
Let k c p(TIY} ~ p(TY),
~(TIY).
The equation
(X-T)x = 0 produces
the
following
implications:
(x-TY)x = 0 => x = 0 => x e Y =>(X-T]Y) x = 0 => x = O. Hence X-T is injective.
Next, let x e X be arbitrary.
There exists y g X
such that
( x - T Y ) ; - x. Then (X-T)y - x e Y and hence there
is
a vector
u ~ Y defined
by
u = R(X;T[Y)[(X-T)y Furthermore,
- x].
we o b t a i n (X-T)u = (X-T)R(X;TIY)[(X-T)y-x]
= (X-T)y-
x
and hence (X-T) ( y - u ) Thus, that
X-T i s
surjective
and by the previous
= x. argnament, b i j e c t i v e .
This proves
X e p(T). (ii):
Let X ~ p(T) N
p(TY).
It is clear that X-TIY is injective.
If
y ~ Y is arbitrary then there is an x E X with y = (X-T)x. Passing
to the quotient
s p a c e X/Y, t h e h y p o t h e s i s
on X g i v e s
X e p(TIY ) . (iii):
Let X e p(T)n
o(TIY)-
The e q u a t i o n (X-T Y) x = 0
implies
that
( X - T ) x e Y a n d h e n c e we h a v e x = R ( X ; T ) ( X - T ) x = R ( X ; T I Y ) ( X - T ) x ~ Y.
Thus x = 0 a n d h e n c e
X-T Y i s
injeetive.
x e Y and c o n s e q u e n t l y
13
Next, let x e X/Y be arbitrary.
For every y s Y, there is a unique u e Y
such that (k-T]Y)u = (I-T)u = y. Also, there is a unique v e X which verifies equation
(X-T)v = x. Summing up, the last two arguments, there is a unique ~ ~ X/Y which verifies equation (%-TY)I : ; , and this proves that k-T Y is surjective.
The bijectivity of ~-T Y implies that
I c p(T Y) and this concludes the proof. [] The
spectJu~in~lu~ion p r o p ~ y
(1.18)
o(T]Y) C o(T), for Y E Inv(T)
will play an important role in the spectral decomposition problem.
There are some
necessary and sufficient conditions for this property to hold.
1.15. Proposition. Given T statements are eq~valent:
for every
(1)
c(TIY) C
o(T);
(il)
o(T Y) ~
o(T);
R(I;T)Y C Y ,
(iii)
Proof.
~ B(X),
(i) <=> (ii) (i)
Y ~ Inv(T)
the following
I e p(T).
follows from properties (ii) and (iii) of Proposition 1.14.
=> (iii): Let y e Y. For I E p(T) C
p(TIY), k-TIY is surjective
and hence there is an x e Y such that y : (%-T)x. The injectivity of
I-TIY
implies (iii).
(iii) => (i): For I e p(T),
I-T[Y
is injective because otherwise the
inclusions I e Op(TIY) C qp(T) C o(T) contradict the hypothesis. Thus, for every
y e Y
there is a unique
x s X
which verifies y = (~-T)x, and hence x = R(I;T)y e Y. Consequently,
I-TIY
is bijective and it follows that
p(T) c p(TIY). []
14
I. 16. Corollary. Given T ¢ B(X) with the SVEP, for every Y ¢ I n v ( T ) , the followin 9 i m p l i c a t i o ~ hold: (i)
o(y,T) = o(y,TIY),
for all y ¢ Y => o(TIY) c a ( T ) ;
(ii)
o(y,T) = o(y,TIY),
for all y ¢ Y <=> {yT(l):l g p ( y , T ) } C Y .
Proof.
(i):
With the help of Theorem 1.9, we obtain
~(y,TIY) C
c(TIY) = U yeY (ii):
U
~(x,T) = ~(T).
xeX
If for all y ~ Y, we have p(y,TIY) = p(y,T) and if X ~ p(y,T) then
yT(X) = yT[y(X) ¢ Y. C o n v e r s e l y , i f yT(X) ¢ Y f o r a l l X ¢ p (y,T) t h e n (X-TIY)yT(X) and hence
p(y,T) c
p(y,TIY).
= (X-T)ZT(X) = y,
Now, Proposition
I.i0 concludes the proof. []
One of the above implications can be strengthened.
In fact, Y being a sub-
space, by (ii) we obtain o(y,T) = ~(y,TIY),
for all y ¢ Y => c.l.m.
{yT(l): ~ ¢ p ( y , T ) } C Y .
Given T E B(X) and Y ~ Inv(T), can any bounded component of p (T) properly and simultaneously
intersect
I. I7. Proposition. component of p(T) then
~(T[Y) and
p(TIY)?
Given T ¢ B(X), l e t Y ¢ I n v ( T ) .
~ h e ] ~ ~(TIY) N G = ~
Proof.
The answer is no.
I f G ~ a~zy bounded
or G C o ( T ] Y ) .
Suppose t h e r e i s a bounded component G o f ~ (T) such t h a t o(T[Y) N G ~ ~ and G ¢ ~ ( T [ Y ) .
Then t h e r e i s a ~ ¢ G such t h a t ¢ ~[o(T[Y)] C Oa(TIY) c
~a(T) C o(T),
but this is a contradiction. []
§ 2.
Anagytica£1y inva~iant subspac~. For further meaningful applications we must sacrifice some generality.
of the invariant subspaces employed in spectral decompositions spectral inclusion property Kariotis
(1.18).
[i] in the following
Most
satisfy the
We shall use the terminology of Bartle and
15
2.1.
Definition.
Given T ~ B(X), Y E I n v ( T ) /S c a l l e d a v-space f o r T i f
(2. i)
o(TIY) C ~(T). Some equivalent defining conditions for ~-spaces are given by Proposition
2.2.
Proposi~on.
If
Y
(2.2)
1.15.
is a v-space for T ~ B(X) then
o(T) = o ( T ] Y ) U °(TY) •
Proof.
In view o f ( 2 . 1 ) ,
Proposition
1.15 ( i i )
and P r o p o s i t i o n
1.14 ( i ) imply
(2.2). []
2.3.
Theorem,
Given T ¢ B(X), l e t
neighborhood D of g ( T ) .
f:D ÷ C be a function a n a l y t i c on an open
I f Y i s a v-space for T then Y i s a v-space for f ( T ) .
Furthermore, we have f(T) IY = f(T[Y) and f(T) Y = f(TY).
Proof.
Let Y be a v-space for T.
By Proposition
1.15, Y is invariant under
R(X;T) and by the functional calculus Y is invaria~t under f(T).
Therefore,
f[T) tY = f [ T I Y ) , and in view of (2.1), the spectral mapping theorem implies the following inclusions
o[f(T)]Y] = o[f(TIY)] = f[~(TIY)] C f[o(T)]
=
o[f(T)].
^
Next, f o r x ¢ X/Y, by t h e c o n t i n u i t y the help of Proposition
o f t h e c a n o n i c a l map X + X/Y and w i t h
1 . 1 5 , we o b t a i n s u c c e s s i v e l y :
f(T)Yx = f ( T ) x = ~ -1~
=
1
2~i
~ = ~1 f f(X)R(X;T)xdX F f f(l)R[l;TY)xdt F
f f(X)R(t;T)Yxdx F
=
= f(TY)x'
where F i s an a d m i s s i b l e c o n t o u r s u r r o u n d i n g o(T) and c o n t a i n e d i n p(T) C
P(TY) •
Since x is arbitrary in X/Y, we have f(T) Y = f(TY). []
2.4.
Theorem.
Given T E B(X), l e t
f:D ÷ C be a f u n c t i o n i n j e c t i v e and
a n a l y t i c on an open neighborhood D of o ( T ) .
If Y ~
a v-space for f(T) then Y i s
a v-space for T. Proof.
Let Y be a v-space for f(T).
R[I;f(T)]. (Dunford
By Proposition 1.15, Y is invariant under
Apply Dunford's theorem on composite operator-valued [i], Dtmford and Schwartz
functions
[I, VII. 3.12]) to the composition f-lof.
For an admissible contour F which surrounds o(T) and is contained in D lip (T), we have
]6
i f-l[f(T)] = 2~i
f f-i [f(1)]R(k;T)dl = ~ i P
f X R(l;T)dl = T. r
On the other hand, we have =
f-l[f(T)]
~1
I
f-l(1)R[l;f(T)]dl
.
f(r) Combining
the above results
it is easy to show that Y is invariant
TY = f-l[f(T)]Y
= ~1
f-i (k)R[i;f(T)]Ydl
I
under
T:
CY.
f(r) Now, we conclude the proof through the following inclusions f[~(TIY) ] -- ~[f(TIY) ] = ~[f(T) I Y ] C o[f(T)] = f[~(T)], o(TIY) C ~ ( T ) .
2,5.
Proposition.
G i v e n T ¢ B(X), i f
[]
o(T)
do¢4 not s e p a r a t e
t h e p / a n ¢ then
every i n v a ~ i a n t subspace i s a v-space f o r T. Proof. y ~ Y
Any Y ~ I n v ( T ) and
Ill
>
IIT
is
invariant
under R(I;T)
II , we h a v e R ( ~ ; T ) y ~ Y.
that @ (T) is simply connected, R(X;T)y s Y on all of
D(T).
for
Ill
>
Proposition.
Thus, for implies
it follows by analytic continuation that Thus Y is invariant under R(I;T) for all ~ e p(T).
Then Proposition 1.15 implies the spectral inclusion property
2.6.
El T II
Since the hypothesis
Every h y p e ~ v a r i a n t
(2.1).
[]
subspace u n d e r T e B(X) / S a
v-space f o r T. Proof.
If Y is hyperinvariant
under T then it is invariant under R(X;T) on ~(T)
and then Proposition l.iS concludes the proof. [] More generally, ~-space for T.
any subspace invariant under both T and R(X;T) on
In particular,
p (T) is a
if E e B(X) is a projection commuting with the given
T then EX is a ~-space for T. In order to make the SVEP more useful, we proceed by introducing and studying the first important class of v-spaces,
2.7.
Definition.
Given T ¢ B(X), a subspaae Y e Inv(Z) is c a ~ e d
an~yt~-
c a l l y i n v a r i a n t under T i f f o r every function f : D + X a n a l y t i c on some open D C C, t h e condition (l-T)f(1)
implies that f(1)
~ Y
g Y on D
on D.
We denote by AI(T) the family of analytically invariant subspaces under T.
17
2.8.
Proposition.
Every analytically i n v ~
subspace is a ~-space for
T e B(X).
P,toof.
Let
Y e A I ( T ) and l e t
y ~ Y be a r b i t r a r y .
y = (X-T)R(X;T)y on Definition 2.7 implies that R(X;T)y ¢ Y on
Since
p(T),
p(T) and then Proposition 1.15
concludes the proof. []
2.9.
P/topos~t~Lon. If T ¢ B(X) has the SVEP and Y ~ AI(T) then o(y,T) = o(y, TIY) , for all y c Y.
Proof.
Let y ~ Y and
X ~ p(y,T).
implies that y(X) c Y on
2. I0,
Corollary.
Then
p(y,T) and Corollary 1.16 (ii) concludes the proof. []
Given T a B(X) with the SVEP, l e t Y E A I ( T ) .
c.l.m.
(~(~):y
Then
~ Y, x c O ( y , T ) } Q Y .
The following result gives an important characterization of analytically invariant subspaces and has many key applications in the spectral decomposition theory.
inv~
2.1 I. Theorem. ~d~ Tiff
Proof.
Given T e B(X), a s ubspace Y ~ Inv(T) /6 analytically the coinduced operator TY has ,the SVEP.
Assume that T Y has the SVEP and let f:D ÷ X be analytic and satisfy
condition (X-T)f(X) E Y
on an open D C C.
By the natural hor~morphism, it follows that (x-TY) f(X) = 0 on D and then by the SVEP, f = 0.
Hence f(X) s Y for all X c D.
Conversely, assume that Y c AI(T),
Let f:D ÷ X/Y be analytic on D and
satisfy condition (2.3)
(x-TY) f(X) = 0 on D.
Without loss of generality we may assume that D is connected.
Let
~(~) = ~ ~n(~-~0~n, with ~ ~ X/Y n=0 be the Taylor series of f in a neighborhood of a point X 0 a D. can choose an ~ an such that
For every n we
~8
J~
<
II an ,~
_
II a
II
n
+ i.
This is possible because in the topology of the quotient space, !I all =
inf a
c
II a II a
Then lira n
+
[Ian II i / n
lim
< _
co
n
II an II i / n + 1
-~ oo
and hence oo
f(~) =
~
an(X-XO ) n s ?(~.)
n=O is analytic in a neighborhood D' ( C
D) of ~0"
Now (2.3) implies that
(~-T)f(%) e Y on D' and since Y is analytically invariant it follows that f(~) s Y.
Consequently,
f(%) = 0 on D' and on all of D, by analytic continuation.[~
2.12.
Corollary.
Given T ~ B(X), l g t Q be q ~ i n l l p o t e n t
commuting with T.
Then every a n a l y t i c ~ g l y i n v a r i a n t subspace under T which i s i n v a ~ i a n t under Q is a n a l y t ~ a l l y i n v a r i a n t under T + Q. Proof.
Let
Y c AI(T) be invariant under Q.
By Theorem 2.11, T Y has the SVEP.
QY being quasinilpotent and commuting with T Y, the sum T Y + QY = (T + Q)Y has the SVEP by Corollary 1.12.
r+q.
Now by Theorem 2.11, Y is analytically invariant under
[] 2.13.
CoroZlary.
Given T s B(X), l e t
B(X) which c o n v e r g ~ uniform£y t o T.
subspace Y i s a n ~ y t i ~ a l l y l n v a r i a n t wria~und~ T. Proof.
(Tn} be a sequence of o p e r a t o ~ i n
I f f o r e v ~ y n, Tn commu~es with T and a under each Tn then Y I S ~ a l y t i c a l l y
£n-
The fact that Y is invariant under T follows directly from the continuity
of the o p e r a t o r s .
By Theorem 2 . 1 1 , e v e r y TY h a s t h e SVEP. n
of Tn to T implies the uniform convergence cf T Y to T Y. n the sequence {T~} i m p l i e s t h a t
The u n i f o r m c o n v e r g e n c e
Theorem i.ii applied to
TY has t h e SVEP and t h e n by Theorem 2 . 1 1 ,
Y ~ AI(T).[-] Some simple examples of analytically invariant subspaces now follow.
19
2.14.
Example.
Le~t T ~ B(X) have t h e SVEP.
I f E i s a bounded projection
in X which commutes with T then E× i s a n ~ y t i c a l l y i n v ~ L a ~ t under T. Proof.
Let f : D + X be a n a l y t i c
and s a t i s f y
condition
( X - T ) f ( X ) ~ EX on an open DC3 C. Since E is bounded,
the function g:D + X defined by g(k) = (I-E)fCX)
is analytic on D.
Moreover,
since E commutes with T, it follows that (X-T)g(X) = 0 on D.
By the SVEP, g(%) = 0 on D and hence f ( X ) = E f ( X ) , f o r a l i X ~ D.
2.15.
Example.
[]
The k ~ n e l of every T s B(X) with the SVEP £6 a n ~ y t i c a l l y
i n v ~ t i a n t und~ T.
Proof.
Let f:D + X be analytic on an open D C C (%-T)f(X)
and satisfy condition
~ Ker T, for all I e D.
Ken 0 = T(%-T)f(X)
= (X-T)Tf(%)
on D
and by the SVEP, Tf(X) = 0 on D. []
2.16.
Example.
(i)
Y + Z
(ii)
TY
Proof. (2.4)
(i):
Given T e B(X), let Y c AI(T) and let Z = Ker T.
is analytically
is analytically
Then
invariant under T;
invariant under T.
Let f:D ÷ X be analytic and satisfy condition (X-T)f(X)
For X e D, there are sequences
e Y + Z on an open D C C . {yn }
(X-T)f(I)
and =
{zn}
lim n
÷
in Y and Z, respectively
such that
(Yn + Zn)" oo
Then (X-T)Tf(X)
=
lim n
Since Y c AI(T), Tf(X) g Y on D. hence f(%) e Y + Z on D.
÷
Ty n E Y on D. oo
It follows from ( 2 . 4 )
that
~f(X) E Y + Z, and
20
(ii):
Let f:D ÷ X be analytic and satisfy condition (k-T)f(k) E TY on an open D C C .
Since TY C
Y and Y s AI(T), it follows that f(k) ¢ Y on D.
Ilence for each
XeD, lf(l) = (I-T)f(X) + Tf(%) e T--Y + T Y C T - ~ .
[]
The analytically invariant subspaces satisfy certain types of transitivity properties.
2.17. Proposigion. Given T ¢ B(X), let Y,Z s Inv(T) with Y C Z . The following properties hold. (i) If Y ~ AI(T) then Y c AI(TIZ); i f Y ~ AI(T[Z) and Z ~ AI(T) then Y s AI (T).
The quotient space Z/Y / 6 analyticaZly invariant under TY i f f Z analytically invaria~ u n d e r T (TY d e n o t e s t h e c o i n d u c e d o p e r a t o r o11 X/Y). (ii)
Proof.
(i) (ii):
is left to the reader. Let Z/Y be analytically invariant under T Y and let f:D ÷ X be
analytic on an open D C C and satisfy condition (X-T)f(%) e Z on D. In the quotient space X/Y the map I ÷ [(l) is analytic on D and (I-TY)f(I) s Z/Y on D. Then, by hypothesis, Conversely,
f(X) c Z/Y and hence f(%) ~ Z on D.
Thus Z e AI(T).
assume that Z ~ AI(T) and let f:D ÷ X/Y be analytic and verify (x-TY)f(I) ¢ Z/Y on D.
We may assume that D is connected.
Fix ~0 in D.
By an argument used in the
second part of the proof of Theorem 2.11, f can be lifted to an X-valued function f analytic on a neighborhood D ' ( C D )
of
~0' i.e.
f(l) ¢ f(X) on D'.
Then
(k-T)f(k) E Z on D', and the hypothesis on Z implies that f(k) c Z on D'.
Passing to the quotient
space X/Y, we have f(k) ~ Z/Y on all of D, by analytic continuation. []
2.18. Proposition. subspace Y = Y1 (~ Y2 Yi ¢ AI (Ti).
Given T i e B(Xi),
let Y.I ¢ Inv(Ti) , i = 1,2.
is analytically /nvar/an,t under T = T 1 ~
T2
The iff each
21
Proof.
First, assume that each Y. c AI(T). In view of Theorem 2.11, each I Y. coinduced (T) i on Xi/Y i has the SVEP. Then Theorem 1.13 implies that T Y = (T) Y1 @
(T) Y2
Conversely,
has the SVEP.
Again Theorem 2.11 proves that Y ~ AI(T).
assume that Y ~ AI (T).
Each Yi in the direct sum decomposition
of Y is the range of a projection in Y commuting with TIY.
By virtue of
Example 2.14, each Yi is analytically invariant under TIY and hence under T i by Proposition 2.17 (i). [] Next, we investigate for the stability of analytically under functional
2.19.
calculus.
Lemma. Given z ~ B(X), l e t
f : D + X iS a nonzero m ~ y t i c
(2.S)
invariant subspaces
First we need a lemmm. Y be a n a l y t i c a l l y i n v a r i a n t under m.
If
f u n c t i o n on an open connected s e t D such t h a t (2,-T)fCl) = 0 on D
then D C O p ( T I Y ) .
Proof,
Let G be a nonempty component of D N0(T]Y)
with f(~) ~ 0. f(~) e Y.
Y being analytically invariant,
so that there is some ~ e G
it follows from (2.5) that
]hen (~-T)f(~) = (~-r]Y) f(~) = 0
implies that f(~) = 0.
2.20. hood G of Proof.
Theorem. ~(T).
This contradiction concludes the proof. [~
Given T ~ B(X), l e t
f:G + C
be a n a l y t i c on an open neighbor-
Then a n y Y ~ AI(T) / 6 a n a l y t i c a l l y i n v ~ r i a n t under f ( T ) .
We may assume that G is connected.
Let Y c AI (T).
Then Y is invariant
under the resolvent and by the functional calculus, Y is invariant under all functions of T which are analytic on some open neighborhood of
o(T).
Let g:D ÷ X be analytic and satisfy condition (2.6)
[X-f(T)]g(k)
c Y on an open D C C .
If D N p[f(T)] # ~ then the assertion of the Theorem follows at once. assume that D C o [ f ( T ) ]
= f[o(T)].
(2.7]
For a fixed k s X-f (Z) : 0
has at most a finite number of roots in
o(T).
If we discard the multiple roots
(i.e. the zeros of f'(z)), we have the simple roots a disk D I C D .
Therefore,
D, the equation
zi,Z2 ..... ~n of (2.7) in
By Rouche's theorem, there is a disk D 2 C D
(2.7) has the same number of roots
ZI(I)' ~ 2 ( 1 ) " " '
I
such that equation
~n (k) for every k c D 2.
22
Note that the functions
~i(k),
(I < i < n) are analytic on D 2.
Now we can factor
k-f(~) as follows: (2.8)
k-f(~) = [~-~l(k)] [~-~2(k)]... [~-~n(k)]hk(~),
where h k
is analytic in ~ and nonzero on G for k ¢ D 2.
By the functional calculus
hk(T) is invertible in B(X) because 0 # h k[q(T)] = ~[h X(T)]. The functional calculus applied to (2.8) gives k-f (T) = [T-~ I(X)][T-u 2(X)]...[T-u n(k)]h k(T) and then, by virtue of (2.6) we obtain [k-f(T)]g(k) = [T-~I(X)][T-u2(k)]...[T-~n(X)]hx(T)g(X ) c Y. Y being analytically invariant under T, we obtain hk(T)g(k)
¢ Y.
Y being invariant under hk(T)-i , we have g ( k ) = hk(T ) l h k ( T ) g ( k ) -
a Y on D2,
and hence by analytic continuation g(k) e Y on D. []
2.21.
Coroi6ary.
neighborhood D of a ( T ) .
Proof.
Obviously,
variant. to Y
=
Given T e B(X), l e t
f : D ÷ C be a n a l y t i c on an open
I f T has t h e SVEP then f ( T ) has t h a t prop6rty.
any T has the SVEP iff the zero subspace is analytically in-
Hence the assertion of the Corollary follows from Theorem 2.20 applied
{0}
2.22.
.
[]
P~oposition.
Given T e B(X),
let
f : G ÷ C be an~gytx'c on an open
neighborhood G of ~(T) and nonconstant on e v ~ y component of G. SVEP i f
f ( T ) has t h a t property.
Proof.
Suppose that T does not have the SVEP.
Then T has the
Then there is a nonzero function
g:D ÷ X analytic on G such that (2.9)
(k-T)g(k) = 0 on D.
The assumption on g implies that G C q ( T ) . function hk:G
+ C satisfying
(see e.g.
For every k 8 D there is an analytic (l.10 and (1.11) where gk plays the
role of hk) (2.10)
f(~.)-f(,,l)
:
( k - p ) h k ( v ) , k ¢ D, I~ ¢ G.
Applying the functional calculus to (2.10), we obtain
23
(2.11)
f(1)-f(T) = (l-T)hl(T),
In view of (2.11), equation
I s D.
(2.9) becomes [f(X)-f(T)]g(k)
= 0 on D.
Since f is nonconstant on D C G, there exists %0 e D such that f' (lO) ~ 0. there is a disk D' with center at %0 such that f-i exists on f(D').
Then
The composite
function gof -I is analytic on f(D') and verifies equation [~-f(T)](gof-l)(~)
= 0 on f(D').
By the SVEP of f(T) it follows that (gof-l)(~) = 0 on f(D') and this implies that g(1) = 0 on D'. By analytic continuation,
we have g(1) = 0 on D
but this contradicts the hypothesis on g.
The contradiction
implies that T has
the SVEP. [] 2.23.
Theorem.
Given T e B(X), let f:D ÷ C be a
open neighborhood D of
on an
If
and Y i s invarian£ u n d ~ T then Y ~ A I ( T ) .
Y c AI[f(T)]
Proof.
function analytic
o(T) and nonco~v~tant on every component of D.
By Theorems 2.3 and 2.11, f(T) Y = f(T Y) has the SVEP.
It follows from
Proposition 2.22 that T Y has the SVEP and then Theorem 2.11 implies that Y e AI(T). []
2.24.
Definition.
Given T ~ B(X), Y e I n v ( T ) / S said to be T-absorbent i f
for any y c Y and a l l I ~ o ( T ] Y ) , t h e equation (2.12)
(h-T) x = y
has a l l s o l u t i o n s x i n 2.25.
Proposition.
Y. Given T ~ B ( X ) , every T-absorbent space Y i s a ~-space
for T. Proof.
If ~(TIY) ~ ( T )
then for some I e p(T) ~ ~(TIY), R(l;T)Y ~ Y
and consequently not all solutions of equation
(2.12) belong to Y. [-~
24
The implication between the T-absorbent and the analytically invariant subspace is given by the following
2.26. Theorem. analy~ca~lyinv~ Proof.
If T c B(X) has the under T.
every T-absorbent subspaee is
SVEP t h e n
Let Y be T-absorbent and let f:D ÷ X be analytic and satisfy condition (l-T)f(l) e Y on an open D C C .
We can assume that D is connected. D.
Therefore, assume that D N g(l)
Since g(l)
=
If D C
o(TIY) then by definition f(l) e Y on
@(TIy ) ~ 9.
Denote
(k-T)f(k), k g D n P ( T I Y ) .
c Y, we can write g(k) = (X-T)R(k;TIY)g(l)
and then we have (k-T)[f(k)-R(X;TIY)g(l)]
= 0 on D g p(TIY ).
By the SVEP of T, f(k) = R(X;TIY)g(I ) on D N @(TIY), and h e n c e
f(x) ~ Y on D f) p(TIY). Thus it follows by analytic continuation that f(X) ~ Y on all of D. ~] The converse of this property does not hold.
A counterexample for the
converse is given in Appendix A.I. There is no direct implication between analytically invariant and hyperinvariant subspaces as it can be s e e n from the following examples.
2.27. invarian£.
Example.
An analytically invarianX~ subspace which is not hyp6r-
Let T ~ B(X) with the SVEP have an eigenspace Z of dimension greater than I. Each nonzero x c Z spans a one-dimensional invariant subspace Y C Z. is not hyperinvariant.
Clearly Y
Now let f:D ÷ X he analytic and satisfy condition (k-T)f(k) e Y on an open D C C
.
There is a complex-valued function g analytic on D verifying equation (2.13)
(k-T)f(k) = g(k)x, X g D.
Let ~ ~ D be the eigenvalue for Z and hence for Y. have
In view of (2.13), we
25
(X-T)(a-T)f(k)
= ( a - T ) g ( X ) x = g ( X ) ( a - T ) x = O.
The SVEP o f T i m p l i e s (~-T) f(X) = 0 on D and then for X # a, we have
(X-a)f(X) = (X-T)f(k) Thus i t
follows that
2.28. invari~.
Y a AI(T).
Example.
[]
A hype~mva~iawtsubspace w ~ c h i s
Let T be the Hilbert every X e C with
e Y.
space adjoint
of the unilateral
IX] < 1 i s an e i g e n v a l u e X =
X
For ~ e C arbitrary
u n d e r T.
tn+l(p)
Now l e t
= ~tn(~)-kn,
Since x spans the eigen-
X be fixed such that and i n d u c t i v e l y
1 Ixt < g .
write
n ~ O.
We h a v e
tl(P )
= ~2-i,
t2(~ ) = ~( 2 i)_ ~ =
3
t3(~ ) = ~( 3
= ~ 4 -~2-~k-k2,
. . .
. . . . . . . .
_X)_k2
.
. . . . . . . . . .
k+l
tk(P ) = p
k-i -p
_i,
°
. . . . . . . . . . .
k-3x2
. - ..
-p
xk-i
.
-
T h e n , f o r e a c h k, irk(u) I < (})k+l + k(})k-1 The f u n c t i o n
defined
< (k+l)2-k+l.
by t h e s e r i e s oo
x(p) = is analytic contained
on {u e C : I ~ I i n Et b u t
1 < ~ }.
It
Then
o
of £ (0,~).
1 I~[ < g p u t t 0 ( u ) = ~
with
on ~ 2 ( 0 , ~ ) .
to the eigenvector
Xnen,
orthonormal basis
s p a c e E l , Ex i s h y p e r i n v a r i a n t
shift
of T corresponding
n=0 w h e r e (e n} i s t h e n a t u r a l
not m~alytica~ly
X tn(~)e n n=0 is seen that
the range of
x(~)
is not
26
(~-T)x(~) =
~ ~tn(~)e n tn+l(~)e n = n=0 n=0
[Utn(~ ) - t n + l ( U ) l e n =
~ lne n =xeE n=O
n=O Thus Ek is n o t a n a l y t i c a l l y
§ 3.
i nvariant
1 .
undel- T.
Spec~ralmax~alspaces. We continue to specialize the invariant subspace so that a successful theory
could be built on it. Given T a B(X) and F e F, define the family of invariant subspaces
I n v ( T , F ) = {g a Inv(T)
: o(TIY)~
F}.
If Inv(T,F) is directed and has a maximal element Y then Y satisfies the condition expressed by the following
3. I.
Definition.
Given T ~ B(X), an i n v a r i a n t subspace Y i s called spec-
t r a l maximal space of T i f for any Z a I n v ( T ) , t h e i n ~ i o n o(TIZ) C
o(TIY)
i m p l i ~ Z C Y.
We d e n o t e by SM(T) t h e f a m i l y o f s p e c t r a l
3.2. for T.
Proposition.
Given T ~ B(X), l e t
maximal s p a c e s o f T. {Yi}i ¢ a be a family of v-spaces
Then Y=
~ YI i ~ ~
iS a v-space for T and (T I ~ ic~
Proo f .
yi) ~
~ i~a
o(TIYi).
For every i a ~, Proposition 1.15 implies that R ( I ; T ) Y i C Y
i and hence
R(k;T)YCY. By P r o p o s i t i o n
1 . 1 5 , Y i s a v - s p a c e f o r T.
Then f o r e v e r y i e
Y C Y i i m p l i e s o(TIY ) C o ( T I Y i ) .
3.3.
Theorem.
[]
Given T E B(X>, for e v ~ y F ~ F the subspace W = c.l.m.
{ U Y:Y ~ Inv(T,F)}
i s hype~invariant. Proof.
Let x a W.
Y. e Inv(T,F). 3
Then x is the norm-limit of finite sums
Let S ¢ B(X) commute with T and k E p(S).
[jyj with yj c Yj and Then for every
27
Y ~ Inv(T,F) the subspace Z = R(~;S)Y is invariant undor T.
Moreover, it is easy
to verify TIZ -- R(~;S)(TIY)(~-S)rZ, for ~ ~ ~(S). Thus the following similarity transformation holds
TIZ =
(s.i)
R(%;SIZ)(TIY)(%-S]Z), for % e p(SIZ ) .
By (3. i), o(TIe ) = o(r]Y). Therefore, R(%;S)yj ~ W and since R(%;S) is continuous and W is closed, we have Sx = ~
1
f 7, R(k;S)xd% e W, F
whore the admissible contour F C p (S) surrounds o(S).
Thus W is invariant under
S. []
3.4. Proof.
Corollary.
Every s p ~
maximal space of
T ~ B(X) / s
~yperinvcu~iant.
Clearly, if Y e SM(T) then Y = c.l.m. { U z:z ~ Inv [T,o(TIY)]} .
Now for F = o(TIY ), Theorem 3.3 concludes the proof. [] In view of Corollary 3.4, Example 2.27 shows that not every analytically invariant subspace is spectral maximal for a given T c B(X).
3.5. for T.
Corollary.
Every spectral maximal space of T e B(X) .66 a ~-space
3.6. Proposition. Given T ~ B(X), an a~bitrary i n t e r s e c t i o n of spectral maximal s p a c ~ of T ~ again a spectT~al maximal space of T. Proof.
Let
{Y- }-
C
SM(T) and denote y=
~ iE~
Y.. 1
By Corollary 3.5 and Proposition 5.2, for every Z e Inv(T) with o(TIZ) C wo have o(TIZ) C
o(T[
~
Yi) ~
~-~
o(TIYi).
Then
o(TIZ) Co(TIYi),
for all i ~ ~.
o(TIY),
28
For every
i ¢ ~, Yi being spectral maximal, Z
C{'~ i ~
we have Z C - Y i and hence
Y. = Y. I
[-]
Theor~mu Every spectral maximal space of Proof.
Let
Y ~ SM(T).
T ~ B(X),
is
T-absorbent.
Assume to the contrary that
Fix y e Y and X ~ a(TIY).
there is a solution x ~ Y to the equation (X-T)x
= y.
The linear m a n i f o l d Z = {z ~ X:z = y + ~x, y ~ Y, a ¢ C} is closed because Y is closed. To obtain a contradiction, p-T[Z is bijective.
A painless
verification
shows that Z s Inv(T).
we shall ascertain that for every p ~ p(TIY ),
Let ~ ~ p(TIY ).
For z ~ Z; the equation
(p-T) z = 0
implies 0 = (p-T)y
+ o~(p-T)x = [(p-T)y
+ c~(X-T) x ]
+ c~(p-~,)x
and it follows that c~(~-X)x Since
p ~ X
and
x ~ Y, we h a v e
c~ = 0 a n d h e n c e (p-T)y
Now p e p(T[Y)
implies
E Y.
= 0.
that y = 0 and hence
For the p r o o f of the surjectivity
of
z = 0. p-TIZ
we have to find a vector
z' = y' + a'x ~ Z, with y' ~ Y and
~' E C
such that (3.2) Rewriting
(p-T)(y'
+ ~'x) = y + ~x.
(3.2) as (p-T)y'
+ a ' ( X - T ) x + ~'(p-1)x = y + ~x
we find that (3.3)
(v-T)y'
+ c~'(X-T)x
and (3.4)
~'(~-I)
=
~.
= y
29
From (3.4) we obtain
~_~
•
Since ~ ¢ PeT]Y) and y-~'(X-T)x e Y, we can solve (3.3). and
Thus u-T[Z is bijective
hence oCTIZ) C o(T1Y).
Then Z C Y
but this contradicts the assumption x ~ Y. []
Not every T-absorbent subspace is spectral maximal for
a
given T e B(X)
as it can be seen from the following
3.8.
Example.
A T-a~sorbent subspace which ~
not s p e c t ~
maximal.
Let X = C[O,I] denote the Banach space of continuous complex-valued functions on [0,I] endowed with the norm II x 11 =
sup [0,1]
Ix(t)], x ~ C [ 0 , 1 ] .
Define T e B(X), by t Tx(t) = f x(s)ds, x ~ X, t e [0,i] . 0 If R(I;T) exists, let denote G(t) : R(X;T)xCt). Then (X-T)G(t) =
x(t),
or XG(t) - 7 0
(3.5)
G(s)ds = x ( t ) .
For ~ ~ 0, equation (3.5) has the solution (3.6)
GCt) : 7
1
et/X ft x(s)e-S/%ds + ~1 xCt) . 0
In order to check (3.6), perform an integration by parts on t t 7 G(s)ds = --~ 1 f 0 0
s
eS/~" f x~r)e -r/~r 0
t 1 drds + ~ f x(s)ds 0
t o get t = i et/% f G(s)ds ~ 0
t I x(s)e-S/kds. 0
30
Then
t ( X - T ) R ( X ; T ) x ( t ) = ( X - T ) G ( t ) = XG(t) - f G ( s ) d s = x ( t ) . O Thus we h a v e
o(T) = {0}. Now l e t 1 z = {x ~ x: x I [ o , y ] = o}.
After a moment a reflection, we deduce that Y is a subspace invariant under both T and R(X;T) and hence a(TIY ) C a ( T ) .
Then o(T) = a(TIY) = {0} .
But since Y # X, Y is not spectral maximal for T.
Since Tx e Y implies x s Y
by continuity, Y is T-absorbent.
3.9.
Theorem.
I f T ~ B(X) has the SVEP then every s p e c t r a l maximal space of
T is an~yt~c~ly inv--. Proof.
Let Y c SM(T) and let f:D ÷ X be analytic and verify condition (X-T)f(X) ~ Y on an open D C C .
We may suppose that D is connected. f(X) e Y on D.
If D C o ( T I Y )
then Theorem 3.7 implies that
Therefore assume that D f~p(TIY) # ~ .
For X ~ D N p(T[Y),
let
g(X) = (X-T)f(I). Since
g(l) e Y, we can write g(X) = ( X - T ) R ( X , T [ Y ) g ( X ) .
Then, by t h e SVEP o f T, f ( X ) = R(X;TIY)g(X ) on D n p ( T J Y ) .
Thus f(X) e Y on D N p(TIY) and on all of D by analytic continuation. [] The opposite implication does not hold in general as it can be seen in Appendix A.i.
5. I0.
Corollary.
/~e.~ T c B(X) have t h e SVEP.
I f Y i6 any s p e c ~ z ~ maximal
space of T then c.l.m. {y(X):y ~ Y, X s p ( y , T ) } C Y . Proof.
By Theorem 3.9, Y c AI(T) and then the assertion of the Corollary follows
from Corollary 2.10. []
31
3. If.
Theorem.
Let T e B(X) have t h e SvEp.
I f for F e F, XT(F) /S ~ o ~ e ,
then XT(F) i s a s p e c t ¢ ~ maximal space of T and (3.7)
o[T/XT(F)]C F no(T).
Proof.
First we prove inclusion (3.7).
Let ~ ¢ Fc.
Then o(x,T)~ F and by Proposition 1.5 (v), o[x(l),T] (3.8)
(X-T)x(~)
and x(~) c ~(F).
Fix an arbitrary x ¢ ~ ( F = o(x,T) C
F.
Hence
= x
Equation (3.8) proves that ~-TI~(F ) is surjective.
Since
T I ~ ( F ) has the SVEP (Proposition i. I0), Corollary 1.3 implies that
s p[T[~(F)]
and hence o[T[XT(F)] C F.
Moreover, XT(F ) b e i n g h y p e r i n v a r i a n t ( P r o p o s i t i o n 1 . 8 ) , i t
follows from
Proposition l.iS that
o[TI XT(F) ] ~ ~(T). Next, l e t Y ¢ Inv(T) be such t h a t o(TIY) C o[TIXT(F)]For every x ¢ Y, with t h e h e l p o f (3.7) and P r o p o s i t i o n 1.10, we o b t a i n successively
o(x,T) C ~ ( x , T [ Y ) C and hence x ¢ ~ ( F ) .
~(T[Y) C ~ [ T I X T ( F ) ] ~ F
Thus Y C X T ( F ) and therefore ~(F)
is spectral maximal
f o r T. [ ] A simple example of an o p e r a t o r f o r which XT(F) i s c l o s e d for F ¢ F i s the hyponormal operator T on a Hilbert space with void residual spectrum. The local resolvent of this operator satisfies the first order growth condition
d[X,o(x,T) ] '
for all
~ ~ p(x,T),
(Stampfli [2]). 3.12.
Corollary.
[email protected] T e B(X) have t/ae SVEP.
I f for r ~ F, Y = XT(F ) i s
~ o s e d then for every y ~ Y, o(y,T) = o(y,T[Y). Proof.
By Theorem 3.11, Y E SM(T) and then Theorem 3.9 coupled with Proposition
2.9 concludes the proof. []
32
3.13.
Proposit~n.
L e t T ~ B(X) have t h e S ~ P
and l e t Y ~ SM(T). Then for
e v ~ y F e F, we have Y N XT(F) = Y T I y ( F ) .
Proof.
L e t x ¢ Y fl XT(F). By C o r o l l a r y
3.10,
x(X) ¢ Y f o r e v e r y X ~ F c and t h e r e -
f o r e o ( x , T [ Y ) C F. Thus, we have y fl X T ( F ) C Y T [ y ( F ) . Conversely,
f o r x e YTty(F) P r o p o s i t i o n
1.10 i m p l i e s
o(x,TIY) C F
o(x,T) C
and h e n c e x e XT(F). Then x e Y NXT(F)
and
YTIy(F) C 2" N XT(F). [ ]
3.14. Example. Given T e g ( x ) , l e t ~ be a s p e c t r a l s e t of e(T) and denote by E (~) t h e corresponding p r o j e c t i o n . Then E (z)X i s a s p e c ~ maximal space of T. Proof.
L e t Y ~ Inv(T) b e s u c h t h a t o(TIY) C o[T[E(~DX] = T .
Let y ~ Y b e a r b i t r a r y .
For C = {X: Ikl =rl T]I + i}
and for an admissible
contour
F which surrounds
T
and is contained
in T c ~ p(T)
we have successively
1 f R(X;TIy)ydX = Y - - 2 -1~ c/ R(),;W)yd), = 2 ~ T C 1 - 2~i / R(X;TIY)YdX r
Hence it follows
The spectral
=
1
2-gf /r
that y e E(T)X and consequently
maximal
spaces
satisfy
3.!5.
Theorem.
(i)
I f Y ~ SM(T) then Y ~ SM(TIZ);
R(X;T)ydX
= E(~)y.
we have Y C E(T)X. []
some transitivity
properties.
Given T s B(X) and Y,Z e Inv(T) ~6£h y c Z, we l~ve
(ii)
If
Z e SM(T)
(iii)
If
Y,Z s
aM.d
then SM(TY).
Y ~ SM(TIz)
SM(T) then
z/Y
¢
Y ¢ SM(T);
38
Proof.
(i) : Let
Y1 ~ Inv(TIZ) be such that
o[(TIZ) I Y i ] C o[(TIZ) IY] or, equivalently,
a(T]Y 1) C o(T[Y). Since Y1 ¢ Inv(T) and Y ¢ SM(T), i t follows that Y1 C Y. (ii):
Let YI ¢ Inv(T) with
~(TIY1) c o(T[Y). Then
(3.9)
a(TIYI) C o(TIZ).
Since Z g SM(T), (3.9) implies that Y I C Z
and hence YI g Inv(TIZ)"
o[(TIZ) IYI] = o(T]Y I) Co(T/Y)
We have
= o[(TIZ) IY],
and since Y ¢ SM(TIZ ) it follows that YI C Y. (iii):
Denote the quotient space Z/Y = Z and let Z1 e Inv(TY) satisfy
condition
e(TYIz1 ) (ZI ~{TYIz). Putting Z 1 = {x c X:x + Y ~ Z1 } , by (i), Y ~ SM(TIZI).
Applying Proposition 2.2 to the operator TIZ 1 with the
restriction (T[ZI)IY and the coinduced (TIZI)Y, we obtain successively: °(TIZ I) = °[(TIZ I) IY] U °[(TIzI)Y] = °(TIY) O °(TYIzI ) C c°(TIZ) U °(TYI ~) =
°(TIZ) U o[(TIZ) Y] =
°(TIZ)"
Note that (i) implies Y c SM(TIZ ) and then o[(TIZ) Y] ~ a(TIZ) follows from Proposition 1.15 (ii). Since Z ¢ SM(T) we have Z I C Z and consequently Z1 c Z. [] 3.16.
Let T ¢ B(X) have the SVEP and let F 1 and F 2 be closed
Theorem.
d i s j o i n t subsets of C.
If XT(F1 U F2) /s closed then each XT(Fi), (i = 1,2) /S
closed and
XT(FI U F2) Proof.
=
~rCFl) ®
XT(F2).
Let us denote Y = XT(F 1 U F 2) and Yi = ~(Fi)' i = 1,2.
By Theorem 3.11,
34
o(T[Y) C F 1 U F 2If,
for i = 1,2,T i is the spectral set defined by T i = F i ~ o(T]Y i)
and E i = E(~i) is the corresponding projection, then we have (3.10)
Y = ElY @
Let y e Y.. l
Then
o(y,T) ~ F . .
E2Y.
Also,
i
o(y,T) C ~(y,TIY i) C o(TIY i) and since y ¢ Y, with the help of Corollary 3.12 we can write a(y,rlY ) = o(y,T) C F i 0 o(TIY i) = T iThis implies y e E.Y and hence i
(3.Ii)
Y~E.Y,
i = 1,2.
I
Conversely, if y ¢ EiY, then o ( y , T ) C o[(T]Y)]EiY] and h e n c e y e Y.. x
C
Ti C
Fi ,
Thus EiYCYi,
i = 1,2.
Now, i n view o f (3.11) we o b t a i n
EiY = Yi' i = 1,2 and the conclusion of the proof follows from (3.101. [] NOTES AND COMMENTS. The single-valued extension property appeared in Dunford [2,3] and received a systematic treatment in Dunford and Schwartz [I, Part III].
It also forms a
preliminary topic of Colojoara and Foia~ [3]. The proof of Proposition 1.5 (v) first appeared in Colojoara and Foias [i] and then in [3, 1.1.2] . Theorem 1.6 first was published in Apostol [2] and it was independently proved by Battle and Kariotis [i] .
In addition to its great significance in the local spectral theory,
Theorem 1.6 is followed by the straightfomcard Corollary 1.7 (Bartle and Kariotis [i]) thus simplifying a more technical proof originally given by Colojoar~ and Foias [2,3]. Proposition 1.2 and Corollaries 1.3, 1.4 appeared in Finch [I]. Theorem 1.9 was proved by Sine [i]. For the more general case of noncommuting operators, Theorem i.ii was proved by Vasilescu [I] and Corollary 1.12 by Colojoara and Foias [i]. Theorem 1.13 appeared in Colojoara and Foia~ [2,3], Proposition 1.14 was partially proved by Bacalu [i] and it can be retraced in J.L. Taylor [i]. Some references for Proposition 1.15 are Scroggs [I], Bartl~ and Kariotis [i]. Proposition 1.17 was proved by Scroggs [i].
35
A comparative study of ~-spaces and ~-spaces, the latter being defined by the property expressed by Proposition 2.9, was dene by Bartle and Kariotis [i]. Bartle and Kariotis have the credit for Definition 2.1, Theorem 2.3, Proposition 2.5 and Proposition 2.6 [ibid.]. For the proof of Theorem 2.20 we borrowed a technique used in Colojeara and Foias [2,3]. For Corollary 2.21 and Proposition 2.22 we simplified the original proofs of Colojoara and Foias [2,3]. J
The concept of analytically invariant subspace (Definition 2.7) was introduced by Frunz~ [2] who also proved the important Theorem 2.11 [ibid.].
Proposi-
tion 2.19 is due to Bacalu [i]. The concept of T-absorbent subspace (Definition 2.24) without the restriction of being invariant under T was introduced by Vasilescu [2]. The illuminating presentation of the spectral maximal space as the maximal element of the directed family Inv(T,F) was conceived by Vasilescu [2] in an attempt to generalize the notion.
The concept of spectral maximal space first appeared in Foias [2].
The closure M(F,T) of the linear manifold XT(F) for T ~ B(X) and F c F appeared in Bishop [i] under the name of strong spectral manifold.
XT(F ) being
closed for F closed represents Condition C among Dunford's sufficient conditions for an operator to be spectral (Dunford [2], Dunford and Schwartz [i, Part III]). Later, Condition C was used by Battle [i], Bartle and Kariotis [i] and Stampfli [1,2] for localization theorems and local spectral theory of some special operators. For Corollaries 3.4, 3.10 and Theorem 3.11 we simplified the original proofs of Foia~ [2], Colojoara and Foias [3]. Example 3.8 was borrowed from G. Shulberg's doctoral thesis.
Theorem 3.7 is an adaptation of a lemma by
Frunz~ [2]. Proposition 3.13 was proved by Apostol [6], Corollary 3.12, Theorem 3.15 appeared in Apostol [3] and Theorem 3.16 in Apostol [5]. Example 3.4 can be found in Colojoara and Foia~ [3, 1.3.10] . Finally, it should be mentioned that most properties of analytically invariant subspaces were obtained in the doctoral dissertation work by Lange [2].
CHAPTER II THE GENERAL SPECTRAL DECOMPOSITION Everything is now prepared for the study of the spectral decomposition problem which really makes invariant subspaces important.
Historically, this
problem evolved from very special operators for the need of providing the selfadjoint boundary value problem in a Hilbert space with a complete orthonormal set of eigenfunctions.
Dunford's extensive theory on spectral operators showed that
the spectral decomposition of linear operators can go beyond the classical spectral theory of self-adjoint and normal operators. Since then more and more general classes of linear operators which admit a certain type of spectral decomposition have been discovered.
Such operators
decompose the underlying space into a finite linear sum of proper invariant subspaces such that the spectrum of the given operator restricted to each invariant subspace is contained in a given subset of the complex plane.
The specific
property that the invariant subspaces have in common determines the type of the spectral decomposition and subsequently confines
the operator to a given class
from the spectral theoretic point of view. It was found that all examples of operators known to admit a spectral decomposition have the single-valued extension property.
We raised the question:
Is the single-valued extension property an intrinsic element of the spectral decomposition?
In order to answer this question we defined axiomatically the
spectral decomposition of an operator T ~ B(X) in terms of unspecified invariant subspaces.
Then we obtained an affirmative answer.
We also found that the
spectral decomposition in those general terms implies a special structure of the spectrum of the operator:
the spectrum is entirely the approximate point spectrum.
As a by-product, we obtained some basic elements for a functional calculus.
This
chapter will follow step-by-step the development of the general spectral decomposition problem by reproducing most of a paper by the authors (Erdelyi and
Lange [1]). Although the theory of general spectral decomposition can be extended to unbounded linear operators on a Banach space as well as to more general topological vector spaces, the present chapter will be confined to operators in B(X).
§ 4.
Operators with s p e c t r a l decomposition properties. We begin with a preliminary property which will give us, when necessary, an
alternative way of defining the subsequent spectral decomposition.
37 4.1.
Proposition.
Let M be a l o c a l l y compact Hausdorff space and l e t K be
a compact subset of M. For e v ~ y open cover {Hi} In of
K suchtha~the
s e t s H.~ are r ~ i v ~ y Hi C
Proof.
{Gi} ~ of
Let k s K be arbitrary.
Gi,
K t h e r e i s an open cover
compact and
i : 1,2,...,n.
Then k s G i for some i, say for i = il,i 2 .... ,i k-
There are relatively compact open neighborhoods
V(lil), V(li2) ..... V(kik) of k
such that V(li ) C Gi , ] ]
j = 1,2 ..... k.
Putting k Vk = ~'] V(%i. ) j=l ] we
have
Vx C
Gi, f o r each Gi which c o n t a i n s X. The f a m i l y
open cover of a compact s e t K, t h e r e i s a f i n i t e
H i = [_){Vk. : VX. C ] ] Then it follows that
H. C G. 1
subcover
{VI})t~K b e i n g an {Vx.}m~ of K. ]
Let
Gi}, i = 1,2 ..... n.
and
1 n
KcU i=1
H.. V1 1
The spectral decomposition of the underlying space X is formulated as a linear sum of an unspecified number n of invariant subspaces. The proofs of all properties given in this chapter go through under the assumption n=2, which makes the operator more general but its relation to other spectral decompositions less suitable.
4.2.
Definition.
T ~ B(X) / s said to have t h e s p e c t r a l decomposition pro-
r n p e r t y (abbrev. SDP and f o r n=2, 2-SDP) i f for every open covet ~Gi} 1
t h e r e i s a system
n
(i)
X :
(ii)
~(TIY i) C
(4.1)
Condition
of
~(T),
{Yi}l of i n v ~ t i a r ~ subspac~ under T with t h e p ~ p ~ r ~ i e s : ~ Yi' i=l G i,
i = i,2 ..... n.
o(TIYi) c [i ~
i : 1,2 ..... n.
(ii) can be substituted by (ii,)
38
In fact, it is obvious that (ii) implies (ii'). there is an open cover {Hi} 1
Conversely, by Proposition 4 . 1 ,
of ~(T) with H.~G., 1
i = 1,2 ..... n. l
Definition 4.2 implies the existence of invariant subspaces Yi(l < i < n) which satisfy (i) and o(T[Yi)
4.3.
Proposition.
c
Gi,
HiCHiC
i = 1,2 .....
n.
I f T h ~ t h e SDP and 0 i s an i s o l a t e d point of t h e
spectrum then T i s t h e sum of an i n v e r t i b l e and a quasin2lpotent o p ~ o r . Proof.
Since 0 is an isolated point of o(T) there is a positive integer n such
that {x ~ c: o < I x [
< L} cp(T). n
Consider the following open cover of C: G 1 = {X e C: Ill >
_7_i} n+l
•
G2
{I c C: Ixl < i} n
"
By the SDP, there are invariant subspaces YI" Y2 which perform the spectral decompos it ion X = Y1 + Y2' o(TIY i) C G i, i = 1,2. That is T
(4.2)
:
TIY1 + TIY2.
Since 0 ~ 0(TIYI), TIY I is invertible and TIY 2 is quasinilpotent having its spectrum
~UIY2) c {o). [] 4.4.
Lemma. Let T have t h e SDP.
I f G is any open s ~
such t h a t
G N ~(T) ¢
then t h e r e i s a nonzero Y ~ Inv(T) such £ h ~ o'(TIY) C G. Proof.
Let H be a second open set such that G,H cover o(T) and o ( T ) ¢
By (4.1) there are Y,Z E Inv(T) satisfying
H.
39
(4.3)
X = Y + Z
o(TIZ) C H .
o(T[Y) c G , Then Y #
{0) because otherwise Z = X and
o(T) = o ( T I Z ) (22 H which is impossible by the choice of H. [] This le~aa has a larger range of application.
For instance,it holds true
if we replace (4.3) by the weaker condition X=Y+Z. A generalization of a theorem by Fog~el [i, Theorem 5] now follows.
4.5.
Theorem.
If T h ~ the SDP then ¢~(T) = o a ( T ) -
P~.*~of. Suppose that z(T) ¢ Za(T ) .
Then
G = [oatT ) ]c is open and
GNc~(T)
¢ ~l .
By Le~aa 4.4 there is a nonzero Y ~ Inv(T) such that
o(T[Y) C G. Then there exists X a G such that
X e 3o(T[Y) C Oa(TIY) C Oa(T) but this
is a contradiction.
[]
The t i m e h a s come t o g a t h e r operators
w i t h t h e SDP.
First
the pieces
n e e d e d f o r t h e p r o o f o f t h e SVEP f o r
we h a v e t o o v e r c o m e t h e l a c k o f t h e s p e c t r a l
inclusion property. 4.6.
a~h
Lepta.
Let T have the SDP.
There ~
a spe~
deeompos~o~
(4.1)
the p r o p e r t i ~
(4.4)
Proof.
o(TIY i) C c ( T ) ,
i = 1,2 .....
g n Let { i } l b e an open c o v e r o f c ( T ) .
o(T) w h i c h m e e t s G i , c h o o s e a c l o s e d d i s k Fki C Vk ~
Gi .
n.
For e a c h b o u n d e d co~,,ponent Vk o f
40
If necessary,
by intersecting every G. with a relatively compact open neighbor1 q(T), we can have an open cover of o(T) such that the number of the
hood of
Fki'S be finite.
Put H i = Gi -
U
Fki, i = 1,2,...,n. k
Then {Hi} 1
covers
~(T) and by the SDP there is a system {Yi}l C Inv(T) which
performs the spectral decomposition n [ Yi' i:l
X =
o(TIY i) c H i C In the given circumstances,
4.7.
iem~a.
Given T
G i, i = 1,2 .... ,n.
Proposition c
B(X), l e t
1.17 implies properties
f:D ÷ X
(4.4).[~
be a nonzero f u n c t i o n , a n a l y t i c
and s ~ i s fying condition (X-T)f(k) = 0 on an open D c C .
I f f o r some open nonvoid U C D, Y s I n v ( T ) ~ (4.5)
such .that
{f(l):l a U} C Y ,
£hen D C ~ p ( T ] Y ) .
Proof.
Since for f analytic, D is locally connected , we can assume that D is
connected.
Define H = {l c D:f(1), f'(1), f"(1) .... ~ Y}.
H has the following properties:
(a)
H / ~.
Let k0~: U. For r > 0
sufficiently small, the circle r = {t ~ c:
,,/t-Xol
: r} C U,
and then {f(l):~
~ r} c Y.
By Cauchy's formula f(n)(l O)
= n: ~
I F
f(X)dk (i-ko)n+l '
we have f(n)(kO) e Y, n = 0,i,2,... where f(O)(kO) denotes f(lo).
(b)
H is open.
4'1
Let k 0
~ H.
~hen f(%0) , f'(%0) .... ~ Y.
Since f, f', f" .... are analytic, they
admit power expansions in an open neighborhood V(%0) of ~0' and hence f(n)(x) ~ Y, k ~ V(X0), n = 0,1 . . . . Thus it follows that
v(~ 0) c H. (c)
H is closed in D: H = ~ n=0
(d)
[f(n)]-l(y).
H~ D COp(T]Y).
Note that for every k ~ D, the vectors f(n)(%) are not all zero because otherwise f = 0.
Let m = rain {n:f (n) (>,) # 0}.
If m = O, Tf(X) = Xf(%), and if m > O, Tf (m)(k) = kf (m)(l). In either case, for X c D (] H, f(m)(l) is an eigenvector of TIY with respect to the eigenvalue
X.
By properties
Thus
~ e Op(TIY ) and property
(d) holds.
(a), (b), (c), H = D and then property
(d) concludes the
proof. []
4.8.
Lemma. ieY~ YI' Y2 be s u b s p a c ~ of X such t h a t X = Y1 + Y2
and l e t
f:D ÷ X be a n a l y t i c on an open D C C.
Then for every ~ ~ D t h e r e i s a
neighborhood v ( c D) of ~ and a n a l y t i c functions f.:V+ I
Y., i = 1,2 I
such t h a t (4.6) Proof. equipped
f(~) Define with II Yl
the continuous
= fl(~)
+ f2(~),
map P: Y1
@
N ~ V.
Y2 ÷ Y1 + Y2 by P(yl(~Y2)
the norm (~
Y2 I[ = II Yl lJ + I] Y2 bl ' Yi ~ Yi'
i = 1,2.
= Yl + Y2'
42 P being surjective,
by the open mapping
that for every x ~ X there
Y satisfying
theorem
Yi
=
E)
Y2 ~ Yi
is a constant
k > 0 such
E)
Y2
conditions Py = x and lJ yll
Then every x s X can be written (4.7)
there
exists
x = Yl + Y2' with
<
k
II Xll
as
II ylll
+ II Y2
II _< k iI x II , Yi s Yi'
i = 1,2.
For k fixed in D, let
(4.8)
f(v) =
be the Taylor
series
for r > 0 sufficiently
expansion
~ (li-k)ng(n), n=O
g(n) a X,
of f in a neighborhood
of I contained
in D.
'llmn,
small we have v
= {-~:l,~-~, I
<
r} c
D
and
(4.9)
sup r
n
II g(n) II < ~.
n
In view of (4.7), we have
(4. ]0)
g(n) = gl(n)
+ g2(n),
with gi(n)
e Yi' i = 1,2
k
,
and (4.ii)
il gl(n) II +
By conditions
(4.9) and (4.11),
(4.12)
and
4.9. property. Proof.
II <
and defines
Yheor~n.
=
an analytic
(4.12) produce
II g(n) II
We may suppose
n = 0,I ....
X (~-~')ngi(n) n=O
function
on
V
with values
in Y.. I
Then
(4.8),
(4.6) .[]
Every operator wZth ,the
SDP
has the single-valued extension
Given T with the SDP, let f:D + X be analytic
(4.13)
for
for i = 1,2, the series
fi(~)
converges (4.10)
g2(n)
(k-T)f(k)
= 0
that D is connected
and verify
equation
on an open D C C
and contained
in o(T),
since for D N p(T) # 9,
f(X) = 0 on some open set and hence on all of D, by analytic D' be an open disk in D and let H I and H 2 be open half-planes
continuation. covering
Let
~(T) such
43
that D' - HI # @"
In view of Le~mm 4.6,there are subspaces YI' Y2 s Inv(T)
which perform the following spectral decomposition: X = Y1 + Y2' (4.14)
o(T]Y i) C H i N o(T), i = 1,2.
By Lewma 4.8, there is an open disk V C D' - HI and there are analytic functions fi :V ÷ Yi' i = 1,2 such that f(l) = fl(l) + f2(l), for all ~ ~ V. Then (4.13) implies that for I e V, g(X) = (k-T)fl(k) = (Z-~)f2(~) s Y1 ~ Y 2 In view of Proposition 1.15,
= Y"
(4.14) implies that for X s V C p(TIYI),
fl(X) = R(X;TIY1)g(X ). Next, we propose to show that fl(X) e Y on V. For I with IXI > 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 : <
R(X;T[Y2)Y~,U
>
Next, we show that v is bounded. Yl @
:
<
R(X;TIY2)Y2,U
>
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
<
~[ KI~ R(X;TIY 2) /l'tl u U ]~ x~ . To s e e t h a t
, l e t x e XT(FC ) be a r b i t r a r y .
v e --XT(FC)
x = Yl + Y2 w i t h Yi a Y i '
Then, from
i = 1,2
we obtain
Y2
:
x-Yl e XT(FC ) A Y1 C XT(FC),
and since XT(FC ) is hyperinvariant,
R(X;TIY2)y 2 ¢ XT(FC ) . Thus, it follows that
< x , v > = < 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 <-!n ' 3 n' n
x(t)
=
O,
t3
l
for
-t-
<
n
%
1
I
n
;
1
+n'n-
n converges uniformly to a function x E X defined by x(t) = t3-t, t ~ [-i,I]. The limit function x s Y~F(G).
We clearly have o(x,T) = supp x = [-i,i]
and then
~(TJY)
= [-1,1] ~
K
-
109
Y is not absorbent
because for 0 ~ o(TIY),
the equation
Ty = x has the solution
y(t)
= t 2- 1
and it is easily seen that y ~ Y. In view of Theorem 3.7, Y is neither spectral maximal.
A.2.
The set-spectra of decomposable operators.
A. 2. I. Definition. Given T ¢ D(X), a Hosed subset F of o(T) i s called a set-spe~L~um of T i f t h e e e x / s t A Y ~ SM(T) such t h a t o(T[Y) = F
or, equivalently, i f ~[T]XT(F)]
= F.
A.2.2. Theorem. Given T c D(X), for e v ~ y closed subset F of ~ ( T ) , t / f e t e i s a l a r g e r set-spe~trwn F 0 contained in F . F 0 has the following proper~i~ (A. 2.13
F0 = o[TIXT(F)] ,
(A.2.2)
XT(F0) = XT(F ) .
Proof.
F 0 a s d e f i n e d by ( A . 2 . 1 )
be any s e t - s p e c t r u m The hypothesis
contained
is a set-spectrum
i n F. Our o b j e c t i v e
contained
i n F. Let o(TIY)
i s t o show t h a t
on Y implies
Y = XT[o(TIY)] C XT(F), and h e n c e we h a v e
o(TIY) c c~[TIXT(F)] = F o. In order to obtain (A.2.2),
it suffices to prove the inclusion
~r(F) c Xv(Fo). Let x ¢ XT(F ) b e a r b i t r a r y .
Then we h a v e O(x,T) C ~[T]KI,(F)]
and hence
x c XT(F0). []
= F0
o(TIY)C
F 0.
110
A. 2.5.
Theorem.
Given T e D(X), l e t F C o(T) be ~losed and denote by F I t h e
i n g e r i o r of F i n t h e topology of a ( T ) .
Then for t h e l a r g e s t set-spectJmm F0 con-
tained i n F we have (A.2.3)
F I C F0 C F. Let X e F I .
Proof.
For e v e r y n e i g h b o r h o o d
V(X) o f
o ( T ) , t h e s e t V(X) O F I i s a n o n v o i d s e t open i n
X open i n t h e t o p o l o g y o f
o(T).
By P r o p o s i t i o n 7 . 2 , t h e r e
e x i s t s Y ~ SM(T) and hence a s e t - s p e c t r u m FV = o(T[Y) such t h a t ~ FI.
~ Fv c v ( x )
We have FVC
FI c F -
Since F 0 is the largest set-spectrum contained in F, FV C F 0 and hence Fv C V(X) A F0. Thus, X ¢ ~0 = F0' and i n c l u s i o n s
A. 2.4.
Co~l~y.
(A.2.3) f o l l o w . [ ]
Given T ~ D(X),
i f F i s a closed subset of
~(T) SUch
that F=
then F ~
a set-Speegrum of T,
A.2.5.
Corollary.
In p a r t i c u l a r , every F I ~
Corog6a~j.
a set-spectrum of T.
Given T e D(X), for every closed F C o ( T ) ,
l a r g e s t set-spectrum contained i n F.
A. 2.6.
FI ,
1££ F0 be the
Then F-F 0 has void i n t e a i o r i n o ( T ) .
Given T ~ D(X), for every F ~ F, F 1 = ~G) - F
a set-spectrum of T. a. 2. 7.
Corollary.
Given T e D(X), l e t Y ~ SM(T).
Then t h e r e e x l s t s
Z e SM(T) such t h a t o(T IZ) = o(TY).
(A. 2.4)
Proof.
By Proposition ii.i0, o ( T Y) = o G )
- ~(rlY)
and then Corollary A.2.6 implies that o(T Y) is a set-spectrum of T.
z = x.r [o (T Y) ] , and ( A . 2 . 4 ] f o l l o w s . [ ]
Then
111
A. 2.8.
Theorem.
o(T) with the G i c {Yi};c
T ¢ B(X) i s decomposable i f f
for e v ~ y cover {Gi} ~ of a system
o(T) open in the topology of ~(T), t h ~ e i s
SM(T) s a t ~ f y i n g the following spectral decomposit~n I1
X=
~ Y. i--i i,
°(TIYi) = G i '
i = 1,2 . . . . . n.
P r o o f . The " o n l y i f " p a r t i s o b v i o u s . Assume t h a t T ¢ D(X) and l e t {Hi}1n open c o v e r o f e ( T ) . There i s a c o r r e s p o n d i n g system
be an
{Zi} 1 G SM(T) such t h a t
n
X=
~ Z. i=l i,
o(TIZ i) C Hi ,
i = 1,2 . . . . . n.
For e v e r y i,
G-,
H. N o ( T )
=
1
1
i s a s e t - s p e c t r u m o f T and Yi = XT(Gi) ~ SM(T) w i t h o(TIYi) = Gi" Moreover, t h e i n c l u s i o n s e(TIZ i) G Hi ~ o(T) C Gi = °(T[Yi) imply that
ZicYi,
i = 1,2 . . . . . n.
[]
In view of Proposition 5.6, any intersection of set-spectra is again a setspectrum. For arbitrary unions of set-spectra we also have a corresponding closure property. To obtain it we need the following
A. 2.9.
Lcmma. Given T e D(X), for every subs~% W of X, I I
F = ~_) a(x,T) xcW
i s a set-spectdvam of T.
Proof.
First, we show that W C XT(F ) . Indeed, for every x E W, we have X ¢ XT[O(x,T)] C XT(F)-
112
Next, with the help of Theorem 1.9 we obtain successively
oETI r F I =
C)
o x,T
cx,T
XeXT(F )
A.2.10.
Theorem.
= F. [ ]
xeW
Given T e D(X), l e t
{F}ae A
be m~ arbitaary family of
set-spectra for T. Then F = ~_~ F ~cA
is a set-speetrum of T.
Proof.
For every ~ E A, there is a spectral maximal space Y
of T with
F
xeY
xsY
By denoting
w=k_) Y
we obtain
F = ~
Fa
~eA By Lemma A.2.9, F is a set-spectrum of T.
= ~)
e(TIY ~) = k,..] O ( x , T ) .
acA
x~W
[]
Part of this section is based on Erdelyi [2]. Theorems A.2.2 and A.2.3 with the Corollaries A.2.4 and A.2.S
are due to Berruyer
[I]. Theorem A.2.8 and Lemma
A.2.9 are among the results obtained by Bacalu [I].
A. 3.
The approx~nate point specgrum and the single-valued e~ension property. Several examples as for instance the unilateral shift operators show that
the SVEP does not imply the equality between the approximate point spectrum and the entire spectrum of an operator. A condition for such a property to hold is given in this section of the Appendix. It is well-known that the approximate point spectrum contains the boundary of the entire spectrum. An extension of this property to the local spectrum now follows.
113
A.3.1.
Theorem.
Let T have t h e SVEP. I f x i s any nonzero vector i n X then 8o(x,T) C ea(T). ~
Proof.
Let X 0 ¢ 3o(x,T).
Then the local resolvent
if X 0 is an isolated point of o(x,T)
x is unbounded near XO" In fact,
and x were bounded
in a n e i g h b o r h o o d
V of X 0
then by
(X-T)x(X)
~ e V n p(x,T),
= x, ~
X0 w o u l d b e a r e m o v a b l e s i n g u l a r i t y
~
o f x. But t h e n x e x t e n d s
t o X0 a n d t h i s
~
contradicts
the maximality
of x.
I f ~0 i s n o t an i s o l a t e d
singularity
then it
is
~
an essential
singularity
of x and again x is unbounded near
T h u s we may c h o o s e a s e q u e n c e
{Xn) C p ( x , T )
such that
X0. Xn ÷ X0
and
~
llX(Xn)l[
+
~ . Then ~
x(~n) Yn =
is a sequence of unit vectors
IiX(Xn) tl
in X such that
X
(Xn-T)y n =
_ + llX(Xn) ll
0
when n ÷ ~.
Thus
( ~ o - T ) Y n = (),O-~.n)Yn + (Xn-T)Y n approaches
A.3.2.
zero as
n ÷
Defini~n.
: x + g(x,T),
and hence
)~0 e O a ( T ) .
[]
Let T have the SVEP. We say t h a t t h e s e t - v a l u e d mapping
x ¢ X, / S l o ~ z e d i f
f o r every open G C C w i t h
G 0 o(T) # t h e r e i s a nonzero x ~ X such t h a t o ( x , T ) C G.
A.3.3.
Theorem.
I f T has t h e SVEP and
%(T) Proof. x
Let ~ ¢ e(T).
¢ X such that n
By hypothesis,
:
o : x ÷ g(x,T)
i s l o c a l i z e d then
o(T).
for every n, there is a nonzero vector
114
1 O(Xn,T) C Dn : {~ : ] ~ - k I < ~ }. Since the local spectrum,
resolvent
is unbounded near any boundary point
f o r e v e r y n we c a n c h o o s e kn ~ P ( X n ' T ) d[k n, ~(Xn,T)]
of the local
such that <
I
and ~
IIXn(ln);l
÷
~ when
n ÷ =.
Then for each n, ~
(~n-T)Xn(kn) = x n and we have unit vectors ~
Yn
Xn(k n) ~ II Xn(k n) rl
such that (k-T)y n = (k-Xn)y n + (Xn-T)Yn Hence
k ~ Oa(T )
and the proof is complete.
÷ 0
as
n + ~.
[]
For operators with the SDP the hypotheses of Theorem A.3.3 are satisfied. In fact, T with the SDP has the SVEP and then Lemma 4.4 implies that o : x ÷ o(x,T) is localized.
A. 4.
Some open problems. The importance of the SVEP in spectral decompositions has been seen through-
out this work. The pathology lurking in operators which do not enjoy this property is brought out by certain "residual" parts of the spectrum which obstruct the construction of a satisfactory spectral theory. A thorough analysis of this case from the spectral decomposition standpoint was made by Vasilescu
[2,3,4] with a
follow-up by Bacalu [i].
Problem I. Does the general asymptotic spectral decomposition imply the SVEP ? The duality theory for decomposable operators has a most satisfactory conclusion expressed by Corollary 12.20. It seems that the class of quasidecomposable operators comes closest to the decomposable operators from the spectral decomposition standpoint. A hint that some kind of duality theory may exist for the even more general weakly and analytically decomposable operators is the fact that the
115
SVEP of the given T implies a(T*) = ~a(T*), (Corollary 1.4).
Problem 2.
What can be s ~ d abo~ the dual of a qua~ideeomposable operator?
Can any ~ind of d u a ~ t y t h e o r y be eo~t~ucted for the class of quasi-decomposable operators? The foregoing comment can be expressed for operators with the SDP giving rise to
Problem 3.
How much can be said a b o u t a d u a l i t y theory for operators with
the SDP? The axiomatic definition of the general spectral decomposition (Chapter II) was rewarding in properties and obviously there is much more to be said about it. One would expect that the decomposable operators defined in terms of the strongly specialized spectral maximal spaces form a proper subclass of the operators with the SDP.
Problem 4. Give an example of an operator with the SDP which ~ not decomposable. As it was mentioned, all properties of operators considered in Chapter II hold for operators with the 2-SDP. The question arises whether there is any other property of operators with the SDP which does not hold for operators with the 2-SDP. In other words, we formulate our
Problem 5.
Does the 2-SDP imply the SDP?
The spectral theory of weakly decomposable operators gained some remarkable properties by the additional decomposable spectrum hypothesis. The weakly decomposable operators de not seem to possess automatically decomposable spectra as the quasidecomposable operators do.
Problem 6. Is there a proper cla~s of operators b ~ e e n t~e weakly- and the qua~i-de~mposable oper~ors which possess the decomposable spectrum property? The concept of spectral capacity in its original, weak and strong formulations characterizes a specific spectral decomposition.
Problem 7. Is there a suitable generaliza~Lon of the spectral capacity adaptable to the general spectral decomposition (i. e. to operators wi~h the SDP} ?
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Appl.
AUTHOR INDEX
Albrecht, E.
69, 72, 116
Apostol, C.
6, 34, 72, 80, 104, 105, 116
Bacalu, I.
34, 35, i12, 114, 116
Bartle, R.G.
6, 14, 34, 35, 116
Berruyer, J.
112, 116
Bishop, E.
35, 105, 116
Colojoar~, I.
34, 35, 69, 71, 104, 117
Dunford, N.
I, 2, 5, 15, 34, 36, 49, 54, 55, 60, 105, 117
Erdelyi, I.
36, 49, 105, 112, 117
Finch, J.K.
3, 34, 117
Foguel, S.R.
39, 117
Foias, C.
34, 35, 49, 69, 71, 73, 104, 105, i17~ 118
Frunz~, S.
35, 104, 105, 118
Halmos, P.R.
1
Jafarian, A.A.
71, 72, 104, 118
Kariotis, C.A,
6, 14, 34, 35, 116
Lange, R.
35, 36, 71, 72, 104, 105, 117, 118
Maeda, F.-Y.
104, 119
Miller, F.R.
49, 117
Plafker, S.
104, 119
Radjabalipour, M.
69, lOS, 119
Schwartz, J.T.
II, 5, 15, 34, 49, 54, 55, 117
Schwartz, L.
104
Scroggs, J.E.
34, 119
Shulberg, G
35
Sine, R.C.
34, 119
Stampfli, J.G.
6, 31, 35, 119
Taylor, J.L.
34, 119
Vasilescu, F.-H.
34, 35, 36, 72, 104, 105, 114, 118, 119
SUBJECT INDEX Admissible contour
2,6
Algebra Analytic functions Banach
I, 5
i
Analytic spectral decomposition
50 ff.
Asymptotic spectral decomposition Banach algebra
S0 ff.
1
Bounded below (operator)
4
Canonical decomposition of a spectral operator embedding
49, 59
90
map between a Banach space and its quotient space 15, 92, 93, 94 Decomposable operator
69, 73 ff.
spectrum
64 ff.
Functional calculus
i, 2, 6, 7, IS, 21, 22, 36, 45, 46, 51, 57, 79, 80
General spectral decomposition 36 Localized set-valued mapping
ff.
112, 113
Operator analytically decomposable closed linear 3, 4, 97 coinduced
I, ii, 12, 15, 17, 32, 33, 82, 85, 92 ff.
decomposable
69, 73 ff.
2-decomposable dual
50 ff.
ff.
83, 84, 88, 89, 90, 91, 92, 93, 94
4, 44, 45, 84, 85, 86, 88, 89, 90, 91, 96, 114
generalized scalar 104 generalized spectral hyponormal
multiplication
106, 108
quasidecomposable quasinilpotent scalar
104
31 69 ff.
10, 18, 38, 58, 59
54, 59
spectral
49, 58, 59
strongly decomposable 2-strongly decomposable
80, 81, 82, 86, 96 95
weakly decomposable $5 ff. Partial isometry
Perturbation Quotient norm
49
54, 58 18
122
Resolution of the identity Set-spectrum
54, 59
78, 79, 109, ii0, iii
Similarity transformation
55
Single-valued extension property
3 ff.
Space absorbent
23, 24, 28, 29, 95, 106, 108
~- 35 v- 15, 16, 17, 35 quotient
ii, 18, 20
spectral maximal
26 ff.
Spectral capacity 60 ff. capacity (strong)
97 ff.
capacity (2-) 61, 86, 87 capacity (weak)
60 ff.
decomposition property distribution
inclusion property projection set
36 ff.
104 13, 39, 46
32, 46, 49, 76
32, 76
Spectrum approximate point continuous
4, 36, 39, 50, iii ff.
31
local S ff. point
40, 41, 57
Subspace analytically invariant hyperinvariant
14, 16 ff.
8, 16, 24, 25, 26, 27
Support of a function
106, 107, 108
of a spectral capacity
61, 68
