Spectral Theory of Banach Space Operators: Ck-classification, abstract Volterra operators, similarity, spectrality, local spectral analysis

183 Pages·1983·2.19 MB·English

by  Shmuel Kantorovitz (auth.)

Lecture Notes ni
Mathematics
Edited yb .A Dold and .B Eckmann
Subseries: Forschungsinstitut rQf Mathematik, HTE ZLirich
2101
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Ck-classification, abstract Volterra operators, similarity,
spectrality, local spectral analysis.
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Berlin Heidelberg New York Tokyo 1983
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Shmuel Kantorovitz
Department of Mathematics, ytisrevinU-na14raB
Ramat-Gan, Israel
AMS Subject Classifications (1980): 4?-02, 46 H 30, 4?A 60, 4? A 65,
47A55, 47 D05, 47 D10, 47 D40, 47 B47, 47A10
ISBN 3-540-126?3-2 Springer-Verlag Berlin Heidelberg New York oykoT
ISBN 0-387-126?3-2 Springer-Verlag New York Heidelberg Berlin Tokyo
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oT Ira, Bracha, Peninah, Pinchas, dna Ruth.
Tabl:e of Content
,O I ntroduct ion.
1. Operational calculus.
2. .selpmaxE
3. First reduction. 15
4. Second reduction. 20
5, Volterra elements. 25
6. ehT family S + .VE 38
7. Convolution operators in L p. 49
8, Some regular semi groups. 59
9. Simi larity. 65
10. Spectral analysis 73
11. ehT family S + ,VE S unbounded. 82
12. Similarity (continued). 99
13. Singular cn-operators. 123
1~. Local analysis. 146
Notes and references. 171
Bibl iography. 174
index. 177
.O I ntroduct ion.
We may view selfadjoint operators ni iH lbert space as the best understood
properly infinite dimensional abstract operators. If we desire to recuperate some
of their nice properties without the stringent selfadjointness hypothesis, we are
led to a "non-selfadjoint theory" such as Dunford's theory of spectral operators
;5 Part llI or Foias' theory of generalized spectral operators 9,4, to mention
only a few, and ti si not our purpose to describe here any one of these. Our basic
concept, as ni Foias' theory or distribution theory (as opposed to Dunford's), will
be the operational calculus (and not the resolution of the identity). However there
will be very little overlapping between 4 and the present exposition. Indeed, we
shall go ni an entirely different direction: starting ni an abstract setting, we
shall reduce the general situation ot a very concrete one, and we shall then concen-
trate on various problems within this latter framework or its abstract lifting.
These notes are based on lectures given ta various universities ni 1981, and
present ni a unified (and often simplified) way results scattered through our papers
since 1964.
We proceed now with a more specific description of the main features of this
exposi t ion.
Let K be a compact subset of the real line ,R and denote by HR(K) the
algebra of all complex functions which are "real analytic" ni a (real) neighborhood
of ,K with pointwise operations and the usual topology. A ci__sab algebra A(K) si
a topological algebra of complex functions defined ni a (real) neighborhood of ,K
with pointwise operations, such that HR(K) c A(K) topologically. fI A si a unital
complex Banach algebra, an A(K)-operational calculus for a E A si a continuous
representation "T A(K) ~- A carried by K (that ,si T(f) = 0 whenever f E A(K)
vanishes ni a neighborhood of K), such that T(t) = a (where t denotes the
function t * t on .)R When such an operational calculus exists, we say that
a si of class A(K). nI that case, the spectrum o(a) of a si necessarily
contained ni ,K and TIHR(K) coincides with the classical analytic operational
calculus for .a This means that we are concerned with the latter's extension
to wider basic algebras, for appropriate elements a EA. For example, fi A
si the Banach algebra B(H) of all bounded linear operators on the Hilbert space
H, and A(K) = C(K), the continuous functions ni a neighborhood of K (or on
K), then T E B(H) si of class C(K) fi and only fi T si similar to a self-
adjoint operator with spectrum ni .K Since any bounded (linear) operator with
spectrum ni K si of class HR(K), we are really interested ni intermediate
topological algebras A(K), contained topologically between HR(K) and C(K),
and ni the study of the corresponding classes A(K) of elements of class A(K)
(in A). A mild assumption on the topology of A(K) shows that ti suffices to
consider the intermediate algebras Cn(K) of all comF~lex functions defined ni a
real neighborhood of ,K and continuously differentiable there up to the order n
(with the usual topology). More specifically, we have the following "first reduction"
(§3): if A(K) si a homogeneous normable basic algebra, and fi a E A(K) , then
there exists n > 2 such that a E Cn(K).
A simplified (equivalent) way to consider cn-operational calculi si to take
the Banach algebra cna,13 as the domain of the representation ,T where 31,a
si a closed interval containing ,O without loss of generality. For n ) ,I an
example of an operator of class cno,B (which si not of class cn-Io,B) si
the operator T defined ni Ca,13 by T = M + n ,J where :M f(x) ~- xf(x), and
n n
X
"J f(x) .- f f(t)dt. Our "second reduction" (§4) consists Jn showing that T si
0 n
a kind of universal model for elements of class C n (in any Banach algebra A).
Let L a denote the "left multiplication by a" operator ni A L( x = ax, xEA).
a
Then a E A si of class 3i,~0nc fi and only fi there exists a continuous
linear map :U Ca,B ~- A, normalized by the condition IU = a nl n ,.v and inter-
twining L a and T . L U = UT When this si the case U si unique, and si
n a n"
related to the cna,B-operational calculus for a by a kind of Taylor formula.
This so-called "weak representation for a" on Ca,13 is a useful tool in produc-
ing more concrete characterization theorems for elements of class C n but its
main conceptual importance is in bringing down the abstraction level to the
in~.tial C-case (what are all the possible maps U: Ca,13-* A ?) and to the con-
crete operator T on Ca,13. In this exposition the weak representation
n
motivates our effort to extract the basic mechanism making T = M + nJ into an
n
operator of class C .n Observing that J satisfies the commutation relation
M,J = j2 with the operator M of class C, we first study general spectral
properties of s + ~v (~ E )C with ,s v E A satisfying the so-called Volterra
relation s,v = v .2 tI si shown that if o(s) c ,R then q(s + kv) = o(s)
for all integers k, and if s is of class C m then s + kv is of class c m+Ikl
This result is then generalized to "perturbations" s + cv with ¢ complex (§6).
Let S,V E B(X), where X si a Banach space. Suppose (.Standing Hypothesis):
S,V = V ,2 and V = V(1), where {V(¢); ~ E C +} si a regular semigroup of
operators on X (cf. Definition 6.0) whose boundary group {V(iq); ~r E R} (cf.
Theorem 6.1) satisfies llv(in) II ,K ~Inl with v < ,rT Set T = S + ,V~ and
--
let m,k denote non-negative integers. nI this general setting, we show that fi
S si of class C m with real spectrum, then o(T )= o(S) for all complex ,~
l¢eRI
and T si of class C m+k ni the strip < .k fi S si of class ,C then
T is of class C k if and only if IRe¢l < .k This latter result si shown to be
applicable ni particular to the operators T¢= M + J¢ ni CO,N or LP(o,N)
0( < N < % I < p < oo). Some needed tools from the theory of convolution opera-
tors are included ni ,7§ and are then appl led ni 8~ to prove the regularity of
certain semi groups , and ni particular~ of the Riemann-Liouville semigroup
;¢J{ ¢ E C +}, where
(J~f) (×) = r(~) -1 f× (x - t) r~-lf(t)dt,
0
acting in either CO,N or LP(o,N) (0 < N < ,oo 1 < p < =).
A by-product of the theory presented ni §6 si the following similarity
itS
result (valid under the Standing Hypothesis when I e II = 0(I)). For ,~ @ E C,
T and T ar@ similar if Re¢ = Re~, and only fi IRe¢ I = IRe~ ,I nI the concrete
case where T¢ -- M + Jc in either CO,N or LP(o,N) (0 < N ,oo< I < p < ,)o~
this can be strengthened to" T~ and T are similar if and only if Re¢ = Rem,
In ,3~ we discuss various refinements of this theorem and of the preceding
ck-classification result. Let S,V satisfy the Standing Hypothesis ni~( particular,
this forces V to be quasinilpotent). Suppose f EI H(O) r taht..( ,si f si analytic
at 0 and f(O) si real). If S is of class C m with real spectrum, ti si shown
that S + f(V) si of class C m+k for Ref'(O) I <k. nI case m = ,O S + f(V)
si of class C k if and only if IRef'(O) <_k, If f,g E H(O) r are such that
f(O) = g(O) and Ref'(O) = Reg'(O), then S +f(V) si similar.to S + g(V), and
l lstiel
conversely, if --0(I) and S + f(V) is similar to S + g(V), then
f(O) = g(O) and I Ref'(O) I = I Reg'(O) -I In particular, on either CO,N or
LP(o,N) (0< N < ,oo I < p< )oo and for f g E H(O) M +f(J) is of class C k
' ' r'
Ref'(o) l
if and only if __< k, and is similar to M + g(J) if and only if
f(O) = g(O) and Ref'(O) = Reg'(O).
Since M si trivially spectral of scalar type on LP(o,N), the latter result
(with g - )O implies ni particular that M + ¢J si spectral (of scalar type) if
Rel~ = O. A more general problem is to determine the values of ~ for which M + ¢J
(or its abstraction T¢ = S + )Vc si spectral (not necessarily of scalar type),
that ,si possesses a Jordan decompostion A + ,N with A and N commuting,
A = ;XE(dX) for some spectral measure ,E and N quasinilpotent. One of the
results proved ni §10 si the following general theorem: fi S,V satisfy the
Standing Hypothesis, and S si a scalar type spectral operator with real
spectrum, then T = S + cV si spectral fi and only if Re¢ = ,O This applies
ni particular to M + J~. acting ni LP(o,N), and illustrates the richness
of the class of cn-operators (n > )O as compared to the class of spectral
operators within the family {T~ = M + ~J; ¢ EC}. cn-operators correspond to
IRe~I
the strip < ,n while spectral operators correspond to the imaginary axis!
In §11, some of the results of Sect ,snoi 6 and 9 are extended to the case
when S si an unbounded operator with domain D(S) in .X Such a situation
x -¢ (x-t)
arises for example when S = M and V = J¢: f(x) + .re f(t)dt (c > 0),
0
both acting in LP(o,®). The Banach algebra methods used in the bounded case are
not available. However, when iS generates a strongly continuous group of opera-
tors S('), methods of the theory of semigroups step ni Tnstead, and we obtain
adequate versions of the main results. With a slightly modified "Standing
Hypothesis", ti si shown for example that S + CV si similar to S + ~V if
ll).(s only
Re~ = Re0J, and when II = 0(1), if IRe~ I = IReual. There si also a
natural way to define cn-classes among group generators. If S si of class C ,m
I¢eRI
then S + {V si of class C m+k for all ~ ni the strip < .k These results
apply ni particular to M + CJ acting ni LP(o,o~) I( < p < 0% ~ > 0). In §12,
similarity is further discussed ni various unbounded settings. We show for example
that M + {J and M + ~J, acting ni LP(o,~o) I( < p < )oo with their maximal
domain, are similar whenever ~0,:z are non-zero complex numbers with Re{ = .0CeR
As previously observed, cn-operators are rarely spectral. tI si proved
however ni §13 that "singular" cn-operators in reflexive Banach space are spectral.
Localized versions of the operational calculus, the spectral decomposition,
and the Jordan canonical form, are obtain ni §13-14 for arbitrary operators with
real spectrum.
.I Opera_tional Calculus
Let K be a compact subset of the complex plane ,C and denote by H(K)
the topological algebra of all complex functions f defined and analytic ni some
neighborhood f~£ of K (depending on .)f The operations are defined pointwise.
A net {f~} E H(K) converges to zero fi there exists a fixed neighborhood ~f of
K ni which all the functions f are analytic, and f~ ~- 0 uniformly on every
compact subset of .~ fI A si a Banach algebra with identity I and a E A
has spectrum o(a) contained ni ,K there exists a continuous representation
of H(K) on A (that ,si an algebra homomorphism of H(K) into A sending the
constant function I to the identity I E )A such that T(X) = a (X denotes
he er the function X ÷ X on C).
T si unique, and may be expressed by means of the Riesz-Dunford integral
I f f(~)(Xl-a)-IdX f( E H(K))
T(f) =T~T" r
where T = )K,f~f(T si a finite union of oriented closed Jordan curves such that
K si contained ni the union of the interiors (cf. 11 , Theorem 5.2.5). The map
T si called the analytic operation_al calculus for a. We often write f(a) instead
of T(f) when f EH(K),
For special elements a EA, the analytic operational calculus may be
extended to topological function algebras which properly contain H(K).
1.1 Definition. A basic algebra A(K) si a topological algebra of complex functions
defined ni a neighborhood of ,K with pointwise operations, which contains )K(lf
topologically. An A(K)-operational calculus for a E_A si a continuous representa-
tion ~ : A(K) +A carried by K (that si T(f) = 0 whenever f EA(K) si zero
ni a neighborhood of K), such that T(~ = a.
When such an o,c, exists, one says that a si of class A(K). The set of all
the elements of class A(K) ni A si denoted by A(K)A, or briefly, by A(K) ,
Let JK be the closed ideal of lla f E A(K) which vanish ni a neighborhood
of .K Since T si carried by ,K ti induces a representation on A of the
quotient algebra A(K)/J K of "K-germs of A(K)-functions", For the sake of