9
Pitman Research Notes in Mathematics Series
Shmuel Kantorovitz
Semigroups of operators and spectral theory
AAA
NNW LONGMAN
330
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V Petkov and R Lazarov
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J Chadam and H Rasmussen 282 Free boundary problems in fluid flow with applications
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K D Elworthy and N Ikeda 285 Dynamical systems
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B D Sleeman and R J Jarvis 290 K-theory for real C'-algebras H SchrSder 291 Recent developments in theoretical fluid
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G P Galdi and J Necas 292 Propagation of a curved shock and nonlinear ray theory P Prasad
293 Non-classical elastic solids
M Ciarletta and D lesan 294 Multigrid methods
J Bramble 295 Entropy and partial differential equations W A Day 296 Progress in partial differential equations: the Metz surveys 2 M Chipot 297 Nonstandard methods in the calculus of variations C Tuckey 298 Barrellcdness, Baire-likc- and (LF)-spaces
M Kunzinger 299 Nonlinear partial differential equations and their applications. College dc France Seminar. Volume XI H Brezis and J L Lions 300 Introduction to operator theory
T Yoshino
301 Generalized fractional calculus and applications
V Kiryakova 302 Nonlinear partial differential equations and their applications. College de France Seminar Volume XII
H Brezis and J L Lions 303 Numerical analysis 1993 D F Griffiths and G A Watson 304 Topics in abstract differential equations
S Zaidman 305 Complex analysis and its applications
C C Yang, G C Wen, K Y Li and Y M Chiang 306 Computational methods for fluid-structure interaction
J M Crolet and R Ohayon 307 Random geometrically graph directed self-similar multifractals L Olsen 308 Progress in theoretical and computational fluid mechanics G P Galdi, J MAlek and J Necas 309 Variational methods in Lorentzian geometry A Masiello 310 Stochastic analysis on infinite dimensional spaces
H Kunita and H-H Kuo 311 Representations of Lie groups and quantum groups
V Baldoni and M Picardelb 312 Common zeros of polynomials in several variables and higher dimensional quadrature Y Xu 313 Extending modules
N V Dung, D van Huynh, P F Smith and R Wisbauer
314 Progress in partial differential equations: the Metz surveys 3
M Chipot, J Saint Jean Paulin and I Shafr r 315 Refined large deviation limit theorems V Vinogradov
316 Topological vector spaces, algebras and related areas A Lau and I Tweddle 317 Integral methods in science and engineering
C Constanda 318 A method for computing unsteady flows in porous media R Ragbavan and E Ozkan
319 Asymptotic theories for plates and shells
R P Gilbert and K Iliackl 320 Nonlinear variational problems and partial differential equations
A Marino and M K V Murthy 321 Topics in abstract differential equations If
S Zaidman 322 Diffraction by wedges B Budaev 323 Free boundary problems: theory and applications
J I Diaz, M A Herrero, A Lilian and J L Vazquez
324 Recent developments in evolution equations
A C McBride and G F Roach 325 Elliptic and parabolic problems: Pant-e-Mousson 1994
C Bandle, J Bemelmans, M Chipot, J Saint Jean Paulin and I Shafrir
326 Calculus of variations, applications and computations: Pont-k-Mousson 1994
C Bandle, J Bemelmans, M Chipot, J Saint Jean Paulin and I Shafrir 327 Conjugate gradient type methods for ill-posed problems M Hanke 328 A survey of preconditioned iterative methods
A M Bruaset 329 A generalized Taylor's formula for functions of several variables and certain of its applications J-A Riestra 330 Semigroups of operators and spectral theory
S Kantorovitz
Shmuel Kantorovitz Bar-Ilan University, Israel
Semigroups of operators and spectral theory
mom A
LONGMAN
Copublished in the United States with John Wiley & Sons Inc., New York.
Longman Group Limited Longman House, Burnt Mill, Harlow Essex CM20 2JE, England and Associated companies throughout the world Copublished in the United States with John Wiley & Sons Inc., 605 Third Avenue, New York, NY 10158
© Longman Group Limited 1995 All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publishers, or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, W 1P 9HE
First published 1995
AMS Subject Classifications: (Main) 47D05, 47B40, 47A60 (Subsidiary) 47D10, 47A55, 47D40 ISSN 0269-3674
ISBN 0 582 27778 7
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Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn
INTRODUCTION
These "Lecture Notes" were written for a second year graduate course on "Topics in Spectral Theory". They present some aspects of the theory of semigroups of operators, mostly from the point of view of its application to spectral theory, and even more specifically, to the integral representation of operators or families of operators. There is no attempt therefore to cover either the subject of "semigroups" or the subject of "spectral theory" thoroughly. These theories and their many applications to Differential Equations, Stochastic Processes, Mathematical Physics, etc..., have been the subject of many excellent books, such as [D], [ Fat], [G],[HP],[Katl], [P], [RS], and others. We refer the interested reader to these (and other) texts. Nevertheless, in order to permit a smooth reading of this monographtype notes, and/or to make them convenient for a course or seminar, we have made them self contained by including a concise description of the basic facts on semigroups. The Hille-Yosida theory, concentrating on the concept of the generator (or infinitesimal generator) of the semigroup, is presented in Section A (Part I), culminating with the wellknown Hille-Yosida Theorem on the characterization of generators. A semigroup of operators is a function T(.) : [0, oo) - B(X),
(where B(X) denotes the Banach algebra of all bounded linear operators on a given Banach space X), such that T(0) is the identity I and
T(s)T(t) = T(s + t)
q, t > 0.
It is of class Ca if it is strongly right-continuous at 0. This implies strong continuity on [0, oo) and exponential growth. The generator A of T(.) is essentially the right derivative at 0 with maximal domain D(A). It is a closed densely defined operator, and for each x E D(A), the function u = T(.)x is the unique solution of the Cauchy problem on [0, oo): W = Au;
u(0) = X.
The generator is bounded if and only if T(.) is uniformly continuous, and is then an ordinary exponential T(t) = e1A. In general, the generator can be approximated pointwise on D(A) by bounded operators
A,\ :=.[)R(A) - I] (the so-called Hille-Yosida approximations of A), where R(\) := (XI - A)-' is the resolvent of A.
The Hille-Yosida theorem establishes that a closed densely defined operator A generates a C0-semigroup with exponential growth IIT(t)II _< Mean if and only if the resolvent exists for A > a and satisfies for all m = 1, 2, ...
IIR(A)mII < M(. - a)-m. In the context of the Cauchy problem for A, the Hille-Yosida theorem characterizes the Cauchy problems that have a unique solution with exponential growth. A characterization of generators that avoids resolvents uses the concept of dissipativity, introduced by Lumer and Phillips. This approach provides also an elegant perturbation theorem for generators, due to T. Kato. These matters are presented
in Section C of Part I. In Section D, we prove the Trotter-Kato theorem about the equivalence of "generator graph convergence", "Strong resolvent convergence", and "Semigroup strong uniform convergence on compacta". Section E deals in the unified way due to T. Kato with the "exponential formula" lim[nR(n)]n
T(t) = n
t
t
= lim(I - t4)-n n 7L
(in the strong operator topology), and with the "Trotter Product Formula" U(t) = 1im[S(t/n)T(t/n)]n n
(strongly), when S(.), T(.), U(.) are C0-semigroups generated respectively by A, B,
and A+ B. The important Hille-Phillips perturbation theorem (that supplies a condition on a closed operator B that is sufficient for the perturbation A + B of the generator A, to be also a generator) is proved in full detail in Section F. The background material is concluded with a. proof of the classical Stone theorem on unitary semigroups in Hilbert space (p. 38). As a prototype of integral representation theorems, Stone's theorem motivates our theory of the so-called "Semi-simplicity manifold" Z for a given group of operators (Section G). The linear manifold Z is defined by means of an adequate renorming, and it turns out to be maximal for the existence of a spectral integral representation of T(.) on it (Theorem 1.49). The renorming idea is also effective in creating the so-called Hille-Yosida Space (Section B), which is maximal with the property that the part of A in it generates a C0-semigroup (cf. [K51).
Section H touches upon the analyticity problem for semigroups, and gives a variant of a recent result of Liu with a (new) proof based on the exponential formula and normal families (cf [K81). Part I is concluded with the presentation of our recent "non-commutative Taylor formula" for semigroups as functions of their generator [K7].
Part II (pp. 65-114) describes some recent generalizations (published mostly after 1988) of the theory presented in Part I. Pre-semigroups (Section A), also called C-semigroups or regularized semigroups in the literature, have been introduced in germinal form in [DP], and their extensive study was started in [DPg]. Their main importance is in the solution of the Cauchy problem when A is not necessarily a generator (cf Theorem 2.5). The recent monograph [DL4] presents in detail many applications of this theory (and its extensions). In Sections B and C, we extend the concept of the semi-simplicity manifold to apply to operators that are not necessarily generators, provided they have real spectrum (Section B), or at least have a half-line in their resolvent set (Section C). The construction is based as before on the renorming method. The operational calculus on the semi-simplicity manifold is developed for reflexive Banach space. A recent extension to the non-reflexive case is contained in [DLK1]. The related concepts of the Laplace-Stieltjes space and the Integrated Laplace space (cf. [DLK]) for a family of closed operators is defined in Section D, also by the renorming method, with application to the spectral integral representation of semigroups of closed operators, and to the characterization of generators of n-times integrated semigroups (cf [Neu] for the concept and its application to the Cauchy problem). In Section E, we develop the Klein-Laundau theory of semigroups of unbounded symmetric operators, generalizing the classical Stone theorem (cf. [KL]). An analogous theory for cosine families of (unbounded) symmetric operators (cf. [KH31) is presented in Section F. These theories provide a natural approach to Nelson's Analytic Vectors theorem and to Nussbaum's Semianalytic Vectors theorem, respectively. The Klein-Laundau theory has seen many applications to Mathematical Physics, but this subject is beyond the scope of these lectures.
TABLE of CONTENT Introduction
PART I. GENERAL THEORY A. THE HILLE-YOSIDA THEORY
3
B. THE HILLE-YOSIDA SPACE
19
C. DISSIPATIVITY
23
D. THE TROTTER-KATO CONVERGENCE THEOREM
27
E. EXPONENTIAL FORMULAS
31
F. THE HILLE-PHILLIPS PERTURBATION THEOREM
34
G. GROUPS AND SEMI-SIMPLICITY MANIFOLD
40
H. ANALYTICITY
58
K. NON-COMMUTATIVE TAYLOR FORMULA
64
PART II. GENERALIZATIONS A. PRE-SEMIGROUPS
75
B. SEMI-SIMPLICITY MANIFOLD (real spectrum case)
83
C. SEMI-SIMPLICITY MANIFOLD (case R+ C p(-A))
96
D. LAPLACE-STIELTJES SPACE
104
E. SEMIGROUPS OF UNBOUNDED SYMMETRIC OPERATORS
117
F. LOCAL COSINE FAMILIES OF SYMMETRIC OPERATORS
123
Notes and References
130
Bibliography
132
PART I. GENERAL THEORY
A. THE HILLE-YOSIDA THEORY
THE GENERATOR. Let X be a Banach space, and let B(X) denote the Banach algebra of all bounded (linear) operators on X into X.
A function T(.) : [0, oo) -+ B(X) is a semigroup if
T(s)T(t) = T(s + t)
s, t > 0
and
T(0) = I, where I denotes the identity operator. The generator A of the semigroup T(.) is the operator
Ax = lim [T(t)x - x]/t t
o+
with "maximal domain"
D(A) = {x E X;
above
limit exists).
The above limit is the limit in X (with respect to the norm), and is in fact the strong derivative of T(.)x at 0. The "continuity at 0" (or C0) condition is
lim T(t)x = x for to+
all x E X.
This is continuity at zero in the strong operator topology on B(X) (in brief, strong continuity at 0). This will be a fixed hypothesis. 1.1. THEOREM. Let T(.) be a C0-semigroup. Then it is strongly continuous on [0, oo), and there exist constants M > 1 and a > 0 such that IIT(t)II < Meat
for allt>0. 3
PROOF. Let cn = sup{IIT(t)II;t E [0,1/n]} for n = 1,2,....
If cn = oo for all n, there exist t,, E [0,1/n] such that IIT(tn)II > n (for n = 1, 2, 3, ...). Then sup I IT(tn)II = 00, n
and so, by the Uniform Boundedness Theorem, there exists x such that sup IIT(tn)xII = 00n
However the sequence IIT(tn)xII converges to IIxii (by the Co condition, since to 0+), and is therefore bounded. This contradiction shows that there exists an n for
which cn < oo. Fix such an n, and let c = cn. Note that c > IIT(0)II = 11111 = 1. For any t > 0, the semigroup property gives
T(t) = T(1/n)n[t]T({t}/n)n,
where [t] denotes the entire part of t, and {t) its fractional part. Since 1/n and {t}/n are both in [0,1/n], we have IIT(1/n)II c and IIT({t}/n)II < c, so that IIT(t)II < (cn)(t]+i < (n)t+i = Me", where M = cn > 1 and a = nlogc > 0 (we used the fact that c > 1 ). Continuity at t > 0. (1) For h > 0, we have for all x E X IIT(t + h)x - T(t)xII = IIT(h)[T(t)x] - [T(t)x]II
0
as h -p 0, by the Co condition with the fixed vector T(t)x.
(2) For h < 0, write h = -k, with 0 < k < t. Then IIT(t + h)x - T(t)xil = IIT(t - k)(x - T(k)x)II < Me"(t-k) IIT(k)x - xii --+ 0 as h -* 0 by the C,, property. II II
1.2. THEOREM. Let A be the generator of the C,,-semigroup T(.). Then: 1. A is closed and densely defined. 2. For each t > 0, T(t)D(A) C D(A), and
AT(t)x = T(t)Ax = (d/dt)[T(t)x] for each x E D(A). 3. For each x E D(A), the function u = T(.)x is C' on [0, oo), and is the unique solution of the "Abstract Cauchy Problem" (ACP) on [0, co):
du/dt = Au; 4
u(0) = X.
PROOF. For each given x E X, the function T(.)x is continuous on [0, oo), by Theorem 1.1, and has therefore a Riemann integral over any finite interval [0, t]. Denote this integral by xt. Also let Ah = [T(h) - I]/h for h > 0. Then Ahxt = h-'[10 t T(s + h)xds h-1[( f t+h
J0
t
T(s)xds]
ft)T(s)xds
h
=
t+h
h_1
it
0
- h-' fo h T(s)xds
t
as h --40+, by continuity of T(.)x. Hence xt E D(A) and
Axt = T(t)x - x.
(*)
The C0-condition implies that xt/t(E D(A)) -+ x, and therefore D(A) is dense in X. If x E D(A), then for each t > 0, AhT(t)x = T(t)Ahx -> T(t)Ax as h ---> 0+. Hence T(t)x E D(A) and
AT(t)x = T(t)Ax. The left hand side in (**) is also equal to h-' [T(t + h)x - T(t)x], and so the right derivative of T(.)x exists, is equal to A[T(.)x] = T(.)(Ax), and is in particular continuous.
If0
Me°'(t-k)IIAkx
- AxhI + IIT(t - k)Ax -T(t)AxhI -+ 0
as k --+ 0+, since x E D(A) and T(.) is strongly continuous. Thus u = T(.)x is of class C' on [0, oo), and solves ACP. 5
Suppose v : [0, oo) -+ D(A) is differentiable. Then h-1 [T(t
+ h)v(t + h) - T(t)v(t)] _
T(t)Ahv(t) + T(t + h)[h-1(v(t + h) - v(t)) - v'(t)] + T(t + h)v'(t). The first term on the right has limit T(t)Av(t) when h -+ 0, since v(t) E D(A) and The T(t) E B(X). The second term has limit 0, since IIT(t + h)II < last term has limit T(t)v'(t), by strong continuity of T(.). Hence (d/dt)[T(t)v(t)] _ Mea(t+h).
T(t)[Av(t)+v'(t)]. Suppose now that v solves ACP with a given x E D(A), in some interval [0, r]. Fix s E (0,,r]. Then (by the fundamental theorem of calculus): T(s)x - v(s) = f 9(d/dt)[T(t)v(s - t)]dt 0
= f T(t)[Av(s - t) - v'(s - t)]dt = 0, 0
which proves the uniqueness. By the fundamental theorem of calculus and (*),
A fot T(s)xds = T(t)x - x =
T(s)Axds 0t
for t > 0 and x E D(A). Suppose xn E D(A) are such that xn -+ x and Ax,, -+ y in X. If V(.) : [0,r] --> B(X) is strongly continuous, then IIV(.)II is a bounded measurable function and for each x E X, II
f f V(t)xdtll < f IIV(t)IIdtIIxII 0
0
(see below).
Therefore, as n -+ oo, I I
f
t
T(s)Ax,,ds - f t T(s)ydsll < 0
0
f
t
I IT(s)I IdsllAxn - yl l < const.ll Axn - yll
Hence /t
Atx = limAtxn = limt-1 n n
0.
o
/o
t
T(s)Ax,, = t-1 fo T(s)yds -> y
ast -0+. This shows that x E D(A) and Ax = y, i.e., A is closed.
Back to the claim about V(.), the boundedness of IIV(.)Il follows immediately from the strong continuity of V(.) and the Uniform Boundedness Theorem. To 6
prove the measurability of it suffices to show that the set C = it E [0, r]; IIV(t)II > c} is Borel for each c > 0. If t E C, there exists x E X with norm 1 such that IIV(t)xII > c, and by continuity of IIV(.)xII, there is a neighborhood of tin [0,7-1 where I I V (. )x I i > c, and so I I V I > c there. Hence C is open, is lower semicontinuous ( which is so certainly Borel. We got actually that I
stronger than Borel measurability). 1111
TYPE AND SPECTRUM.
By Theorem 1.1, logilT(.)II is bounded above on finite intervals and clearly subadditive. We need the following general lemma on such functions. 1.3. LEMMA. Let p : [0, oo) - [-oo, oo) be subadditive (i.e., p(t+s) < p(t)+p(s) for all t, s in the domain of p) and bounded above in [0, 1]. Then
-oo < inf p(t) = lim p(t) < oo. t-oo t t>o t PROOF. If p(to) _ -oo for some to, then for all t > to, p(t) < p(to) + At - to) _ -oo, and the result is trivial. So we may assume that p is finite. Fix s > 0 and r > p(s)/s. For t > 0 arbitrary, let n be the unique positive integer such that
ns
p(t)/t = p(ns + (t - ns))/t < np(s)/t + p(t - ns)/t < rns/t + sup p/t. [0,9]
Since the hypothesis imply that p is bounded above on any interval [0, s], it follows that limsupt-. 9(-) < r, for any r > p(s)/s. Hence
< inf p(S) < lim inf P(t) limsupp(t) t s>o s t and the lemma follows.I I I
I
In particular, the type of T(.) is (fixed notation!)
w := inf >o
log
T(t)II I = tii-m
log IIT(t)II
t 7
For any non-negative a > w, we clearly have IIT(t)II < Meat
for all t > 0 (where the constant M > 1 depends on a). 1.4. THEOREM. The spectral radius of T(t) is &a.
PROOF. Since the claim is trivial for t = 0, fix t > 0, and let r(T(t)) denote the spectral radius of T(t). By the Beurling-Gelfand formula and Lemma 1.3, we have
r(T(t)) = lim I IT(t)n I I' /n = lime('/n) log IIT(nt)II n n = et
/nt) log IIT(nt)II
= e"'t.I I I I
UNIFORM CONTINUITY.
The next theorem shows that the stronger hypothesis of continuity at zero in the uniform operator topology (that is, in the norm topology of B(X)) yields to a rather uninteresting class of semigroups. 1.5. THEOREM. The semigroup T(.) is norm-continuous at 0 if its generator A belongs to B(X); in that case, T(t) = et' (defined as the usual power series, which converges in B(X)).
PROOF. 1. If A E B(X ), one verifies directly that e1A is a well-defined normcontinuous group with generator A. Since by Theorem 1.2 the generator determines the semigroup uniquely, and A is also the generator of T(.), we have T(t) = et t, so that, in particular, T(.) is norm-continuous. 2. Suppose conversely that T(.) is norm-continuous at 0 (hence everywhere on
[0, oo), by the argument in the proof of Theorem 1.1). We may then consider Riemann integrals of T(.), defined as the usual limits (in B(X)!). For h, t > 0, a calculation as at the beginning of the proof of Theorem 1.2 shows that
[T(t) - I] J T(s)ds = [T(h) - I) fo T (s)ds. 0
8
Since 11h h-1 f " T(s)ds - III -4 0 when h -p 0+ by norm-continuity of T(.), we can fix h so small that the above norm is less than 1, and therefore V := f h T(s)ds is invertible in B(X). Hence,
T(t) - I = f
t
T(s)ds.A,
0
where A :_ [T(h) - I]V-1 E B(X) (the change of order in the calculation is valid, since the values of T(.) commute). Dividing by t and letting t --+ 0, we get t-1[T(t) - I] -i A in B(X), by norm continuity of T(.). Hence A(E B(X)!) is the generator of T(.).IIII
CORE FOR THE GENERATOR
1.6. Let A be any closed operator with domain D(A) in X. The "graph-norm" on D(A) is the norm IXIA := Ilxll+IlAxll
induced on the graph of A by the norm on X2. D(A) is a Banach space under the graph-norm (because A is closed), and we shall use the notation [D(A)] for this Banach space. Any subspace Do dense in [D(A)] is called a "core" for A. Explicitely, a subspace Do of D(A) is a core for A if for any x E D(A), there exists a sequence {x } in Do such that x -+ x and Ax,, -+ Ax (i.e., A equals the closure (A/Do)- of its restriction to Do). Since it is often difficult to determine D(A), it is important (and sufficient in most case) to know a core for A. The following theorem gives a simple useful tool in this direction for the generator A of the semigroup T(.).
1.7. THEOREM. If Do is a subspace of D(A) dense in X and.T(.)- invariant, then it is a core for A.
PROOF. Note first that T(.) is a Co-semigroup in the Banach space [D(A)], since for all x E D(A), when t -+ 0+, IT(t)x - XI A = IIT(t)x - xli + IIT(t)(Ax) - (Ax)II -40. Therefore, for x E Do, Riemann integrals (over finite intervals) of T(.)x make sense in the graph-norm, and belong to Do , the closure of Do in [D(A)]. Let x E D(A). C Do such that x,, -+ x in X. By density of Do in X, there exists a sequence 9
The elements xt and (xn)t (see notation in proof of Theorem 1.2) are in D(A), and for each t > 0
I(xn)t - xtIA =
j T(s)(xt x)dsll
--H][T(t)xn - xn] - [T(t)x - x]II -+ 0
when n -+ oo. Since (xn)t E D. for each n, we have also xt E Do . Finally, by the C°-property of T(.) in [D(A)], t-'xt(E Do !) -+ x in the graph-norm, and so x E Do .III
A useful core for A is the space D°° = D°°(A) of all "C°°-vectors" for A, that is, the set of all x E X for which the function T(.)x is of class C°° on [0, oo). 1.8. THEOREM. 1. D°° = nw 1 D(An ). 2. D°° is dense in X and T(.)-invariant. 3. D°° is a core for A.
PROOF. 1. and 2. imply 3. by Theorem 1.7.
If X E D°°, T(.)x is differentiable at 0, i.e., x E D(A), and (d/dt)T(t)x = T(t)(Ax). Hence Ax E D°°, and so, in particular, x E D(A2). Inductively, x E D(An) and
T(.)A"x (*) for all n = 1,2,3.... Conversely, if x E D(An) for all n, then T(.)x is differentiable and [T(.)x]' = T(.)Ax (cf. Theorem 1.2), so that, inductively, we obtain that T(.)x is of class C°° and (*) is valid. This proves 1. and the T(.)- invariance of D°°.
To prove the density of D°°, we use an "approximate identity" 0 _< hn E C°° with support in (0,1/n) and integral (over R) equal to 1. Given X E X, define xn = f °° hn(t)T(t)xdt. Then xn --+ x in X. It remains to show that xn E D°° for all n. For k > 0,
Akxn = k-'
J
M
hn(t)[T.(t + k)x - T(t)x]dt
-+ - j
k-' [h,,(t - k) - hn(t)]T(t)xdt
h(t)T(t)xdwhen
k --+ 0+. Hence x,, E D(A) and Axn = - f°° hn(t)T(t)xdt. Repeating the argument, we obtain xn E D(A3) for all j and Ajxn = (-1)i f °° hnj3(t)T(t)xdt. The conclusion follows now from 1.1111
10
THE RESOLVENT The verification of the following elementary facts is left as an exercise.
1.9. PROPOSITION. Let A be a closed operator, with domain D(A). Then: 1. If A is bijective, its inverse with domain D(A-') equal to the range ran (A) of A, is closed. 2.
If B E B(X) and a, 0 E C, then aA + /3B, with domain X for a = 0 and
D(A) otherwise, is closed. 3.
If B E B(X), then AB, with its maximal domain, is closed. If B is non-
singular, then BA, with domain D(A), is also closed.
1.10. DEFINITION. The "resolvent set" p(A) of the closed operator A is the set of all complex A for which AI - A is bijective (i.e., one-to-one and onto X). Its complement is the "spectrum" Q(A) of A. The operator R(A) = R(A; A):= (Al - A)-' for A E p(A) is closed (see 1.9) and everywhere defined, and belongs therefore to B(X) by the closed graph theorem. It is called the "resolvent of A". It is useful to observe that A E p(A) if there exists an operator R(A) E B(X) with range in D(A) such that
(AI - A)R(A)x = x
(x E X)
and
R(A)(AI - A)x = x
(x E D(A)).
It is useful to write the above relations in the form R(A)A C AR(A) = AR(A) - I
(*)
(where all operators are with their maximal domain).
1.11. THEOREM. Let A be a closed operator. Then p(A) is open, R(.) is analytic on p(A) and satisfies the "resolvent equation"
R(A) - R.(p.) = (p - \)R(A)R(p). Also IIR(.\)II >
A,o A
PROOF. Let A E p(A), and set b = JJR(A)IL-1. The series
S(S) =
En>o(-1)nR(A)n+1((
- A)n 11
is norm-convergent in B(X) for I( - Al < 6, and so defines an element of B(X). For x E D(A),
S(()((I - A)x = S(()[(( - ),)I + (AI - A)]x = E(-1)nR(A)n+1(( - A)n+1x
+E(-1)"R(A)n(( - A)nx = X. Next, for any x E X, let x,n denote the m-th partial sum of the series S(C)x. Then xm E D(A) (because x,n E ran R(A) = D(A)), x,,, -+ S(()x, and by (*) Axm =
EO
-E0
AS(()x + ((- A)S(()x - x
=(S(()x-x. Since A is closed, it follows that S(()x E D(A) and ((I - A)S(()x = x, and we conclude that ( E p(A) and R(() = S(() for all ( such that I( - Al < S. Hence p(A) is open and R(.) is analytic on p(A). Also, since the disc of radius 6 around A is contained in p(A), we have d(A,v(A)) > 6 lIR(A)II-' Finally, for A, p E p(A) and x E X, (AI - A)[R(A) - R(p) - (p - A)R(A)R(p)]x
= x - [(A - p)I + (Al - A)]R(p)x - (p - A)R(p)x = x - (A - p)R(,u)x - x + (A - p)R(p)x = 0. Since AI - A is injective, the resolvent equation follows. IIII It will be convenient to consider the following general concept. 1.12. DEFINITION. A is a function R(.), defined on an. open set U C C, with values in B(X), and satisfying the resolvent equation in U.
1.13. THEOREM. If R(.) : U -+ B(X) is a pseudo-resolvent, then ker R(A) and ran R(A) are independent of A E U, and R(.) is the resolvent of some closed A with U C p(A) if ker R(A) = 0.
PROOF. Let A, p E p(A). If x E ker R(A), we have by the resolvent equation R(p)x = R(A)x + (A - p)R(p)R(A)x = 0, i.e., x E kerR(p), and so, by symmetry, ker R(A) = ker R(p). 12
If y E ranR(A), write y = R(A)x, and then y = R(p)lx + (p - A)R(,\)xj E ran R(p), so that ran R(A) = ran R(p) by symmetry. Suppose ker R(A) = 0 for some (hence for all) A E U. Then A := Al - R(A) ran R(A) -+ X is closed, and since aI -A = R(A)-' and R(A) E B(X), the operator AI - A is bijective. Thus A E p(A) and R(A) = (Al - A)-'. For any p E U, we have by the resolvent equation R(p) = R(A)[I + (A - p)R(p)] and therefore
(pI - A)R(p) = (p - A)R(p) + (-\I - A)R(A)[I + (A - p)R(p)] = I and similarly R(p)(pI - A) C I. Therefore p E p(A) and R(p; A) = R(p). Conversely, if R(A) = R(A; A) for all A E U (for some closed operator A), then R(.) is a pseudo-resolvent by Theorem 1.10, U C p(A), and ker R(A) = 0 trivially. I
I I I
Another characterization of resolvents among pseudo-resolvents uses the range of R(A).
1.14. THEOREM. Let R(.) : (w,oo) -- B(X) be a pseudo-resolvent such that for all A > w, IIR(A)II
(*) AMw, Then R(.) is the resolvent of a closed densely-defined operator if the range of R(A) is dense in X for some (hence for all)A.
PROOF. The necessity is trivial, since ran R(A; A) = D(A). Sufficiency. For x E ranR(A), write x = R(A)y, and then, for p > w, pR(p)x = pR(p)R(A)y = p " A[R(A)y - R(p)y] -} R(A)y = x
when p -+ oo, by the growth condition. Since ran R(A) is dense in X, it follows from the growth condition that pR(p)x --+ x for all x E X [indeed, let x,, E ran R(A) converge to x. Then IIpR(,u)x - xII <- IIpR(p)xn - xnll + IIpR(p) - III.IIx - xnll.
(M + 1)((x - xnll, The second term on the right is < [Mp/(p - w) + 1j(Ix and the first term --' 0 when p -+ oo (for each fixed n).Therefore
limsupllpR(p)x -xll < (M+1)Ilx - xnII. u-00
13
Letting n -> oo, the conclusion follows].
Suppose X E ker R(A) for some A > w. Then X E ker R(µ) for all p > w, but then x = limµR(µ)x = 0, i.e., ker R(A) = 0, and so R(A) = R(A; A) with A closed (by Theorem 1.13) and D(A) = ran R(A) dense, by hypothesis.I
LAPLACE TRANSFORM.
We show next that the resolvent of A is the Laplace transform of T(.). 1.15. THEOREM. 1. o(A) C (A E C; RA < w}. 2. For RA > w and x E X,
je_)tT(t)xdt.
R(A)x =
3. For c > w,t > 0 and x E D(A),
T(t)x = lim 1 f c+tr e-'tR(A; A)xdA, r- oo 2iri
-ir
where the limit is a strong limit in X.
PROOF. For any a > w, IIT(t)II = O(eat), and therefore the Laplace integral L(A)x defined in 2. converges absolutely for to > a, and defines an operator L(A) E B(X) satisfying IIL(A)II <-
M R,\-a'
If x c D(A),
L(A)(AI - A)x =
J
F{Ae-atT(t)x
- e-at[T(t)x]'}dt
0
j [eT(t)x]'dt = x. On the other hand, for any x E X and h > 0,
AhL(A)x = h-' 14
J0
r e-At[T(t + h)x - T(t)x]dt
rh
= h-1(eah - 1)L(A)x -
J
e`a`T(t)xds
0
->h-.o+ AL(A)x - X.
Since A is closed, it follows that L(A)X C D(A) and (AI - A)L(.A)x = x for all x E X, and we conclude that L(A) = R(A; A) for all A in the half-plane WA > a. Since a > w was arbitrary, Statements 1. and 2. are proved. To obtain 3., we observe that T(.)x is of class C1 on (0, oo) (by Theorem 1.2), and we may therefore apply the (vector version of the) classical Complex Inversion Theorem for the Laplace transform (cf. Theorem 7.3 in [W]). II
I
I
The Laplace integral representation of R(A; A) implies the growth condition IIR(A;A)II A
M
(*)
a
for all A > a (where a > w is fixed). Consider now any closed densely defined operator A with (a, oo) C p(A), which satisfies (*) for all A > Ao (for some A0 > a).
For short, call such an operator an abstract potential. 1.16. LEMMA. Let A be an abstract potential, and consider the bounded operators AA := AAR(A) = A[AR(A) - I]
ford > a. Then as A -+ oo, 1. Aax -pAxforall x E D(A); 2. AR(A) -> I strongly (equivalently, AR(A) -> 0 strongly).
PROOF. For x E D(A) and A > Ao, I IAR(A)xlI = IIR(A)AxII < AMa II AxII -, 0. Since
- a +1
IIAR(A)II = IIAR(A) - III <
0(1)
when A -> oo, and since D(A) is dense in X, it follows that AR(A)x
0
for all x E X. This is equivalent to 2. Next, for x E D(A), Aax = AR(A)(Ax) -' Ax by 2.1111 15
Note that the notation AA in the present context should not be confused with the notation Ah used in previous sections.
When A is the generator of a semigroup T(.) satisfying IIT(t)II < Meat
the
growth property (*) can be strengthened as follows:
For any finite set ofAk>a,
k=1,...,m,
I Ink(Ak - a)R(Ak; A)II < M.
In particular (with all Ak equal A), I IR(A; A)m I I <
M
a)-
(a
for all A > a and m = 1, 2, 3,....
Indeed, for all x E X, IIilk(Ak - a)R(Ak; A)xll = II
f
0
00
J
00
-a)e-Alt,-...-AmtmT(tj +...+-t..)xdt1...dt,,,II
nk(Ak
0
<Mf 0
...
f
00
nk(Ak
-
a)e-(a1-a)tdtll
xll
0
= Milk j(Ak
- a)edtllxll = MIIxII
We prove now that (***) characterizes generators among all abstract potentials.
THE HILLS-YOSIDA THEOREM.
1.17. THEOREM. An operator A is the generator of a Ca-semigroup T(.) (satisfying IIT(t)II < Meat for all t > 0) if (1) it is closed and densely defined; and (2) (a, oo) C p(A) and (***) is valid.
PROOF. We already saw the necessity of (1) and (2). Let then A satisfy (1) and (2). In particular, it is an abstract potential, and so Lemma 1.16 is satisfied. Define Ta(t) = etAX.
16
We have for A > 2a (so that aaa < 2a):
I
(t)II 5 e-atE,
tnn! n
nA2n I IR(A)
II < Me-"tEn n!(A
a)n
- Me t
< Me2at
Also for A - oo, limsup IITA(t)II 5 Meat.
(1)
CLAIM: Ta(t) converge in the strong operator topology (as A -+ oo), uniformly for t in bounded intervals.
For x E D(A) and A, p > 2a, IIT,,(t)x - TA(t)xII = II f (d/ds)[TA(t - s)T,,(s)x]dsII t 0
t
TA(t - s)T,(s)(A, - AA)xdsIj < M2eaattI JAax - AAxII -; 0
= II 0
when A, p --+ oo, by Lemma 1.16, uniformly for tin bounded intervals. Since IIT,\(.)II is uniformly bounded in bounded intervals (by (1)), it follows from the density of D(A) that {TA(t)x} is Cauchy (as A --+ oo) for all x E X, uniformly for t in bounded intervals. Define therefore
T(t)x = slim TA(t)x -00
for x E X (limit in X-norm). By (1), IIT(t)I I < Meat for all t > 0. The semigroup property of T(.) follows from that of T,,(.). The uniform convergence on bounded intervals implies the continuity of T(.)x on [0, oo), for each x E X. Let A' denote the generator of T(.). We have
f TA(s)A,xds. t
Tax - x =
0
For x E D(A), Lemma 1.16 implies (by letting A --+ oo)
T(t)x - x =
f
t
T(s)Axds.
0
17
Dividing by t > 0 and letting t -> 0+, we conclude that x E D(A') and Ax = Ax. Thus, for A > a, AI - A and AI - A' are both one-to-one and onto X, and coincide on D(AI - A) = D(A). Therefore D(A) = D(A'), and the proof is complete. I
I I
I
For contraction semigroups (i.e., IIT(.)II < 1), the Hille-Yosida characterization is especially simple (case M = 1, a = 0). 1.18. COROLLARY. An operator A is the generator of a C0- contraction semigroup if it is closed, densely defined, and AR(A; A) (exist and) are contractions for
all y>0. We call the bounded operators A), the Hille-Yosida approximations of A. From Lemma 1.16 and the proof of the Hille-Yosida theorem, AAx -+ Ax for all x E D(A) and etA, --+ T(t) strongly, uniformly on bounded t-intervals (as A -+ oo).
18
B. THE HILLE-YOSIDA SPACE
The inequalities (**) following the proof of Lemma 1.16 can be used to construct,
for an arbitrary (unbounded) operator A with (a, oo) C p(A), a maximal Banach subspace Z of X such that AZ, the "part of A in Z", generates a Co-semigroup in Z.
1.19. DEFINITION. A Banach subspace Y of X is a linear manifold Y C X which is a Banach space for a norm II.IIy >- 11.11.
If A is any operator on X with domain D(A), and W is a linear manifold in X, the "part of A in W", denoted Any, is the restriction of A to its maximal domain as an operator in W: D(Aw) = {x E D(A); x, Ax E W}.
1.20. DEFINITION. Let A be an arbitrary operator with (a, oo) C p(A) for some real a. Denote IIxIIY
=SUP IIllk(Ak - a)R(Ak; A)xf I,
where the supremum is taken over all finite subsets {A1,...,A,,,} of (a,oo) (the product over the empty set is defined as x). Set
Y= {xEX;IIXIIY
PROOF. Clearly, Y is a linear manifold in X, and its norm majorizes 11.11. In particular, if {xn} is Cauchy in Y, it is also Cauchy in X; let x be its X-limit, and let K = sup,, IIxnIIY. For any finite set {Ak}1
n
so that IIxIIY < K < oo, i.e., x E Y. 19
Given e > 0, there exists no such that Ilxn - xplly < e whenever n,p > no. Therefore for any finite set { A k } as before,
IIl1k(Ak - a)R(Ak; A)(x,, - xp)II <- IIx, - xplly < e
if n,p > no. Letting p --* oo, and then taking the supremum over all finite subsets {Ak}, we obtain Iix,, - xlly < e for all n > no. Thus Y is a Banach subspace of X. If U E B(X) commutes with A, it commutes also with R(A; A) for each A > a. Therefore, for x E Y, IIUxIIY = SUP IIUIIk(Ak - a)R(Ak; A)xII <- IIUIIB(X)-IIXIIY < 00, Ak>a
and so Y is U-invariant and IIUIIB(Y) <- IIUIIB(x)- IIII
1.22. DEFINITION. The Hille-Yosida space Z for A is the closure of D(Ay) in Y. The terminology is motivated by the following
1.23. THEOREM. Let A be an unbounded operator with (a, oo) C p(A) for some real a. Let Z be the Hille-Yosida space for A. Then Az, the part of A in Z, generates a Co-semigroup T(.) in Z that satisfies IIT(t)IIB(z) <- eat. Moreover, Z is maximal in the following sense: if W = (W, I I I w) is a Banach subspace of X such that Aw generates a Co-semigroup in W with the above exponential growth, then W is a Banach subspace of Z. I
PROOF. Since R(A; A) commutes with A for each A > a, the linear manifold Y is R(A; A)-invariant and IIR(A; A)IYIIB(y) S IIR(,; A)IIB(x) < oo by Lemma 1.21. The identities
(Al-A)R(A;A)y=y R(A; A)(AI - A)y = y
(yEY) (y E D(Ay))
show then that R(A; A)IY = R(A; Ay) for all A > a. If y E D(Ay), then y, Ay E Y, so that R(A; A)y(E D(A)) E Y and AR(A; A)y = AR(A; A)y - y E Y, that is, R(,1; A)D(Ay) C D(Ay). Since R(A; A)IY E B(Y), it follows that Z is R(A; A)- invariant, and IIR(A; A)IZIIB(z) <_ IIR(A; A)IYIIB(y) < 00.
(1)
The above identities show then that
R(A; Az) =R(A;A)IZ 20
(A E (a,oo).
(2)
In particuBlar, Az is closed. For all y E Y and all finite sets {Ak} C (a, oo), I IHk(Ak - a)R(Ak; AY)ylI Y =I Ink(Ak - a)R(Ak; A)yIIY
= sup IIH;(µ; - a)R(rii; A)nk(Ak - a)R(Ak; A)yII µj >a < sup I IHr(vr - a)R(vr; A)ylI = I lylly. s,, >a
Therefore Ilnk(Ak - a)R(Ak; AY)IIB(Y) C 1
(3)
for any finite set {Ak} C (a,oo), and the same is true with Y replaced by Z. In particular, taking singleton subsets of (a, oo), we have IIR(\;AY)IIB(Y) <
A
a
(A > a).
Therefore, for all z E D(Ay), IIAR(A;A)z-zIIY = IIR(A;A)AzIIY
A-a
as A --p oo, since Az E Y. Thus AR(A; A)z --a z in Y for all z E D(Ay). For z E Z arbitrary, if e > 0 is given, there exists zo E D(Ay) such that I Iz - zo I I y < e, since D(Ay) is dense in Z by Definition 1.22. Then I1AR(A; Az)z - zIIY <- II(AR(A; Az) - I)(z - zoIIY + IIAR(A; A)zo - zoIIY
'
a + 1)e + IIAR(A; A)zo - zoIIY --+ 2e
as A -+ oo. Hence as A -+ oo,
AR(A; Az)z(E D(Az)) - z in the 11.1ly-norm, and so D(Az) is dense in Z. In conclusion, Az is a densely defined closed operator in Z, which satisfies in Z the condition (**) with M = 1. By the Hille-Yosida theorem (1.17), it follows that Az generates in Z a Co-semigroup T(.) satisfying IIT(t)Ilz < eat for all t > 0.
On the other hand, if W is as in the statement of the theorem, then for any w E W, it follows from (**) (with M = 1) in B(W) that IIwIIY <- sup Ilnk(Ak - a)R(Ak;A)I1B(w)I1w11w c Ilwliw. Ak>a 21
Therefore W is a Banach subspace of Y. In particular D(Aw) C D(Ay). Since Aw generates a Co-semigroup in W,
W = W-closure(D(Aw)) C W-closure(D(Ay)) C Y-closure(D(Ay)) := Z, and we conclude that W is a Banach subspace of Z. 1111
Note in particular the case a = 0: if (0, oo) C p(A), the Hille-Yosida space for A is a maximal Banach subspace such that the part of A in it generates a Cosemigroup of contractions in it.
22
C. DISSIPATIVITY
A useful characterization of generators that avoids resolvents depends on the
numerical range.
1.24. DEFINITION. Let A be an arbitrary (usually unbounded) operator on the
Banach space X. Its numerical range is the set v(A) = {x*Ax; x E D(A), x* E x*, I Ixl I= Ilx* I I= x*x =1}.
Given x E D(A) with Ilxll = 1, define x* on Cx by x*(Ax) = A for A E C. Then 1Ix*II = x*x = 1, and x* extends to a unit vector in X* by the Hahn-Banach theorem. This shows that v(A) is not empty.
1.25. DEFINITION. The operator A is dissipative if tv(A) < 0.
1.26. THEOREM. If A generates a C0-semigroup of contractions, then it is closed, densely defined, dissipative, and AI - A is surjective for all A > 0. Conversely, if A is closed, densely defined, dissipative, and AI - A is surjective for all A > A0 (for some A0 > 0), then A generates a C0-semigroup of contractions.
PROOF. Necessity. Suppose A generates the C0-semigroup of contractions T(.), and let x E X and x* E X* be unit vectors such that x*x = 1. For h > 0, 51, and therefore lx*T(h)xl <
Rx*[h-1(T(h)x - x)] = h-1[R(x*T(h)x) - 1] < 0.
For x E D(A), letting h -p 0, we get Rx*Ax < 0, so that A is dissipative. It is closed and densely defined by Theorem 1.2. By Corollary 1.18, (0,oo) C p(A), so that, in particular, AI - A is surjective for all A > 0.
Sufficiency. For all unit vectors x E D(A),x* E X* such that x*x = 1, and for all A > 0, we have II(AI - A)x112 > lx*(AI - A)x12 = IA - x*Ax12
_ A2 - 2AR(x*Ax) + I
(1)
x*Ax12 > A2
23
because t(x*Ax) < 0. Therefore ,\I - A is one-to-one (for all A > 0) and onto X (by hypothesis) for all A > At,. Thus A E p(A), and I IAR(A; A)II < 1 for all A > A,. This proves that A generates a C,-contraction semigroup, by Corollary 1.18 (it is clear from the proof of Theorem 1.17 that for the sufficiency part, the growth condition on the resolvents is needed for large A only).I i I
I
Note that in (1), we used only some unit vector x* with the needed properties. This allows the following weakening of the hypothesis in the sufficiency part of the theorem. 1.27. THEOREM. Let A be a closable densely defined operator such that A0I-A
has dense range for some \, > 0. Suppose that for each x E D(A), there exists a unit vector x* E X* such that x*x = IIxii and 2(x*Ax) < 0. Then the closure Aof A generates a C,- semigroup of contractions.
PROOF. As in (1), II(AI - A)xil > AIIxII for all A > 0 and x E D(A). Let x E D(A-), and let then x,, E D(A) be such that x,, -+ x and Ax,, -4 A-x. Letting n -+ oo in the inequalities I J (AI - A)xn I I > Al Ixn 11, we obtain
II(AI - A-)xIi ? allxll
(A > 0; x E D(A )).
(2)
In particular, Al - A- is one-to-one for all \ > 0. We claim that A0I - A- is onto X. Indeed, for any y E X, there exist by hypothesis xn E D(A) such that (\0I - A)xn --+ y. Then by (2), Ilxn - xmli
A0' il(A0I - A)(xn - xm)II -+ 0,
so xn -+ x, and necessarily x E D(A-) and (,\,I - A-)x = y. Thus \o E p(A-) and IIR(Ao;A-)II < 1/,\,. By Theorem 1.11, d(A0,a(A-)) > .;A=_)> ,10. Therefore (0,2A0) C p(A-). Inductively, one obtains that R(XI
(0, 2n A.) C p(A-) for all n, and so (0, oo) C p(A-) and AR(A; A-) are contractions for all A > 0, by (2). The result follows now from Corollary 1.18.IIII
The criterion of Theorem 1.26 is effective for certain types of perturbations of generators. 1.28. DEFINITION. Let A, B be (usually unbounded) operators. One says that B is A-bounded if D(A) C D(B) and there exist a, b > 0 such that IIBxII <- allAxil +biixli
(x E D(A)).
The infimum of all a as above is called the A-bound of B. For example, any B E B(X) is A-bounded with A-bound equal to 0. 24
1.29. LEMMA. If A is closed and B is A-bounded with A-bounpl'c < 1, then A + B (with domain D(A)) is closed. PROOF. Note first that the A-boundedness of B means that B (recall that [D(A)] is normed by the graph-norm for A).
B([btA)],X)
Let xn E D(A), xn - x, and (A + B)xn - y. Then IAxn - AxmII = I I (A + B)xn - (A + B)xm - B(xn - xm) I <- iI(A +
I
B)xn - (A + B)xmit + aIIAxn - AxmII + bll xn - xm,l1.
Hence
(1 - a)IIAxn - AxmIl < II(A+B)xn
- (A + B)xmll + bllxn - xmll
0.
Since a < 1, (Axn) is Cauchy, and since A is closed, it follows that x E D(A)(= D(A + B)) and Axn -, Ax. Since B is continuous on [D(A)], also Bx,, -> Bx, and therefore (A + B)x,, -> (A + B)x.III I
We now have the following perturbation theorem.
1.30. THEOREM. Let A generate a C,,-semigroup of contractions, and let B be dissipative and A-bounded with A-bound a < 1. Then A + B generates a Cosemigroup of contractions.
PROOF. Since A generates a C0-contraction semigroup, it is dissipative (1.26). Also B is dissipative with D(A) C D(B). Therefore A + B is dissipative, because
for all x E D(A + B) = D(A) and x* E X* with Ilxll = IIx*II = x*x = 1, R[x*(A+ B)x] _ R[x*Ax] + R[x*Bx] < 0. By Lemma 1.29, A + B is closed, and it is densely defined (D(A) is dense by Theorem 1.2). By Theorem 1.26, it remains to show that \I - (A + B) is surjective for all A > \o (for some A0 > 0). We have
for allA>0 ran (AI - A - B) = [(Al - A) - B]R(A; A)X = [I - BR(A; A)]X.
(*)
However, for all x E X, IIBR(A; A)xII < aIIAR(A; A)xII + bIIR(A; A)xII
< aIIAR(A; A)II.IIxII + allxll +
IIAR(A;
(2a + a)IIxII 25
since AR(A; A) are contractions, by Corollary 1.18.
In case a< 1/2, 2a + -X < 1 for A> Ao, and therefore I IBR(A; A)II < 1, hence I - BR(A; A) is invertible in B(X), and so AI - (A + B) is surjective for A > ao, by (*)-
Consider now the general case a < 1. Let ti = i 2 a . For any s E [0,1] and x E D(A),
(1 - asIIBxII = IIBxII - asIIBxII < (aflAxII + bIIxII) - asIIBxII = a(IIAxjl - sIIBxII) + bIIxII <- aII(A + sB)xII + bIIxII
Therefore, fort E [0, t,], IItBxII <
1
2asIIBxII
2II(A+sB)xlI+2IIxjI,
so that tB has an (A + sB)-bound < z . By the preceding case, if A + sB generates a C0-contraction semigroup for some s E [0, 11, so does A+sB+tB for all t E [0, t1 ]. Starting with s = 0 (for which A + sB = A is a generator by hypothesis), we get that A + tB is a generator for all t E [0, tl ], hence A + tl B + tB is a generator for all such t, etc... . Let n be the first integer such that ntl > 1. A last application of the above argument with s = (n - 1)t1 < 1 gives that A + tB is the generator of a C0-contraction semigroup for all t E [0, ntl], so that, in particular, A + B is a generator. I I I I
26
D. THE TROTTER-KATO CONVERGENCE THEOREM
We consider next a one-parameter family T,(.) of C,,-semigroups (s E [0, c)), with generators A,; let us write T(.) = To(.) and A = A0. BASIC HYPOTHESIS: there exist M > 0 and a > 0 such that j1T,(t)II < Meat
(t > 0,s E [0,c)).
(1)
This implies (cf. (*) preceding 1.16) 11R(A! AS )11 <
M
- A-a
(2)
for all A> a and sE [0,c). We define the following convergence properties:
1.31. DEFINITION. 1. Generators graph convergence on a core D,, for A: for each x E Do, there exists x, E D(A,) such that [x A,x,] -> [x, Ax] in X2 when s -+ 0.
2. Resolvents strong convergence: for each A > a, R(A; A,) -+ R(A; A) in the strong operator topology (when s -, 0).
3. Semigroups strong uniform convergence on compacta: for each x E X, T,(t)x -, T(t)x in X, uniformly on compact t-intervals (when s --+ 0). 1.32. THEOREM (Trotter-Kato). The convergence properties in Definition 1.31 are equivalent.
PROOF. In the following, the numbers 1,2,3 refer to the three types of convergence formulated in Definition 1.31.
Denoting y, = (AI - A,)x, and y = (,\I - A)x for all x E Do, we see that 1. is equivalent to 1t .
[ye, R(A; A,)y,] - [y, R(A; A)y]
for all y E (Al - A)DO (for suitable y and for all A > a). By (2), Property 1'. is equivalent to 1".
[y3, R(A; A, )y] - [y, R(A; A)y] 27
for all y E (AI - A)DO (for suitable y, and for all A > a). 1. implies 2. Since we are assuming 1"., we have in particular
R(t;A.)y - R(A;A)y
(3)
for all y E (AI - A)Do. Since Do is a core for A, for each x E D(A), there exist xn E Do, n = 1, 2,..., such that [xn, Ax,,] -r [x, Ax], hence [xn, (AI - A)xn] [x, (AI - A)x]. In particular, X = (Al - A)D(A) = [(AI - A)D0]-. Thus (3) is valid for y in a dense subspace of X, hence for all y E X, by (2). This is Property 2. 2. implies 1. Given x E Do and \ > a,, choose x, = R(A;A,)(AI - A)x. Correspondingly, we have y = (AI - A)x and y, = (AI - A)x = y, so that by 2.
[y9, R(A; A, )y] _ [y, R(A; A, )y] -* [y, R(A; A)y]
Thus 1". (and so 1.) is satisfied. For the implication 2. => 3., we need the following LEMMA. Let A, B generate the C.-semigroups T(.) and V(.) respectively, both
O(eae)forsome a>0. Then for Re A > a, t > 0 and x E X, I
R(A; B)[V(t) - T(t)]R(A; A)x =
J0
V(t - s)[R(A; B) - R(A; A)]T(s)xds.
PROOF (of Lemma). For .\, t as above and 0< s < t,
d V(t - s)R(A; B)T(s)R(A; A)x = V(t - s)(-B)R(A; B)T(s)R(A; A)x +V(t - s)R(A; B)T(s)AR(A; A)x
= V(t - s)[T(s)R(A; A)x - AR(A; B)T(s)R(.1; A)x] +V(t - s)R(A; B)T(s)[AR(A; A)x - x]
= V(t - s)[R(A; A) - R(A; B)]T(s)x. Integrating with respect to s from 0 to t, we obtain the formula in the lemma.
2. => 3. For all y E X, write [T,(t) - T(t)]R(A; A)y = R(A; A,)[T,(t) - T(t)]y 28
+T,(t)[R(A; A) - R(A; A8)]y
+[R(A; A,) - R(A; A)]T(t)y = I + 11 + III
(s, t > 0,.A > a).
We estimate I for y = R(A; A)x, using the lemma. Thus for 0 < t < r, 11111:5
J0 r Me°lr-", II[R(A; A,) - R(A; A)]T(u)xI du.
The right hand side converges to zero when s -+ 0, by 2.; therefore I -+ 0 uniformly on every compact t-interval. Since R(A; A)X = D(A) is dense in X, and the
operators in I are uniformly bounded with respect to s (and with respect to t in compacta (cf. (1) and (2)), it follows that I converges to zero for all y E X, uniformly on compact t-intervals (when s -+ 0).
By (1)and 2.,for 0
so that II -+ 0 uniformly on compact t-intervals. For y E D(A), we may write
T(t)y = y + JOt T(r)Aydr, and therefore, for 0 < t < r, 1111111:5
+
J0 r
II[R(A; A8) - R(A; A)]yII
II[R(A; A8) - R(A; A)]T(r)AyII dr.
By 2., the first term on the right converges to 0 ass -+ 0. The integrand on the right converges pointwise to 0, and is bounded by -m ear II AyI1 in the interval [0, r].
By Lebesgue's Dominated Convergence theorem, the integral converges to 0, and therefore III -+ 0 uniformly on compact t-intervals. Since D(A) is dense in X and the operators appearing in III are uniformly bounded (with respect to s E [0, c) and t in compacta; cf. (1) and (2)), the above conclusion is valid for all y E X.
We thus obtained that T,(t)x -+ T(t)x (when s -+ 0), uniformly on compact t-intervals, for all x E R(A; A)X = D(A) (A > a fixed); since D(A) is dense in X and 11T,(t) - T(t)I1 < 2Meat are uniformly bounded for all s > 0 and for all t in compacta, Property 3. follows. 3. => 2. For A > a and x E X, R(A; A,)x = f °O e`T,(t)xdt. When s -+ 0, the integrand converges pointwise to its value at s = 0 (by 3.) and is norm-dominated 29
Me-(,\-a)tIIxII E L1(0,oo). By Lebesgue's Dominated Convergence theorem, the integral converges to its value at s = 0, which is precisely R(A; A)x. II I
by
1.33. COROLLARY. (Same "basic hypothesis" (1)). Suppose that for each x in a core D,, for A, there exists So E (0, c) such that x E D(A,) for all s E (0, so ) and A,x -r Ax when s -+ 0+. Then T,(t)x -+ T(t)x for all x E X, uniformly on compact t-intervals (when s --+ 0+). PROOF. Property 1. is satisfied with x, = x for s E (0, so). Therefore Property 3. holds, by Theorem 1.32. 1111
1.34. COROLLARY. Let T(.) be a Co-semigroup, and denote A, = s-1 [T(s)-I]. Then
T(t) = lim0+ etA.
strongly and uniformly on compact t-intervals.
PROOF. Let r > w, and choose a = re'. Let T,(t) = etA, fort > 0, s E (0,1), and To(.) = T(.). For M = Mr, we have IIT(t)II < Me't < Meat and IIT9(t)II =
Me-t/ 3E(t/s)ne'28'/n! = Mexp[ts-1(e*` - 1)] <
Meat
for all t > 0 and s E (0,1). Thus the "basic hypothesis" (1) is satisfied (with c = 1), and since A,x -+ Ax for all x E D(A) (by definition of A), our corollary follows from Corollary
30
1.33.1111
E. EXPONENTIAL FORMULAS
A useful application of Corollary 1.33 is the following
1.35. THEOREM. Let A generate a C0-contraction semigroup T(.), and let F be any contraction-valued function on [0, oo) such that F(0) = I and the right derivative at zero of F(.)x coincides with Ax for all x in a core D,, for A. Then T(.) is the strong limit of F(t/n)n (as n -+ oo), uniformly on compact t-intervals. PROOF. We need the following
LEMMA. Let C be a contraction on X. Then contraction semigroup, and en(C-I)x II
- CnxII
OC-I> is
a uniformly continuous
n1/2Ii(C - I)xII
for all x E X and n = 1, 2,.... PROOF (of Lemma). iiet(C-1)11 =
e-tlle'Cll < e-tetllCll < 1.
n(C-I)x - CnxII < e-nllEk>0(nk/k!)[Ckx - Cnx]II = e nII Eo
n(nk/k!)(Ck-nx - x)II l
ie
< e-nEk>o(nk/k!)IICIk-nlx - xli. Since C is a contraction,
IICmx - xi[ =
II(C'n
+... + I)(C - I)xii < miiCx - xli,
and therefore the last expression is
< [Ek>oe-n(nk/k!)Ik -
xll.
The term in square brackets is the expectation of iK - nl, where K is a Poisson random variable with parameter n. By Schwarz' inequality, since K has expectation and variance n, we have E(IK - nI) < [E(K - n)2]1/2 := a(K) = n1/2.1111 31
Back to the proof of the theorem, consider the bounded operators
A = (t/n)-1 [F(t/n) - I] for t fixed. By hypothesis, Anx -+ Ax for all x E Do. For all unit vectors x E X and x* E X* such that x*x = 1, we have 3t(x*Anx) = (n/t)[3t(x*F(t/n)x) -1] < 0 because Ix*F(.)xl < IIx*II.IIF(.)II.IIxII < 1. Thus An is dissipative, and so, by 1.26, e°An are contraction semigroups satisfying trivially the "basic hypothesis" (with M = 1 and a = 0). By Corollary 1.33, e'An
(1)
-' T(s)
strongly and uniformly on compact s-intervals, when n --i oo. However, by the lemma with C = F(t/n), IletAnx
- F(tln)nxll <_ n1"2II[F(tln) - I]xII = nt1/2IIAnxII
--> 0
(2)
as n -* oo, for all x E Do. Since D,, is dense in X (as a core) and IIetAn -F(t/n)n11:5
2 for all n because both operators are contractions, it follows that (2) is valid for all x E X. By (1) and (2), the theorem follows.IIII
As a first application of Theorem 1.35, we obtain the following "exponential formula".
1.36. THEOREM. Let T(.) be a C0-semigroup, with generator A. Then for all
t>0,
T(t)=lim['R(t;A)]n in the strong operator topology (as n -> oo). PROOF. We consider first the case when IIT(t)II < eat for all t _> 0 (with some
a>0). Let S(t) := e-atT(t). This is a C0-semigroup of contractions, with generator A - aI. We set
F(s) :_
(s-1
- a)R(s-1; A) =
(s-1
-
a)R(s-1
- a; A - al)
(s E (0,1/a)),
and F(0) = I. By Corollary 1.18, F(.) is contraction-valued. By Lemma 1.16,
s-1[F(s)x - x] = A1l,,x - as-1R(s-1; A)x --> (A - aI)x 32
for all x E D(A) = D(A - aI), as s -> 0+. We may then apply Theorem 1.35; thus, in the strong operator topology and uniformly in compact t-intervals,
S(t) = lim F(t/n)n, n and therefore
T(t) = eatS(t) = lim[(1 - n)-1F(t/n)]n
=lim[
n
n n- at
n-at R(n;A)]n t
t
n
t
t
=lim[nR(n;A)]n.
General case. Fix a > w. We have IIT(t)II < Meat for all t > 0 (with a suitable M depending only on a). Renorm X by IxI := sups>0 e-atllT(t)xll. Then IIxii < IxI < Mllxll, i.e., the norms are equivalent, and
IT(t)xl = supe-68IIT(s +t)xii 5 eat supe-auIIT(u)xII = eatlxl. u>0
s>0
We may then apply the preceding case to the semigroup T(.) on the space (X, yielding the result with respect to the 1.1-norm, hence also with respect to the given norm (since the two norms are equivalent).I I I I
Another application of Theorem 1.35 is the following
1.37. THEOREM (Trotter's Product Formula). Let A, B, C generate contraction Ca-semigroups S(.),T(.), U(.) respectively, and suppose that C = A + B on a core Do for C. Then for all t > 0,
U(t) = lim[S(t/n)T(t/n)]n n strongly.
PROOF. Take F(t) = S(t)T(t) in Theorem 1.35. For x E D,, and t > 0,
t-' [F(t)x - x] = S(t)t-' [T(t)x -- x] + t-' [T(t)x - x] - Bx + Ax = Cx, and the conclusion follows from Theorem 1.35.1111
33
F. THE HILLE-PHILLIPS PERTURBATION THEOREM
The next theorem is concerned with perturbations A + B of the generator A by closed operators B in a suitable class. FIXED HYPOTHESIS H1. Let A generate a Ca-semigroup T(.) and let B be a closed operator such that T(t)X C D(B) for all t > 0. By the Closed Graph Theorem, BT(t) E B(X) for all t > 0. Hence, if t, t+h > 0, 0 when h -+ 0, showing that BT(.) is strongly right continuous on (0, oo). It follows that I IBT(.)I I is bounded on compact intervals, and we deduce that BT(.) is strongly continuous for t > 0, as in the proof of Theorem 1.1. Thus I IBT(.)I I is measurable on t > 0 (cf. proof of Theorem 1.2). Also, for any t > e > 0,
BT(t + h)x - BT(t)x = [BT(t)][T(h)x-x]
logIIBT(t)II < logIIBT(e)II t t
-
+(1_ t
log IIT(t-e)II t-e
so that lim sup
log I I BT(t) I < w I
t
t-.oo
Therefore, for any a > w, there exists a constant Ma > 0 such that IIBT(t)II <
-
Ma eat
The non-negative measurable function [IBT(.)II has an integral over [0,1] (that could be infinite). We assume FIXED HYPOTHESIS H2. fo IIBT(t)Ildt < oo.
Note that any B E B(X) satisfies H1 and H2 trivially.
1.38. THEOREM. Let A, B satisfy the hypothesis H1 and H2. Then A + B (with domain D(A)) generates a Ca-semigroup.
More details about the structure of the semigroup generated by the perturbation A + B will be obtained in Lemma 3. and in (**) below.
LEMMA 1. (Hypothesis Hi, H2). For R. > w, D(A)(= R(A; A)X) C D(B) and
BR(A; A)x = f "o e_ tBT(t)xdt
o
34
for all x E X, where the Laplace integral above converges absolutely.
PROOF. The Riemann sums S,, for the integral fb (with 0 < a < b < oo) approximating the Laplace integral of T(.)x are in D(B) by H1, converge to f °, and by linearity of B and continuity of BT(.)x in [a, b), BS,, converge to fb e-a'BT(t)xdt. Since B is closed, it follows that fb E D(B) and B fb = fb e-XtBT(t)xdt. Since 00
f IIe-atBT(t)Ildt < e-xA j IIBT(t)Ildt + f
0°
e-RAtIIBT(t)Ildt < oo
by H2 and the remarks following H1, the Laplace integral of BT(.)x converges absolutely in X to an element L(.\)x E X. By Theorem 1.15, fb e"T(t)xdt(E D(B)) -+ R(A; A)x (when a -+ 0 and b -+ oo), and we observed before that B f°(...) -+ L(A)x. Since B is closed, it follows that R(A; A)x E D(B) and BR(A;A)x = L(A)x.II]I
LEMMA 2. (Hypothesis H1, H2). There exists r > w such that q
jePIBT(t)Idt < 1.
For WA > r,
R(A; A + B) = R(a; A)En>o [BR(A; A)]',
(*)
where the series converges in B(X). In particular, A + B is closed (and densely defined, since D(A + B) = D(A), by Lemma 1.).
PROOF. Fix c > w. By H1, for all A > c, e-atIIBT(t)II < e-`jJBT(t)II E L1(0, oo), and e-at I IBT(t)II --+ O as A -+ oo. By Dominated Convergence, it follows
that f °° e-at II BT(t)I jdt -+ 0 when A -+ oo. We may then choose r > c such that
q < 1. Then, by Lemma 1., IIBR(A; A)II < q < 1 for RA > r, and therefore the right hand side of (*) converges in B(X) to an operator K(A) with range in D(A) = D(A + B). We have for RA > r [AI - (A + B)]K(A) = (Al - A)K(A) - BK(A) = En>o[BR(A;
A]n
- En>o [BR(A; A)]n+1 = I.
On the other hand, for x E D(A), K(A)[AI - (A + B)]x = R(A; A){I + En>1 [BR(A; A)]') [(Al - A) - B]x 35
= x + R(A; A) { -Bx + En>o [BR(A; A)] n [BR(.; A)] [(AI - A) - B]x }
= x + R(A; A){-Bx + En>o [BR(7; A)]'Bx - En>1 [BR(A; A)]?Bx} = x.I I II
The functions f := and g := IIBT(.)II are both in the class of locally integrable functions on (0, oo) (cf. remarks following H1, together with H2). The "Laplace" convolution (
u * v)(t) :u(t - s)v(s)ds j
a function in
u, v E Liodefines
and therefore the repeated convolutions
9(n)=g*...*g n times are in Lia. We consider also the h(n) = f *g(n) , and we set The next lemma will justify the following inductive definition: So(.) = T(.);
Sn(t)x =
J
t
T(t - s)BSn_1(s)xds
0> = f.
(n = 1,2,...).
0
LEMMA 3. For all n = 0, 1, 2,..., Sn(.) are well-defined bounded operators such that, for all x E X, a. S,,(.)x : [0, oo) --+ D(B) is continuous, and for r, q as in Lemma 2. and all
t>0,
IISn(t)II C h(n)(t) < Mertgn;
b. BSn(.)x : (0, oo) -- X is continuous and II BSn(t)I I < g(n+l)(t\ l= /0
PROOF. We prove the lemma by induction on n. The case n observations following HI).
is trivial (see
Assume then the lemma's claims are valid for some n. By b. for n, Sn+1(.) is well-defined and
fb
Sn+l(t)x = lim
J
T(t - s)BS,,(s)xds
(*)
a
as a -* 0+ and b -4 t-. The Riemann sums for each integral over [a, b] are in D(B) by H1, and when B is applied to them, the new sums converge to fa BT(t - s)BS,,(s)xds, because the latter's integrand is continuous on [a, b] by b. (for n). Since B is closed, it follows that each integral in (*) belongs to D(B),
and B fa(...) = fa B(...). The same type of argument with a --> 0+ and b -r t(using again the closeness of B) shows that Sn+1(t)x E D(B) and
BSn+1(t)x = 1 IBT(t - s)BSn(s)xds. 0
36
Since T(.), BT(.) and BSn(.) are continuous on (0, oo) and majorized by L1ocfunctions (using the induction hypothesis), the integral representations for
and BSn+1(.)x imply their continuity on (0, oo) for each x E X. The function Sn+l(.) is even norm-continuous at 0, since IISn+1(t)II <
.l
t * g(n+1)(_ h(n+1)) < Mert 1 9(n+1)(s)ds --+ 0 0
when t -+ 0+, because the integrand is in Lioc. Note that Sn(0) = 0 for all n > 1. The estimates in a. and b. for n + 1 are trivial consequences of the induction hypothesis. For example, h(n+1) _ f * g(n+l) = If *g (n)] * g = h(n) *,q,
so that
h(n+0/t) l
M9n J t er(t-9)g(s)ds = Mgnert rt e-r'g(s)ds < Mertgn+l.IIII 0
0
The exponential growth of Sn(.) (as in a.) shows that its Laplace transform converges absolutely for RA > r. We show next
LEMMA 4. For t\>randxEX, R(A; A)[BR(A; A)]n x = J
e-_'tSn(t)xdt
n = 0, 1, 2,....
0
PROOF. The case n = 0 is verified by Theorem 1.15. Assuming the lemma for n, we have by Theorem 1.15,
R(A; A)[BR(A; A)]n+1 x = f e-atT(t)[BR(\; A)]n+1 xdt 0
00
= J0
e-''tT(t)B.{R(A; A)[BR(A; A)]nx}dt
_ j etT(t)B J eS(s)xdsdt, 0
where we used the induction hypothesis. Since B is closed, the argument we used before (cf. Lemmas 1. and 3.) allows us to move B inside the inner integral, and 37
then do the same with the bounded operator T(t). We then obtain the repeated integral
J J 0
e-a(t+')T(t)BS,,,(s)xdsdt =
0
J e-A' f T(u - )BS,,(s)xd,du o
0
00
= J
e-Ausn+t(u)xdu,
0
where the interchange of integration order is justified by absolute convergence.I I II
We state now a simple characterization of generators as our final
LEMMA 5. An operator A on the Banach space X is the generator of a C,,semigroup if and only if it is closed, densely defined, and for all A > a and x E X, R(A; A)x =
J "o e-atS(t)xdt, 0
where S(.) : [0,00) -r B(X) is strongly continuous and IIS(t)II In that case, S(.) is the semigroup generated by A.
Me°t.
PROOF. Necessity follows from Theorems 1.1, 1.2, and 1.15.
Sufficiency. The series expansion for the resolvent obtained in the proof of Theorem 1.11 shows that (_1)nR(A; A)(n) = n!R(A; A)n+1
The exponential growth of allows us (using a Dominated Convergence argument) to differentiate the Laplace transform of S(.)x under the integral sign, yielding inductively to the formula [R(A;
A)xj(n) = f(_t)neS(t)xdt.
Therefore
IIR(A; A)nxll =
11 [R(.\; A)x]cn-1' II <_ 1 1 (n-1)! (n-1)! 1o
< (MI - i)!
j
et1dt =
tn-le-atlIS(t)IIdt.IIxII
a)n Ixll'
(A
and the Hille-Yosida Theorem applies. If T(.) is the C°-semigroup generated by A, then R(.\; A)x is the Laplace transform of T(.)x, and we conclude that T(.) = S(.) by the Uniqueness Theorem for Laplace transforms.I I I
38
I
PROOF OF THEOREM 1.38. For 0 < t <,r, we have by Lemma 3. IIS,,(t)II < Merrgn, with q < 1. Therefore the series
S(t) := E.>OSn(t)
(**)
converges in B(X)-norm, uniformly on every interval [0, T]. By Lemma 3., it follows that S(.) is strongly continuous on [0, oo), and IIS(t)II < QMi e rt Since Sn(0) = 0 for n _> 1 (cf. Lemma 3.), we have S(0) = I. The exponential growth of I IS(.)I I shows that the Laplace transform of S(.)x converges absolutely for *A > r, and a routine application of the Lebesgue Dominated Convergence Theorem shows that
or 0
e-atS(t)xdt = En>o
f 0
"o e-'\tSn(t)xdt
= EnR(A; A)[BR(A; A)]nx = R(A; A + B)x
by Lemma 4. and Lemma 2., and we conclude now from Lemma 5. that A + B generates the Ca-semigroup S(.) (given explicitely by (**) and Lemma 3.).IIII
39
G. GROUPS AND SEMI-SIMPLICITY MANIFOLD
If the C,,-semigroup T(.) can be extended to R = (-oo, oc) with preservation of the identity T(s)T(t) = T(s + t) (s, t E R), it will be called a Q,-group of operators. When this is the case, the semigroup S(t) := T(-t), t > 0, is also of class CO, since for 0 < t < b,
S(t)x - x = T(-b)[T(b - t)x - T(b)x] -+t-o+ 0 for all x E X (cf. Theorem 1.1). The generator A' of S(.) is -A, because for x E D(A),
t-' [S(t)x - x] = -T(-b)(-t)-' [T(b - t)x - T(b)x] -+t-.o+ -Ax by Theorem 1.2, so that -A C A', and therefore A' _ -A by symmetry. Let w,w' be the types of T(.) and S(.) respectively. Since IIS(t)II = IIT(t)'II IIT(t)II-1, we have
w' = tlim t-' log IIS(t)II ? -slim t-' log IIT(t)II = -w. _00 C-0
Note also that w'
lim t-' log IIT(t)II. t-.-oo
By Theorem 1.15, since a(-A) _ -Q(A), the spectrum of A is necessarily contained in the closed strip
S:-w'
(*)
Also since R(A; -A) = -R(-a; A), the necessary condition IIR(A; -A)" II < M/(Re\ - w')n
for IRA > w' becomes IIR(,\; A)nII < M/[(-w') - Rea]n for RU < -w'. Thus the growth condition on the resolvent outside the strip (*) is I IR(,\; A)n 11 < M/da, 40
where da = d(A, S) denotes the distance of A to the strip S. The sufficiency of (**) (together with the usual conditions that A be closed and densely defined) for A to generate a C,,-group is easily obtained from the HilleYosida Theorem. Indeed, applying the theorem separately in the half-planes ?A > w and W.1 > w', we obtain two C,,-semigroups T(.) and S(.) with the respective generators A and -A. Using Theorem 1.2, we have for all x E D(A):
[T(s)S(s)x]' = T(s)AS(s)x - T(s)S(s)Ax = 0
(s > 0).
Therefore T(s)S(s) = T(0)S(0) = I, and similarly S(s)T(s) = I, i.e., S(s) _ T(s)-1 for all s > 0. Thus A generates the Ce-group T(.). Formally 1.39. THEOREM. The operator A generates a Co-group of operators if and only if it is closed, densely defined, has spectrum in a strip S : -w' < RA < w, and I IR(A; A)" I < M/d(A; S)" I
for all real A
Consider now a bounded group:
IIT(t)II < M
(t E R).
In this case, both types vanish, so that o(A) C iR. When X is a Hilbert space (with inner product (., .)), the case of a bounded group reduces to that of a unitary group: 1.40. THEOREM. Let T(.) be a bounded strongly continuous group of operators acting on the Hilbert space X. Then there exists a non-singular bounded (positive) operator Q such that QT(.)Q-1 is a group of unitary operators.
PROOF. Let B(R) denote the Banach algebra of all bounded complex functions
on R, with the supremum norm, and let LIM be the "generalized Banach limit" functional on it (cf. [DS-I]). Define
(x, y)T = LIM (T(t)x,T(t)y)
(x, y E X),
and let IIXIIT = (x, x)T Z. Then 11-11T 5 41
so that X is a Hilbert space under the equivalent inner product (.,.)T, and there exists therefore a (strictly) positive operator P such that (x, y)T = (Px, y) for all x, y E X. Let Q be the positive square root of P. For s E R and x, y E X fixed, write x = Qu and y = Qv. Then (QT(s)Q-1
x,QT(s)Q-1
y) = (PT(s)u,T(s)v) =
LIMt(T(t)T(s)u,T(t)T(s)v) = LIMt(T(t+s)u,T(t+s)v) = LIM(T(t)u,T(t)v) = (u, v)T = (Pu, v) = (Qu, Qv) = (x, y).lI I
I
We make a short elementary digression about symmetric operators on a Hilbert space X. If A is a densely defined operator on X, we set D(A*) to be the set of all y E X for which there exists a (necessarily unique) z E X such that (Ax, y) = (x, z) for all x E D(A); we then set A*y = z for all y E D(A*). Thus (Ax, y) = (x, A* y)
(x E D(A), Y E D(A* )).
The Hilbert adjoint A* of A is closed [indeed, if ytt E D(A*) converge to y and A*y,, -> z, then (Ax, y) = lim(Ax, yn) = lim(x, A* y E D(A*) and A*y = z].
A densely defined operator A is symmetric if (Ax, y) = (x, Ay)
(x, y E D(A)).
This is of course equivalent to A C A*. In particular, any symmetric operator is closable, and its closure A- (which satisfies necessarily A- C A*) is a closed symmetric operator. We say that A is selfadjoint if A = A*, and essentially selfadjoint if A- = A*. Note that we always have A* = (A-)* and A- = A**, so that A is essentially selfadjoint if and only if A* = A**.
As usual, [D(A*)] denotes the Hilbert space D(A*) with the (Hilbert) graphnorm, induced by the graph-inner- product (x,y)A* := (x, y) +(A*x,A*y)
(x,y E D(A*)).
Since A* : [D(A*)] -> X is continuous, the subspaces
D+ := ker (iI - A* ); 42
D_ := ker (-iI - A*)
are closed subspaces of [D(A*)]; Ax = ix on D+, Ay = -iy on D_, and the subspaces are clearly orthogonal:
(x, y)A = (x, y) + (ix, -iy) = 0
for all x E D+ and y E D_. Again, for x,y as above and z E D(A-) for A symmetric ( so that A- C A*), we have
(z,x)A. = (z,x) +(A*z,A*x) = (z,x) +(A-z,ix) = (z,x) + (z,A*ix) = (z,x) - (z,x) = 0, and similarly for y. Therefore the (Hilbert) direct sum of D+ and D_ is contained in the orthocomplement Y of D(A-) in [D(A*)]. On the other hand, if u E Y, then for all y E D(A), 0 = (y,u)A _ (y, u) + (Ay, A* u),
so that (Ay, A*u) _ (y, -u); hence A*u E D(A*), and A*(A*u) = -u. Therefore (iI - A*)(-iI - A*)u = 0 (with commuting factors!), showing that
(-iI - A*)u E D+;
(ii - A*)u E
for all u E Y. Since u = (1/2i)[(iI - A* )u - (-iI - A* )u], we see that Y is contained in the direct sum of D+ and D_, hence equals it (by the preceding observation). Thus
[D(A*)] = D(A-) ®D+ ®D-.
(1)
The subspaces D+ and D- are called the deficiency subspaces of A, and their Hilbert dimensions are the deficiency indices n+ and n_ respectively. Since it is generally true, for any densely defined operator T, that ker T* equals the orthocomplement of ran T, we have
D+ = [ran (-iI - A)]';
D- = [ran (iI - A)]l,
where the orthocomplement is taken in the space X. In particular, we read from the decomposition (1) that the symmetric operator A is essentially selfadjoint if and only if both ii - A and -ii - A have dense range in X. After this digression about symmetric operators, consider again a C,,-group of
unitary operators T(.), and write its generator as A = iH. Then for all x, y E D(H) = D(A), (Hx, y) = (-i) tlim (t-1 [T(t) - I]x, y) = (-i) lim(x, t-1 [T(-t) - I]y)
= -i(x, -Ay) = (x, Hy), 43
i.e., H is a (densely defined closed) symmetric operator. Also o(H) = -io(A) C R (see above!), so that, in particular, iI - H and -iI - H are onto. By the preceding remarks, this proves that H is selfadjoint. Let e'tH be the operator associated with H by means of the operational calculus for selfadjoint operators: e''BE(ds)x,
eitHx :=
(2)
JE
where E(.) is the resolution of the identity for H. A trivial application of dominated convergence shows that estH is a C,-group with generator iH = A. Therefore T(t) = e`tH. Since any C,-semigroup of unitary operators extends in an obvious manner to a C,-group, we have
1.41. THEOREM (Stone). Let T(.) be a C,-(semi)group of unitary operators in Hilbert space. Then there exists a selfadjoint operator H such that T(t) = etiH for
alltER. Combining this with Theorem 1.40, we obtain
1.42. COROLLARY. Let T(.) be a bounded C. -group of operators in Hilbert space. Then there exists a (unique, not necessarily selfadjoint) spectral measure E(.) on the Borel algebra of R such that
T(t) =
fet3E(ds)
(t E R),
where the integral exists in the strong operator topology.
Using the terminology of [DS-III], the generator of T(.) equals is, where S is
a scalar-type spectral operator with real spectrum, and T(t) = e'ts (using the operational calculus for S). Also QSQ-' is selfadjoint, with Q as in Theorem 1.40.
REMARK. A neat way to deal with the Hilbert adjoint is to consider its graph G(A") C X2, where the cartesian product X2 is a Hilbert space under the inner product ([x,y], [x',y']) :_ (x,x') + (y,y') The operator
j : [x, y] - [y, -x] is a unitary operator on X2, and a simple calculation shows that G(A*) = {JG(A)}-L. 44
One reads from this formula that A* is closed (its graph is closed as an orthocomplement!), and that A* = (A-)*. Also, since J is unitary with J2 = I,
G(A**) = [JG(A*)]1 = JG(A*)1
= J[JG(A)]11 = J[JG(A)]- = J2G(A)- = G(A)- = G(A-). Therefore A** = A-. Stone's theorem can be used to prove Bochner's theorem about Fourier-Stieltjes' transforms.
1.43. THEOREM (Bochner). A continuous function f : R --+ C is the FourierStieltjes transform of a finite positive Borel measure p (i.e., f(t) = f e`tsp(ds)) if and only if it is positive definite, that is, Ej,k f (tj - tk)CjCk
-0
for all finite sequences t j E R and c j E C. PROOF. Let X° be the complex vector space of all complex functions on R with finitely many non-zero values. Let
(4),0 = Et,sE]Rf(t - sM00(s)
,
where f is a given positive definite function, and 0, ?p E X°. This is a pseudoinner product on X° ( i.e., we may have (0, ¢) = 0 for non-zero (k). The set K = (0 E X°; (¢, 0) = 0} is a closed subspace of X°. The factor space X1 = X°/K is a pre-Hilbert space with the inner product (¢ + K, Vi + K) = (¢, 0) (which is well- defined, i.e., independent of the choice of the cosets representatives). Let X be the completion of the pre-Hilbert space X1. Define U° on X0 by [U,001(t) = 0(t - r), r E R. Then (U, 0, Ur 0) = (0, tai). Since Ur maps K into itself, it induces an operator U,1. of X1 into itself, well-defined by U, (0 + K) = U, ¢ + K, and U,'. is a unitary operator of X1 onto itself. It extends uniquely to a unitary operator U, of X onto itself. The family (U,; r E R} is a C°-group on X (where the C°-property follows from the assumed continuity of f). By Stone's theorem, (U,x,x) = f], e`t'(E(ds)x,x) for all x E X, where (E(.)x,x) = IjE(.)xjj2 is a finite + K with 4)°(t) = 1 for t = 0 and positive Borel measure. In particular, for x° vanishing otherwise, we have (UrxoI xo) - (U, x0, x0) -
/ V'O, Y'O) -
f(r),
so that f is indeed the Fourier-Stieltjes transform of the finite positive Borel measure (E(.)x°, xe). 45
The converse is an easy calculation.I II I
It is also possible to deduce Stone's theorem from Bochner's. If T(.) is a C,,-group of unitary operators on the Hilbert space X, then f := (T(.)x, x) is continuous and positive definite (for each x E X):
Ej,kJ (tj - tk)cjck = E(T(tk)*T(tj)x, x)cjck
E(T(tj)x,T(tk)x)cjck = IIEjcjT(tj)xII2 > 0. Therefore, by Bochner's theorem, there exists a family {µ(.; x); x E X} of finite positive Borel measures on R such that (T(t)x, x) =
J
e`tay(ds;x)
for all t E R and x E X. Define
x, y)
x + iky)
(1/4)Eo
(x, y E X).
These are (finite) complex Borel measures, and
(T(t)x, y) =
jeit8(ds;xy)
(t E R; x, y E X).
From this representation and the uniqueness property of the Fourier-Stieltjes transform, it is easy to deduce the existence of a resolution of the identity E such that
T(t) = f e=t'E(ds) = e:ta for the selfadjoint operator A := f sE(ds) with domain D(A) = {x E X; f s2(E(ds)x,x) < oo}. The details are given in an analogous proof below, with X a reflexive Banach space. For the Banach space setting, we start with another theorem of Bochner, characterizing the Fourier-Stieltjes transforms of complex Borel measures.
1.44. THEOREM (Bochner). A function f : R --+ C is the Fourier-Stieltjes K if and only if it is
transform of a complex regular Borel measure p with IIiII continuous and
IEjcjf(tj)I < KIIEjcje t'BII. for all finite sequences of real tj and complex cj. PROOF. See [R2], p. 32.
We consider the normed space P = P(R) of all "trigonometric polynomials"
¢(s) = Ejcjett;' 46
(cj E C;tj E R),
where the sum above is finite, with the supremum norm. Let Y be any Banach space. Given a function f : R -> Y, we define the linear operator B f : P -p Y by
BfO := Eicif(t.i), for ¢ E P as above, and set
IIBfll, where the norm on the right is the operator norm (that could be infinite a priori). IIfIIB
We refer to I I I I B as the "Bochner norm", and consider the vector space
F(Y):= {f:R-,Y;IIfIIB
t, s E R. Then for any f R -+ Y, IIfllB = IIBf1I ? IIBfOtIl = IIf(t)II
(t E R),
so that IIf IIB >_ IIfII.:= sup II1(011. t
(1)
Thus all functions in F(Y) are bounded, and 11.11B is a norm on that space. By (1), if { fn } C FF(Y) is I I. I I B-Cauchy, it converges uniformly in Y to some f. Given
E > 0, let no be such that l lfn - fm I l B < E for all n > no. Then IlEJc7[fn(tJ) - fm(t,)]II < E
for all c ti such that the corresponding 0 has supremum norm equal to 1, and all n, m > no. Letting m - oo, we get IIE,ej[fn(t,) - f(t7)1I1 < E
for all n > no, and all c.,, tj as above. Thus I lfn - f II B < E < oo (for n > no), i.e., fn - f E IF(Y ), and so f = fn - (f. - f) E F(Y), and fn fin the Bochner norm. This shows that F(Y) is a Banach space with the Bochner norm. A simple calculation shows that IF(Y) contains all Fourier-Stieltjes transforms f1e't'm(ds), of Y-valued vector measures m (cf. [DS-I], Section IV.10): if f(t) then for all ¢ as above with II0Ii... = 1,
lIEicif(ti)Il = II L 4,(s)m(ds)I < llmII, so that IIfuIB < IImII < oo, where IImII denotes the "semi-variation" of m. In particular, ]F(Y) contains the constant functions c, and IIcIIB = Ilcll. 47
The Bochner norm is invariant under additive translation f (t) -4f (t + c) and non-zero multiplicative translation f(t) -' f(ct) (c E R). If Y, Z are Banach spaces and U E B(Y, Z), then UF(Y) C F(Z) and IIUfIIB <_
Y*, IIy*II =1). By the Uniform Boundedness Theorem, f E F(Y) if and only if y* f E F(C) for all IIf
E
y* E Y*.
Also, if F -i B(X, Y) for Banach spaces X, Y, then the following are equivalent:
(i) F E F(B(X,Y)); (ii) Fx E F(Y) for all x E X; (iii) y*Fx E F(C) for all x E X, y* E Y*, In addition, IIFIIB = sup{IIFxhIB; x E X, IIxII =1} = sup{IIy*FxIIB;x E X, Y* E Y*, IIxII = IIy*II =1).
If f E F(Y) is weakly continuous, then by Bochner's theorem, there corresponds to each y* E Y* a finite regular complex Borel measure a(.; y*) such that
y* f(t) = fe,i(ds;y*)
(t E R),
and
IIfIIBIIY*II-
The uniqueness of the integral representation implies that for each 6 E 8(R) (the Borel algebra of R), µ(b; .) is a linear functional on Y* with norm < IllIIB Therefore
p(b; y*) = m(b)y*
(y* E Y*, b E S(R)),
where m : 8(R) -+ Y** is such that m(.)y* is a regular finite Borel measure for each y* E Y*. The element ff e't'm(ds) E Y**, well-defined by the relation if eitsm(ds)]y* = r eit. [m(ds)y*]
(y* E Y*),
s
coincides with the element f (t) E X (imbedded in Y** as usual). In this weakened sense, the weakly continuous elements of F(Y) are Fourier-Stieltjes transforms of Y**- valued measures like m. When Y is reflexive, m is a weakly countably additive (hence strongly countably additive, by Pettis' theorem, cf. [DS-I]) Y-valued measure, and we can write
f (t) = f e't'm(ds) 48
(t E R),
so that the weakly continuous functions in F(Y) are precisely the Fourier-Stieltjes transforms of vector measures m as above. We summarize the above discussion formally:
1.45. PROPOSITION. The space F(Y) is a Banach space for the Bochner norm, and contains all the Fourier-Stieltjes transforms of Y-valued measures with finite variation. Every weakly continuous function in the space is the Fourier-Stieltjes transform of a Y**-valued measure m such that m(.)y* is a regular complex Borel measure for each y* E Y*. When Y is reflexive, every weakly continuous function in the space is the Fourier- Stieltjes transform of a strongly countably additive Yvalued measure m (such that y*m(.) is a regular complex Borel measure for each y*), and is in particular strongly continuous as well. 1.46. DEFINITION. Suppose iA generates a C,,-group T(.) on the Banach space X. We set (x E X). IIxIIT =
The "semi-simplicity manifold" for T(.) is the set
Z=ZT= {xEX;IInIIT
which commutes with T(.). Also Z = X if and only if the two norms on X are equivalent).
PROOF. Since IIxIIT > IIT(.)xlloo > IIxII, (Z, II.IIT) is a normed space. We prove
completeness. If {xn} is Cauchy in (Z, II.I IT), it is also Cauchy in X; let x be its X-limit. Then T(t)xn --1 T(t)x for each t E R. By definition of the II.IIT-norm, Therefore T(.)xn -+ fin {T(.)xn} is Cauchy in the Banach space (F(X), that space, and since II.IIB > II.Iloo> f(t) = limn,T(t)xn = T(t)x (limit in X), for each t. Thus T(.)x E F(X), i.e., x E Z, and IIxn - xIIT = IIT(.)x - T(.)xll B -> 0, is a Banach subspace of X. and Z (with the If U E B(X) commutes with T(t) for each t E R, then for each x E Z, we have T(.)x E F(X), and therefore UT(.)x E F(X) and II UT(.)xII n 5 IIUII.IIT(.)xIIB (see above), which is equivalent in our present situation to T(.)[Ux] E F(X) (i.e.,
Ux E Z) and IIUxIIT <
This shows that Z is U-invariant, and
IIUIIB(z) <- IIUIIB(x) If IIT(.)IIB < oo, then for all trigonometric polynomials 0 as above, and for all x E X, IIE;c;T(tj)xIl < IIE3cjT(tj)II.IIxII < oo, i.e., Z = X. Conversely, if Z = X, and therefore IIxIIT < then sup II [EicjT(tj)]xII < oo (supremum over all ¢ as above), and therefore II E;cjT(tj)ll < oo by the Uniform Boundedness Theorem. The 49
equivalence of the norms is a consequence of the Closed Graph Theorem, or explicitely from the above discussion, (x E X).I1II
11X11 :5 IIXIIT <- IIT(.)llsllxll
1.48. DEFINITION. Let W be a linear manifold in X, and let T(W) denote the algebra of all operators with domain W and range in W. A "spectral measure on W" is a function E(.) : 3(R) -> T(W),
such that (i)E(R) = I1W; (ii) for each x E W, E(.)x is a regular, strongly countably additive vector measure; and (iii)E(b n e) = E(b)E(e), for all b, e E 3(R). Note that by [DS-I] (Corollary 111.4.5), E(.)x is necessarily bounded for each x E W. Note also that (ii) is equivalent to (ii') for each x E W and x* E X*, x*E(.)x is a regular complex Borel measure. This follows from Pettis' theorem (cf. [DS-I]). Let B(R) denote the Banach algebra of all bounded complex Borel functions on R. For E as above and h E B(R), the operator r(h) : W -, X is defined by
r(h)x = / h(s)E(ds)x n
(x E W).
We then extend r to the algebra ]Bia,(R) of all complex Borel functions on R that are bounded on each interval [a, b], by letting
-r(h)x = li m f h(s)E(ds)x b
j h(s)E(ds)x
for h E 31o0(R), where lima,b stands for the limit in X when a --i -oo and b --i oo; the domain of r(h) is the set of all x E W for which the limit exists. We are now ready to state our Banach space version of the Stone Theorem.
1.49. THEOREM. Let T(.) be a Co-group of operators on the reflexive Banach space X, with generator iA and 6(A) C R. Let Z be the semi-simplicity manifold for T(.). Then there exists a spectral measure E on Z with the following properties: (1) T(t)x = ffe`t$E(ds)x (x E Z;t E R); (2) E commutes with every U E B(X) commuting with T(.); 50
(3) r (corresponding to E) is a norm-decreasing algebra homomorphism of E(R) into B(Z, such that T(ot) = T(t)/Z for ct(s) = e't s,t E R; (4) If fl(s) = s (s E R), then Az = r(fl)z (where the subscript Z means the part of the relevant operator in Z), that is, (i) D(AZ) = {x E Z; fla sE(ds)x exists and belongs to Z}, and (ii) Ax = fa sE(ds)x (x E D(AZ)). In addition, Z is maximal and E is unique in the following sense: if W is a Banach subspace of X and F is a spectral measure on W for which (3) is valid, then W is a Banach subspace of Z and F(.) = E(.)/W.
PROOF. For each x E Z, the function T(.)x is a strongly continuous element of 1F(X). Since X is reflexive, Proposition 1.45 gives a strongly countably additive X-valued measure m(.; x) on 8(R) such that
T(t)x =
J
eitam(ds; x)
(1)
for all t E R,x E Z; (2)
x)II <_ IIxllT;
and x*m(.; x) is a regular complex Borel measure for each x* E X*. The uniqueness property of the Fourier-Stieltjes transform of regular complex Borel measures and the linearity of the left side of (1) imply that m(.; x) = E(.)x, where E(b) is a linear
transformation from Z to X, for each 6 E B(R). By (1) with t = 0, E(R) = I/Z. We rewrite (1) in the form
T(t)x = fe"3E(ds)x
(t E R,x E Z).
(1')
If U E B(X) commutes with T(.) and x E Z, then Ux E Z by Lemma 1.47, and by (1'),
J
e`t'UE(ds)x = UT(t)x = T(t)Ux =
J
ett'E(ds)Ux,
hence UE(b)x = E(b)Ux for all x E Z, b E 8(R). 1 and with respective parameters For each 0, zb E P with II0IIoo = cj, tj; ck, tk, we have for x E Z, Il EkckT(tk).EjcjT(tj)xll =
IlFk,jckcjT(tk +tj)xll
5 IIxIITIIEk,jckcjets(tk+ti)llo <_ IIXIITll0lloollbllao = IIxlIT.
Therefore
IlEjcjT(tj)xllT 5 IIkilT 51
for all parameters as above. Fix h E B(R),x E Z. Since EicjT(tj) is a bounded operator commuting with T(.), we have by (2) IIE.icjT(ti)r(h)xUI = IIr(h)EJc.iT(t.i)xII
< IIhII.IIE(.)EjcjT(tj)xII <_ IIhII.IIEjc;T(tj)xIIT <_ IIhII.IIxIIT. Therefore IIT(h)xlIT 5 IIhIIooIIxIIT
for all h E B(R), x E Z. In particular
r : B(R) -* B(Z, has norm < 1 ; actually,
I Ir I
I = 1, because
IIr*)xllT = IIT(r)xIIT := IIT(. + r)xII B = IIT(.)xll B = IISIIT,
by the translation invariance of the Bochner norm on IF(R). Taking h = X6 (the characteristic function of b E 8(R)), we get that
IIE(b)IIB z) 51,
so that surely E(b) E T(Z). For t,u E R and X E Z, with all integrals below extending over R, we have
J
e""E(ds)T(t)x = T(u)T(t)x = T(u + t)x = f dus[e'tsE(ds)xJ.
By uniqueness for Fourier-Stieltjes transforms,
E(u)T(t)x = J e't9X6(s)E(ds)x
for all b E B(R), etc. However the left side equals T(t)E(b)x = f e'" F,(ds)E(b)x since E(b)x E Z for X E Z. Therefore, again by uniqueness for Fourier-Stieltjes transforms,
E(ds)E(b)x = XoE(ds)x, so that
E(v)E(b)x = J XoX6E(ds)x = E(v fl b)x 52
(3)
for all o,bEB(R)andxEZ. We conclude that E is a spectral measure on Z. By (3), T(h)T(Xb)x = T(h)E(b)x =
h(s)Xo(s)E(ds)x = T(hX6)x
J for all h E 13(R), b E 5(R), x E Z. By linearity of T, it follows that T(hg) = 7-(h)-r(g) for all h E B(R) and g E B0(R), the subalgebra of simple Borel functions. Next, for g E B(R), choose simple Borel functions gn converging uniformly to g. Then for all
xEZ,
IIT(h9)x - T(h)T(g)xll S IIT[h(9 - 9n))xf l + IIT(h)T(9n - 9)xl) II h(9 - 9n)IIo0IIxIIT +
IIhIIooII7-(9n - 9)xIIT
< 2IIhII.II9, - 9II0IIxIIT - 0 as n -p oo, and Statement (3) of the theorem is proved.
For all t E R, o(itA) = ito(A) C iR, so that R(t) := R(1;itA) is a welldefined bounded operator commuting with T(.). If X E R(t)Z, say x = R(t)z
with z E Z, then x E D(A) n Z, and Ax = (it)-1(x - z) E Z (for t # 0), i.e., R(t)Z C D(Az). On the other hand, if x E D(Az), then z therefore x = R(t)z E R(t)Z. This shows that
(1 - itA)x E Z, and
(0# t E
D(Az) = R(t)Z
Let x e D(Az); write then x = R(t)z with z E Z and 0 74- t E R fixed. By Theorem 1.15,
R(t)z = t-'R(t-1; iA)z = t-' =
J0
r e-'/tT(s)zds
rao
J
e-"T(tu)zdu
(5)
0
for all00tER,zEX.ForzEZ,,wegetf R(t)z =
f
00
e-"
J
e't"'E(ds)zdu
e-"0-tt')duE(ds)z = J(i - its)-'E(ds)z,
(6)
where the interchange of the order of integration is justified by applying on both sides an arbitrary x* E X* and using Fubini's theorem. 53
For real a < b, we then have by (6) and the multiplicativity of -r on B(R) (for x = R(t)z as above):
f s(1 - its)-1E(ds)z
b
Ja
sE(ds)x =
Ja
b
s(1 - its)-'E(ds)z
when a -+ -oo and b -> oo. Writing s(1 - its)-1 = it-'[1 - (1 - its)-1], the last integral is seen to equal it-1 [z - R(t)z] = it-1(z - x) = Ax E Z.
This shows that D(Az) C {x E Z; f]k sE(ds)x exists and belongs to Z}, and Ax = f sE(ds)x on D(Az). On the other hand, if x belongs to the set on the right of (i) (in the statement of the theorem), then denoting the integral in (i) by z E Z, we obtain from the multiplicativity of r for t # 0: R(t)z = R(t) lim a,b
J
a
e
sE(ds)x = lim R(t)
jb
- its)1 E(ds)x =
a,b
J
b
sE(ds)x
a
f
JE s(1 - its)1 E(ds)x = it' [x - R(t)x]
(cf. preceding calculation).
Therefore x = R(t)(x - itz) E R(t)Z = D(Az) by (4), and we conclude that Property (4) in the statement of the theorem is valid. Finally, suppose (W,11.11w) is a Banach subspace of X for which Property (3) (in the statement of the theorem) is valid, with (W,11.11w) replacing (Z, II.IIT) and 16(R) - B(W,11.11w) (induced by F) replacing r. Then for all 0 E P with I ICI I,,,, = 1 and parameters cj, t IIEiciT(ti)xII = II7-'(4>)xII 5 IIr'(b)xllw IIr'(0)IIB(W)IIxIIW 5 IIxIIW
(x E W).
Therefore IIxIIT < IIxIIw, and W is a Banach subspace of Z. Since T(t)x = f e`t'F(ds)x = f e`t'E(ds)x for x E W, the uniqueness property of the FourierStieltjes transform implies that F(.)x = E(.)x for all x E W.IIII
We consider the special case Z = X. By Lemma 1.47, this happens if and only if I I B < oo, and in this case the two norms 1 1 . 1 1 and I I I I T on X are equivalent. Let E be the spectral measure on Z = X provided by Theorem 1.49. Since IIE(b)xllT <- IIxIIT for all x E X, the equivalence of the norms shows that E : 8(R) -+ B(X) is a "spectral measure" in the usual sense, that is, an algebra I
54
homomorphism of the Boolean algebra 13(R) into B(X) such that E(.)x is regular and countably additive for each x E X. Properties (4)(i),(ii) become (i) (ii)
D(A) _ {x E X; fx sE(ds)x := lima,b fa sE(ds)x exists}; and
Ax = ff sE(ds)x
(x E D(A)).
Using the terminology of [DS-III], the operator A is spectral of scalar type (with
real spectrum). The map r defined above is now the usual operational calculus for the scalar-type spectral operator A, and in particular, the semigroup T(.) is precisely eitA, as defined through this operational calculus. Note that when X is a Hilbert space, the condition IIT(.)Iloo < oo was necessary and sufficient for the above conclusions (Corollary 1.42), while our generalization to reflexive Banach space requires the stronger assumption IIT(.)IIB < oo. This latter condition is however necessary as well, by Proposition 1.45 and Lemma 1.47. We formalize the above discussion in
1.50. COROLLARY. Let iA generate a Co-group T(.) on the reflexive Banach space X. Then A is a scalar-type spectral operator with real spectrum if and only if oo. In that case, T(t) = e=tA := ff e`t°E(ds), where E is the resolution of the identity for A. Actually, we can restate this corollary without assuming a priori that iA generates a C,,-group. We need only to assume that A is densely defined, and has real spectrum. Let then
R(t) := (I - itA)-'
(t E IR).
We first establish some identities.
1.51. THEOREM. If iA (with a(A) real) generates a Co-group T(.), then IInIIT = sup IIRnxIIB
(x E X)
(*)
n>O
and
IIT(.)IIB = sup 1IR"IIB. n>0
PROOF. By Theorem 1.36,
T(t)x = n-oo lim Rn(t/n)x
(x E X; t E R).
Therefore for each 0 E P with II0I1OO = 1 and parameters cj,t1, II F-icjT(t.i)xII = l im II EaciRn(ti/n)xI1 55
Each norm on the right is < IIRn(./n)xIIB = IIRnxlIB, by the invariance of the Bochner norm under multiplicative translations, and this implies the inequality < in (*).
On the other hand, the Taylor expansion of the resolvent obtained in the proof of Theorem 1.11 shows that
R(A; A)(n-1) = (-1)n-' (n - 1)!R(\; A)'
(1 E P(A); n > 1).
(1)
The derivatives may be calculated by using Theorem 1.15, when A generates a C,,-semigroup T(.). We then obtain the following Laplace transform representation for the powers of the resolvent:
R(A; A)nx = f e-at[tn-1/(n - 1)!]T(t)xdt,
(2)
0
for all x E X, RA > w, and n > 1. In our case, with the generators iA and -iA of the semigroups T(.) and S(t) T(-t) (t > 0) respectively, a simple calculation leads to the formula e-esn-1T(ts)xds,
r(n)-'
Rn(t)x =
J
0
(3)
for all x E X, n = 1, 2,... and t E R. Since IIT(ts)xIIB = IIT(t)xII B := IIxIIT for each fixed s > 0, we have for all 0 as above, II
ij
(i)II=I'()n IIf
E c Rn t x
1
000
<
ji (j)
e-'sn-1E c T t s xds II
e-ssn-1IIT(ts)xliBds
r(n)-1
Jo
= IIxIIT,
hence IIRnxIIB < IIxIIT for all n _> 0, and (*) follows. The second identity is then an elementary consequence. iii
We can restate now Corollary 1.50 without assuming a priori that iA is a generator. 1.52. COROLLARY. Let A be a densely defined operator with real spectrum, acting in the reflexive Banach space X. Then A is a scalar-type spectral operator if and only if VA := sup IIR"IIB < o0-n>0
(in that case, iA generates the group eitA, which is the Fourier-Stieltjes transform of the resolution of the identity for A). 56
PROOF. If VA < oo, we surely have
IIR"II',. < VA < oo for all n.
Since
)R(A; iA) = R(1/A)
(0
A E R),
we have II[\R(A;iA)1nII < V A
(n = 1,2,...;0 54 A E R).
Also iA is closed (since p(iA) is non-empty) and densely defined (by hypothesis). The conditions of the Hille-Yosida theorem for groups (Theorem 1.39) are therefore satisfied by the operator iA, with w' = w = 0. If T(.) denotes the group generated by iA, we have IIT(.)IIB = VA < oo, and Corollary 1.50 applies to establish that A is a scalar-type spectral operator. Conversely, if A is scalar-type spectral, let E : 13(R) --+ B(X) be its resolution
of the identity. Then iA generates the Co-group T(.) = e'=A := fj e" E(ds). In particular IIT(.)IIB < oo by Proposition 1.45, that is, VA < oo (by Theorem 1.51).III I
57
H. ANALYTICITY
A function F : D -i B(X) (where D is a domain in C) is analytic in D if
F'(z) := him h-1 [F(z + h) - F(z)] exists in the uniform operator topology, for all z E D. This is equivalent to the existence of that limit in the strong operator topology, and in the weak operator topology as well (cf. [HP, Theorem 3.10.1]).
1.53. DEFINITION. The Ca-semigroup T(.) is analytic if it extends to an analytic function (also denoted T(.)) in some sector
So:={zEC;Iargzj <0,Izj >0},
0<0<7r/2,and limT(z)x=xas z->0,zESo. The extended function necessarily satisfies the semigroup identity in S9:
T(z)T(w) = T(z + w)
(z, w E So)
(cf. [HP, Theorem 17.2.2]). In the study of analyticity for C0-semigroups, it may be assumed without loss of
generality that the semigroup is uniformly bounded (consider e-atT(t) instead of T(.)). For simplicity, we consider only the special case of C0-contraction semigroups,
and the possibility of extending them as contraction-valued analytic semigroups. We refer to the literature for criteria applicable to the general case.
1.54. THEOREM. Let T(.) be a C0-semigroup of contractions, with generator A. Then T(.) extends to an analytic contraction semigroup in a sector So if and only if e'aA generates a C0-contraction semigroup for each a E (-0,0).
PROOF. Necessity. For each a E (-0,0), define Ta(t) :=T(te'a'), t > 0. Clearly, Ta(.) is a Co-contraction semigroup. Denote its generator by A,,, and consider a > 0 (the case a < 0 is analogous). By Theorem 1.15, for all s > 0 and
xEX, R(s; A)x = 58
fe8Tc(t)xdt
f00
= e-`"
exp[-se-'"te'"]T(te'")xd(te'").
(1)
0
The function F(z) := exp[-se-'"z]T(z)x is analytic in So (for s, a, x fixed). Denote C. = {z = ae'd; 0 < 0 < a}, oriented positively (a > 0). On Ca, Ilxlie-eacos(a-m) <
IIF(z)II <- Ilxllexp[-sR(ze-'")] =
Ilxlle-sacosa
Therefore the integral of F over Ca has norm < irae-'acos"]fix]] -- 0 when a and when a < b < oo, consider the closed contour
0+
Fa,b :_ [a, b] + Cb - [a, b] e'" - Ca.
By Cauchy's theorem, fra b F(z)dz = 0. Since the integrals of F on Ca and Cb converge strongly to 0 when a -f 0+ and b -4 oo, it follows that the right side of (1) is equal to e-'" f o° exp[-(se-'")t]T(t)x dt. However 2(se-'O) = s cos a > 0, so that, by Theorem 1.15 (for the contraction case), the last expression is equal
to e-'"R(se-'"; A)x = R(s; e'"A). We conclude that A" and e'"A have equal resolvents on R+, and therefore e'"A is indeed the generator of the C0-contraction semigroup T"(.).
Sufficiency. Suppose that for each or E (-0, 0), e'"A generates a C0- contraction semigroup Tc(.). By Theorem 1.36, T"(t)x = lim[ R( ; e'"A)]nx n
t
= lim[I - z A]-nx, n
n
(2)
where z = te'",t > 0.
Denote
Fn(z) :_ [I - nA]-n = [zR(z;A)]-. Since A generates a C0-contraction semigroup, Fn are analytic in W11 > 0, hence in
So (if z := te'" E Sei then t(n/z) = (n/t) cos ¢ > 0). Since FF(z) = [!!R(2; e'"A)]n, and e'"A is the generator of a C0-contraction semigroup, we have IIFn(z)11 < 1 for all z E So (by Corollary 1.18).
For each x E X and x* E X*, the sequence {x*Fn(.)x} of complex analytic functions is uniformly bounded (by llxll. x*11) in Se, hence is a normal family. It has then a subsequence converging uniformly on every compact subset of So to a function f (.; x, x*) analytic in S9. By (2), f(te'";x,x*) = x*T"(t)x
(3)
59
for allxEX,x* EX*,t>0, and aE(-0,0). Define T(z) = Ta(t), for z = teia E Se. By (3), T(.) is analytic in Se. It coincides with the original semigroup on [0, oo) and is contraction-valued in the sector (by definition). It remains to verify that lim IIT(z)x - xII = 0
as z -4 0, z E So (for all x E X). Since I IT(.)-III < 2 in the sector, we may consider
only x in the dense set D(A) = D(e'aA). For such x, writing z = teia E Soi we have (since eiaA generates the C0-contraction semigroup Ta(t) = T(teia)), IIT(z)x - xII = II fo TT(s)e'aAxdsII
tl jAxII = IzI.IIAxiI,
0
and the conclusion follows.
I
I
I
I
COROLLARY 1. Let A generate a C0-semigroup of contractions T(.). Then T(.) extends as an analytic semigroup of contractions in a sector Se (0 < 9 < 7r/2) if and only if cos aQ(x*Ax) - sin as(x*Ax) < 0 (*)
for all unit vectors x E D(A) and x* E X* such that x*x = 1, and for all a E (-0, 9).
PROOF. For all a E (-9, 9), eiaA is closed, densely defined, and for all A > 0, AI-e'aA = eta (Ae-"1-A] is surjective, since 2(Ae-'a) = A cos a > 0 (cf. Theorem 1.26). Therefore, by Theorem 1.26, e`'A generates a C0-semigroup of contractions if and only if it is dissipative, i.e., if and only if t(x*eiaAx) < 0
for all unit vectors x E D(A) and x* E X* such that x*x = 1. This is precisely Condition (*), so that the corollary follows immediately from Theorem 1.54.
When 0 = n/2 (i.e., for analytic semigroups in the right halfplane C+), we may consider "boundary values" on the imaginary axis.
1.55. THEOREM. Let T(.) be an analytic semigroup in C+, and suppose it is bounded in the rectangle Q := {z = t+ is E C; t E (0,1], s E[-1,1]}. Let v := log[supQ
(of course, 0 < v < oo). Then for each s E R,
T(is) := t-.o+ lim T(t + is) 60
exists in B(X) in the strong operator topology, and has the following properties: (1) T(i.) is a C0-group; (2) T(is) commutes with T(z) for all s E R, z E C+; (3) T(t + is) = T(t)T(is) for all t > 0, s E R; (4) T(.) is of exponential type < v in the closed right halfplane, i.e., I IT(z)II 5 Ke" 1z1
(Rez > 0);
and (5) If A is the generator of {T(t); t >_ 0), then iA is the generator of the bound-
ary group {T(is); s E R}. PROOF. See [HP], Theorem 17.9.1 and its proof. COROLLARY 2. Suppose that the generator A of the Ca- semigroup of contractions T(.) has real numerical range (i.e., v(A) C R, cf. Definition 1.24). Then T(.) extends as an analytic semigroup of contractions in C+. In particular, the boundary group {T(is);s E R} exists, and is a C,,-group of isometries (with generator iA).
PROOF. Condition (*) of the previous corollary reduces here to cos a R(x*Ax) < 0 (for all parameters in their proper ranges), which is trivially satisfied (since Ia] <
0 < r/2, and by Theorem 1.26 applied to A). Observe finally that a group of contractions consists in fact of isometries. COROLLARY 3. Let A be a closed densely defined operator. Then the following are equivalent: (1) A generates an analytic semigroup of contractions in the sector So (0 < 9 < it/2); (2) IIzR(z; A)I I < 1 for all z E Se;
(3) zI - A is surjective for all z E So, and R[e`"v(A)] < 0 for all a E (-9, 0). PROOF. Writing z = te`", we see that Condition (2) is equivalent to
(2') IItR(t;e'"A)II <1forallt>0andaE(-0,0), and (2') is equivalent to (1), by the Hille-Yosida Theorem (for contraction semigroups) and Theorem 1.54. Assume (3). For all a E (-0, 0), e'"A is closed, densely defined, and for all t > 0,
tI - e'aA = e'"[te-"I - A] is surjective. The inequality in (3) means that e'"A is dissipative, and (1) follows from Theorem 1.26 and Theorem 1.54. Conversely, if (1) holds, then Theorems 1.54 and 1.26 imply that e'"A is dissipative and tI - e'"A is surjective for all a E (-9, 0), and this is equivalent to (3). COROLLARY 4. Let A be a closed densely defined operator such that zI A is surjective for Rz > 0 and v(A) C (-oo, 0]. Then A generates an analytic 61
semigroup of contractions T(.) in the right halfplane. In particular, the boundary group {T(is)} exists, and is a C,,-group of isometries (with generator iA). PROOF. For real numerical range, the inequality in Corollary 3 (3) reduces to cosy x*Ax < 0 (for all parameters in their proper ranges), which is satisfied by hypothesis. The conclusion follows then from Corollaries 2 and 3.
In case X is a Hilbert space, let 7r : X* ---> X be the canonical isometric antiisomorphism of X * onto X given by the Riesz representation x*x = (x, zr(x* )). Then v(A) _ {(Ax, lr(x* )); x E D(A), x* E X*, Ilxl I= I Ix* II = (x, lr(x* )) = 1).
However, writing 7r(x*) = y, we have (for x, x* as in the above formula): Ilx-y112=IIxI12-2t(x,y)+IlylI2=1-2+1=0,
so that y = x, and v(A) = {(Ax, x); x E D(A), Ilxll =1}.
Therefore A is dissipative if and only if (x E D(A)).
R(Ax, x) < 0
The inequality in Condition (3) of Corollary 1 becomes
R[e'"(Ax, x)] < 0
(x E D(A)).
We then have COROLLARY 5. Let A generate a C,,-semigroup of contractions T(.) in Hilbert space. Then T(.) extends as an analytic semigroup of contractions in a sector So (0 < 0 < 7r/2) if and only if
R[e'"(Ax, x)] < 0
(x E D(A), a E (-8, 8)).
COROLLARY 6. Let A be a closed densely defined operator in Hilbert space, such that zI - A is surjective for Qz > 0 and (Ax, x) < 0 for all x E D(A). Then A generates an analytic semigroup of contractions in Rz > 0; the boundary group {T(is)} is a unitary Co-group (with generator iA), and A is selfadjoint. 62
PROOF. By Corollary 2, A generates an analytic C0-semigroup of contractions in 3?z > 0. Since (Ax, x) is real for all x E D(A), A is symmetric. The boundary group {T(is);s E R} in that corollary (with generator iA) satisfies (cf. Theorem 1.36):
T(is)x = lim[n R(n; iA)]"x = lim[ n R(n ; A)]nx n
s
n
8
2s
2s
for all s > 0 and x E X. Since -iA is the generator of the C0-semigroup {T(-is); s > 0}, we also have T(-is)x = linm[ R(n ; -iA)]nx = lim[
is R(n; A)]"x.
s = R(z-; A), and in particular, R(z; A) is normal. Since A is symmetric R(z; A)` Therefore, for all x, y E X and s > 0, (T(-is)x, y) = lira([nR(n ; A)]n*x, y) = lim(x, [.R(n ; A)]ny) = (x,T(2s)y), n 2s n is
2s
2s
i.e.,
T(is)* = T(-is) = T(is)-' for all s > 0 (hence for all s E R). Thus the boundary group is unitary; by Stone's theorem (Theorem 1.41), its generator iA has the form iH with H selfadjoint, that is, A is sel fadjoint.IIII A more direct way to prove the selfadjointness of A goes as follows. Suppose y E X satisfies ((iI - A)x, y) = 0 for all x E D(A). Then i(x, y) = (Ax, y) for all x E D(A). Take x = R(s; A)y (E D(A)!) for some s > 0. Then i(R(s; A)y, y) = (AR(s; A)y, y) = ([sR(s; A) - I]y, y)
The left side is pure imaginary, while the right side is real (since the bounded operators appearing there are both selfadjoint). Therefore (R(s; A)y, y) = s(R(s; A)y, y) - (y, y) = 0,
hence (y, y) = 0 and y = 0. This shows that iI - A (and similarly, -iI - A) has dense range, which means that A is essentially selfadjoint (cf. "digression" preceding Theorem 1.41). Since. A is closed, it is actually sel f adjoint.
Note that the relation T(t)x = limn[ R(i ; A)]"x shows that the operators T(t) are selfadjoint (for A symmetric). The longer discussion given above illustrates the "method of analytic continuation to the imaginary axis", that will be used in Section
i
2.37 in the more general case of a "local semigroup" of unbounded symmetric operators to produce a selfadjoint operator H such that each T(t) is a restriction of a-tx 63
K. NON-COMMUTATIVE TAYLOR FORMULA
In this section, we consider a C0-semigroup T(.) as a function of its generator
A, when A varies in the set of all generators of C0-semigroups. The notation T(.) = T(.; A) will be used to exhibit the generator A of the semigroup. In order to get a feeling about a possible Taylor formula relating T(.; B) with T(.; A) and derivatives of the semigroup with respect to A (at the point A), we consider first the case of uniformly continuous semigroups (i.e., the variable generator varies in B(X)). This case can be formulated in an arbitrary complex Banach
algebra A with identity I, and we may consider analytic functions on it, more general than the functions ft(A) := etA, t > 0, A E A. As before, we denote the resolvent set of an element S E A by p(S), etc... We start with the following elementary
1.56. LEMMA. Let S,T E A and z E p(S) fl p(T). Then f o r all n = 0,1, 2,...,
R(z; T) = E;'=o[R(z;S)(T - S)]'R(z;S) +[R(z; S)(T -
S)]"+1R(z;T).
PROOF. If Q E A is such that I - Q is invertible in A, then one verifies directly the "geometric series addition formula"
(I - Q)-1 = EJ._0Q'' + Q"+1(I - Q)-1,
(1)
n = 0,1, 2,.... For z c- p(S) fl p(T ), we take
Q := R(z; S)(T - S) = R(z; S)[(zI - S) - (zI - T)]
= I - R(z; S)(zI - T), so that I - Q = R(z; S)(zI - T) is indeed invertible in A with inverse equal to R(z; T)(zI - S). Substituting in (1) and multiplying on the right by R(z; S), the lemma follows.
The formula of the lemma simplifies as follows when S, T are commuting elements of A: 64
1.57. LEMMA. Let S, T E A commute, and let z E p(S) fl p(T). Then
R(z; T) = E, 0R(z; S)i+1(T - S)i +R(z; S)"+1R(z;T)(T - S)"+i,
n=0,1,2,.... Given arbitrary elements A, B E A, we consider the commuting multiplication operators LA, RB E B(A) defined by LAU = AU;
RBU = UB,
(U E A).
We then have
1.58. LEMMA. Let A, B E A and z E p(A) fl p(B). Then for all n = 0, 1, 2,..., R(z; B) = E, oR(z; A).i+' (RB - LA)'I +R(z; A)"+' [(RB - LA)"+1I]R(z; B) and
R(z; B) = Ej'=o[(LB - RA)'I]R(z; A)i+' +R(z; B)[(LB RA)"+1I]R(z; A)"+'
-
PROOF. We apply Lemma 1.57 to the commuting elements S = LA and T = RB of the Banach algebra B(A). If z E p(A) fl p(B), then z E p(LA) fl p(RB), R(z; LA) = LR(z;A), and R(z; RB) = RR(z;B). Therefore
RR(z;B) = E7
LA)' +
LA)"+l
Applying this operator to the identity I E A, we obtain the first formula of the lemma. The second formula is deduced in the same manner, through the choice S = RA and T = LB in Lemma 1.57.1111
The non-commutative Taylor formula for analytic functions on the Banach algebra A uses the Riesz-Dunford analytic operational calculus. Let f be a complex analytic function in an open neighborhood 1 of the spectrum a(B) of B E A. If K C Sl is compact, we denote by F(K, Sl) any finite union of positively oriented simple closed Jordan contours in fl, that contains K in its interior. The element f (B) E A is defined by
f(B) = 217ri 1 f(z)R(z;B)dz, 65
where r = r(o(B), Il), and the definition is independent of the choice of such r (cf. [DS I]).
1.59. THEOREM. Let A be a complex Banach algebra with identity I. Let A, B E A, and let f be a complex function analytic in a neighborhood St of o(A) U o(B). Then f o r n = 0,1, 2, . o
f(B)
fiA) , (RB -LA)3I+L"(f,A, B)
and -RA)il.f0)'A)
f(B) = E' 0(LB
7
+R.(f,A,B),
where the "left" and "right" remainders L and R. are given by the formulas
L,,=
1 f f(z)R(z; A)"+' (RB - LA)"+' I.R(z; B)dz tai
and
R,,= tai 1 f f(z)R(z; B)(LB - RA)"+'I.R(z; A)"+'dz, with r = r(u(A) U o(B),1). PROOF. For r as above, the Riesz-Dunford operational calculus satisfies
f(i)(A) =
tai f rf(z)R(z;A)i+'dz,
and the theorem follows from Lemma 1.58 by integration. I I I I
Note that (RB - LA)-I = Ek=0 Wk
(-A)kBi-k,
with a similar formula for (LB - RA)'I. When A, B commute, these formulas reduce to (B-A)i, and the Taylor formula of the theorem reduces to its "classical" form
(i)A
f(B) = E o f j!
+21 7rz 66
)
(B - A)i
/ f(z)R(z; A)"+'R(z; B)dz.(B - A)"+1
r
When f is analytic in a "large" disc, the "Taylor formula" of Theorem 1.59 implies a non-commutative Taylor series expansion:
1.60. THEOREM. Let A be a complex Banach algebra with identity I, and let .4, B E A. Suppose f is a complex function analytic on the closed disc {z E C; IzI < 2IIAII + IIBII}.
Then
f(B) = E0
f(i) (A)
f(i) (A)
(RB - LA)j I = E;=0(LB - RA)Y1.'
,a!
7!
with both series converging strongly in A.
PROOF. It suffices to prove that the remainders Ln and R converge strongly to 0 in A. Fix r > 2IIAII + IIBII such that Cr(:= the positively oriented circle of radius r centered at 0) and its interior are contained in the domain of analyticity St of f. Clearly o(A) U o(B) lies in the interior of Cr, so we can take r = Cr in Theorem 1.59. Let Mr := maxZEC, If(z)I. On Cr, we have IIR(z; A)II = IIEn°_o z (r - IIAII)-1, and similarly for R(z; B). Therefore
III
IILnII <
rA1r
(r - IIAII)n+1(r - IIBII)
LA)n+IIII,
II(RB -
with a, similar estimate for R (replace the last factor by II(LB - RA)n+IIII)However
II(RB - LA)n+IIII = 11E;=+01n
A)iBn+1-iII
5 (IIAII +
IIBII)n+1
and similarly for the letters R, L interchanged. Therefore IILnII <
rMr I IIAII + IIBII ]n+1 r - IIBII r - IIAII
0
as n -* no, because IIAII + IIBII < r - IIAII, and similarly for R,,.IIII
Taking f(z) = ft(z) := et: (for t > 0 fixed) in the "Taylor formula" of Theorem 1.59, we obtain t? = o!( RB
etB = etAEn
- LA)iI + Ln,
(1)
67
with the appropriate expression for the remainder L,,. This is the "non-commutative Taylor formula" we wish to generalize to the case of strongly continuous semigroups. For (generally) unbounded operators A, B, we use the (suggestive) notation
(B - A)131 :_ (RB - LA)jI := Ek=o
()(_A)kBi_k
with maximal domain nJ
D((B - A)1i]) =
D(AkBi-k ). I
I
k=0
The dense T(.; A)-invariant core for A consisting of all the C°°-vectors for A (cf. Theorem 1.8) is denoted by D°°(A). The type of T(.; A) is w(A) (cf. Section 1.3). We can state now 1.61. THEOREM. Let A, B be generators of Ca- semigroups such that D°°(B) C D°°(A). Fix a > max[w(A),w(B)]. Then for it = 0, 1, 2,... and c > a,
T(t; B)x = T(t; A)E
o
t-i (B
7
- A)1'1 x + L,, (t; A, B)x
for all x E D°O(B) and t > 0, where the "n-th remainder" L is given by L,,
A B)x =
1 -27ri
ffO+a et zR(z A)"+'(B-A)I"+1]R(4`,B)xdz; O
the integral converges strongly in X as a Cauchy Principal Value and is independent of c > a. PROOF. Let A, and B,, be the Hille-Yosida approximations of A and B respectively (s, u > a; cf. Sections 1.16 and 1.18). We recall that there exists M > 0 and r > a such that IIetA,
IIeVB
I I < Meat;
I I < Meat
(2)
for all s, it > r and t > 0. In particular, A, and B. have their spectra in the closed halfplane {z E C; Rz < a}, and
IIR(z; A,.)II < M 68
a
(3Rx > a),
(3)
with a similar estimate for R(z; B,,,) (for all s, u > r). Recall that, in the strong operator topology, etA' -+ T(t; A)
(t > 0),
(4)
and
R(z; A,) -+ R(z; A)
(Rz > a),
(5)
ass - oo. Also, for allxED(A), A,x
Ax
(6)
(cf. Sections 1.16, 1.17, 1.32).
By (3), it follows from (5) that for all j E N and Rz > a, R(z; A, )J -+ R(z; A)1
(7)
in the strong operator topology, as s -+ oo. By (6), in the strong operator topology, A,R(z; A)
AR(z; A),
(8)
for each z E p(A).
By (3) and the definition of A IIA,R(z; A)II = IIsAR(s; A)R(z; A)II = sIIR(s; A)[AR(z; A)]II
sIIR(s; A)II.IIAR(z; A)II < sM a IIAR(z; A)II < 2MIIAR(z; A)II for all s > a. This uniform boundedness together with (8) imply that for all m E N, [A9R(z;
A)]'
-'s
[AR(z; A)]'
(8')
in the strong operator topology. Since
D(Am) = R(z; A)'X
(m E N; z E p(A)),
writing x E D(Am) in the form x = R(z; A)'y for a suitable y E X, we obtain (since A, commutes with R(z; A)):
A; x = As R(z; A)'y = [A,R(z; A)]my --+ [AR(z; A)]' y = A' x. Thus for all m E N,
A; x -+A'x
(x E D(Am)).
(9) 69
For 0 < k < j, x c D(Bi-k ), and s > r fixed, it follows from (9) that
(-A,)kBj-kx -,
(-A,)kBi-kx
(as it ---, oo). Therefore
(Bu - A,)[jl x
(10)
(B - A,)[j]x
for all x E D(Bi ). Hence
R(z;A.)i+'(Bu
- A,)[jlx -u-oo R(z;A9)j+l(B -A,)[i]x
(11)
for all x E D(Bi ), Rz > a, s > r, and j = 0,1, .... If x E D((B - A)[3]), then for 0 < k < j, Bj-kx E D(Ak), and therefore (9) implies that (-A,)kBJ-kx - (-A)kB3-kx as s -, oo, hence (B - A,)[jl x
(B - A)[j]x.
(12)
Together with (3) and (7), this implies that R(z; A,)m(B - A,)[jl x
R(z; A)m(B - A)[jl x
(13)
for a.llxED(B-A)[jl),mEN,and tz>a. If x E no D((B - A)[j] ), then surely x E D(Bn), and it follows from (11) and (13) that
lim lim Ej=OR(z; A,)j+'(B,, - A,)[jl x = E 0R(z; A)j+1(B
9-OQU-00
- A)[jl x
(14)
for Rz > a. On the other hand, by Lemma 1.58, for all x E X, the left hand side of (14) is equal to R(z; Bu)x - R(z; A,)n+1(Bu - A,)[n+']R(z; Bu)x. (15)
If x E D(B'n-')(= R(z; B)'-'X) (for any m E N), writing x = R(z; B)'n-1 y for a suitable y E X, we have Bu R(z; Bu)x = [BuR(z; Bu)][BuR(z;
B)]"'-1 y.
The operators in the first bracket on the right are equal to zR(z; Bu) - I, and are therefore uniformly bounded (with respect to u) by 1+ R Z a (by (3)), and converge (as it -* oo) to zR(z; B) - I = BR(z; B) (by (5)) in the strong operator topology. The second bracket converges to [BR(z; B)]"-'y, by (8') for B. It follows that
B...R(z; Bu)x - [BR(z; B)][BR(z; B)]'n-' y = B'R(z; B)my = BmR(z; B)x 70
for all xED(B"n-1) and9z>a. Therefore, for s > r fixed and x E D(B'), the right hand side of (15), which is equal to R(z; Bu)x -
Ek+1(n
k
/
R(z;
A9)"+1(-A,)kBu+1-kR(z; B.)x
converges as u - oo to R(z; B)x - R(z; A,)"+1(B
- A9)["+1]R(z; B)x
(cf. (5)).
If x E D(B") is such that R(z; B)x E D((B - A)["+1]), it follows from (13) that the last expression converges to
R(z; B)x - R(z; A)"+1(B - A)["+1]R(z; B)x asS-->00.
We then conclude from (14) that the following generalization of Lemma 1.58 (first formula) is valid: LEMMA 1. Let A, B be generators of Co-semigroups. Then for Rz > max[w(A),w(B)],
R(z; B)x = Ey 0R(z; A)i+1(B
- A)[i] x + R(z; A)"+' (B - A)["+1]R(z; B)x
for all x En,-=o D((B - A)[J]) such that R(z; B)x E D((B - A)["+1]) (i.e., for all x in the maximal domain of the right-hand side).
Assume now that D°°(B) C D°O(A). If 0 < k < j and x E D°°(B), then
B'-kx E D°°(B) C D°°(A) C D(Ak), so that x E D((-A)kBJ-k). Hence x E n;=o D((B - A)[3]) for all n. Since also R(z; B)x E D°°(B), we have R(z; B)x E D((B - A)["+1]) as well, and the formula in the lemma is valid for all x E D°°(B). We need the following generalization of the second formula in Theorem 1.15.
LEMMA 2. Let A generate the Co-semigroup T(.; A). Then for t > 0, c > w(A), and j = 0, 1, 2,...,
1
rc+ir
j
etzR(z; 9)J+1xdz = t ]T(t; A)x lim r-co 21ri Jc-ir 7
(x E D(A)). 71
PROOF ( of Lemma 2). Since R(z; A)i+' _ ZR(z; A)(3), we may integrate by parts j times to show that the integral appearing in the lemma is equal to -etzEk-o (j - k - 1)!tkR(z; A)i-kxlc+ir Jc-ir j!
ti +: J
The "integrated part" has norm
Jc-ir fir
etzR(z;A)xdz.
(16)
< ect ,k=0tk(I IR(c + iT; A))-kxII + I IR(c - iT; A)J-kill )
If x E D(A), write x = R(A; A)y for some A with WA > a. Then since j - k > 1, IR(c + iT; A)i-kxll = I IR(c + iT; A)i-k-'
MJ-kllyll
< (c
R(A; A)y - R(c + iT; A)y
C+2T-A 1
II
1
- a)Jk'Ic+aT-AIc-a+ la-a --+r-.oo 0,
and similarly for c - ir. Therefore the integrated part in (16) converges to 0 when r -* oo. By Theorem 1.15, the integral in (16) converges to 27riT(t; A)x (for s. E D(A)), and the lemma follows. If x E D°°(B) and t > 0, we have by Lemma 1 c+ir
I-ir e--R(z; A)n+' (B - A)!n+'IR(z; B)xdz et:R(z; et-R(z; A)i+' (B - A)[il xdz. B)xdz - E'=0 f c
c+ir
-ir
o+ir
-ir
(17)
However, for x E D°°(B), we surely have x E D(B), so that the first term on the right converges to 27riT(t;B)x, by Theorem 1.15 (when T -+ oo). We observed
above that BJ-kx E DO°(A) for all 0 < k < j, and therefore (-A)kBJ-kx E D'''(A), and so (B - A)!il x E D°°(A) C D(A). Hence, by Lemma 2, the sum on the right of (17) converges to t)
2iriT(t; A)E o t-j(B
- A)Iil x.
(18)
This shows that the remainder L in Theorem 1.61 converges (as a "Cauchy Principal Value"), and its "value" is independent of c > a, and is equal to
T(t; B) - T(t;
72
t' (B - A)!il x.l III
J
PART II. GENERALIZATIONS
A. PRE-SEMIGROUPS
We consider the following elementary properties of a Ca- semigroups S(.):
Property 1. S(.) : (0,oo) -+ B(X) is strongly continuous and S(0) is injective.
Property 2. S(t - u)S(u) is independent of u, for all 0 < u < t. Property 3. There exists a > 0 such that e-atS(t)x is bounded and uniformly continuous on [0, oo), for each x E X.
Property 1. is contained in Theorem 1.1 (together with the trivial injectivity of S(0) = I). Property 2. follows from the semigroup identity. Property 3. follows from Theorem 1.1 and the estimate lie
-a(t+h)S(t + h)x - e-atS(t)xll
< e-atjjS(t)jj.jje-ahS(h)x - x1I < Mlle-ahS(h)x - xl i. 2.1. DEFINITION. A pre-semigroup is a function S(.) with the properties 1. and 2. If Property 3. is also satisfied, the pre-semigroup is exponentially tamed. By 2., equating the values of S(t - u)S(u) with the value at u = t, we see that
S(t - u)S(u) = S(0)S(t)
(t > u > 0).
(1)
Writing t = u + s in (1) , the identity is equivalent to
S(s)S(u) = S(0)S(u + s)
(s, u > 0).
(1')
In particular, the values of S(.) commute.
2.2. DEFINITION. The generator A of the pre-semigroup S(.) has domain D(A) consisting of all x E X for which the strong right derivative at 0, [S(.)x]'(0), exists and belongs to S(0)X, and
Ax = S(0)-1[S(.)x]'(0)
(x E D(A)). 75
Note that if T(.) is a C,,-semigroup with generator A, and C E B(X) is injective and commutes with T(.), then S(.) := CT(.) is a pre-semigroup with S(O) = C and with generator A. We first generalize Theorem 1.2 as follows
2.3. THEOREM. Let A generate the pre-semigroup S(.). Then: 1. A commutes with S(t) for all t > 0. 2. A is closed with S(0)X C D(A)-. 3. For each x E D(A), u:= S(.)x is of class C' and solves
(ACP)
u' = Au;
u(0) = S(O)x
on [0, oo).
PROOF. For t > 0, h > 0, and x E D(A),
S(h)[S(t)x] - S(0)[S(t)xj = S(t)[S(h)x - S(0)x] = S(0)S(t + h)x - S(0)S(t)x. Dividing by it and letting h -+ 0, we get that the strong right derivative at 0 of S(.)[S(t)x] exists, equals the strong right derivative of S(0)S(t)x at t, and equals S(t)S(0)Ax = S(0)S(t)Ax E S(0)X. Therefore S(t)x E D(A) and A[S(t)x] S(0)-' [S(0)S(t)Ax] = S(t)Ax. This proves 1. Also for 0 < it < t (with t fixed), letting K := supo<,
Ilh-'[S(0)S(t - h)x - S(0)S(t)x] + S(0)S(t)Axll 11h' [S(0)S(t - h) - S(0)S(t)]S(h)x + S(0)S(t)Axll
h) - S(t)II.IIh-'[S(h)x - x] - S(0)Axll +IIS(0)ll.IIS(t)[S(0)Ax] - S(t - h)[S(0)Ax]jl.
The first term on the right of the inequality equals IIS(0){-h-'[S(0)S(t + h) - S(0)S(t)] + S(t)Ax}II -+ 0
when h -+ 0, as observed before. The second term on the right is
< 2KII
h-' [S(h)x - x] - S(0)Axll -+ 0
by definition of A. The third term on the right -+ 0 by continuity of S(.)[S(0)Ax]. 76
Thus we proved that S(0)S(.)x is differentiable on [0, oo) for each x E D(A),and
(t > 0; x E D(A)).
[S(O)S(.)x]'(t) = S(O)S(t)Ax
(2)
Integrating from 0 to t and using the boundedness and injectivity of S(0), we obtain
f S(s)Axds = J t
S(t)x - S(0)x =
0
t
AS(s)xds
(x E D(A)).
(3)
0
For h, t > 0 and for all x E X, we have
h-' [S(h) - S(0)] r t S(s)xds = S(0)h-' Jo
t+h
= S(0)[h-' it t
S(s + h)xds 0
/
t
S(s)xds]
0
S(s)xds - h-' fo S(s)xds] -a S(0)[S(t)x - S(0)x] E S(0)X,
showing that fo S(s)xds E D(A) and
AJ t S(s)xds = S(t)x - S(0)x
(x E X).
(4)
0
In particular, for all x E X,
S(0)x =t0+ lim t-'
!t
J
S(s)xds E D(A)-.
0
We show now that A is closed. If x E D(A), x --f x, and Ax,, -+ y, then with K as before (for t fixed) and L = supnjlAx,,jj, we have JIS(s)AxnII < KL for all n and s E [0, t], and S(s)Ax,, -+ S(s)y pointwise. By dominated convergence and (3), t
S(t)x - S(0)x = lim[S(t)x,, -
limJ 0
t
J S(s)yds. 0
Dividing by t and letting t --a 0+, we obtain that the right hand side converges to S(O)y E S(0)X, so that x E D(A) and Ax = y, as wanted. We read also from (3) that S(.)x is of class C' and solves ACP on [0, oo) for each x E D(A).I1II
A partial converse of Theorem 2.3 is the following
2.4. THEOREM. Let S(.) have Property 1. and commute with A, and either D(A) is dense or p(A) is non-empty. If S(.)x solves ACP for each x E D(A), then S(.) is a pre-semigroup generated by an extension of A. 77
PROOF. For x E D(A) and 0 < u < t,
du S(t - u)S(u)x = -AS(t - u)S(u)x + S(t - u)AS(u)x = 0, and Property 2. follows on D(A), hence on X in case D(A) is dense, because S(t - u)S(u) E B(X). In case p(A) is non-empty, fix A E p(A). Since R(A; A)x E D(A) for all x E X, and since R(A; A) commutes with S(.) (because A commutes with S(.)), we have
R(A; A)S(t - u)S(u)x = S(t - u)S(u)R(A; A)x = S(0)S(t)R(A; A)x = R(A; A)S(0)S(t)x,
and therefore S(t - u)S(u) = S(O)S(t), i.e., Property 2. is satisfied. Let then A' be the generator of S(.). By hypothesis, [S(.)x]'(0) = A[S(0)x] S(0)Ax E S(O)X for all x E D(A), that is, A C A'.IIII A generalization of Theorem 2.4 is the following
2.5. THEOREM. Let A, B be (unbounded) operators such that (i) 0 E p(B); (ii) D(B) C D(A); and (iii) B commutes with R(A; A) for some A > 0. Then ACP for A has a unique C1-solution on [0, oo) for each x E D(B) if and only
if an extension of A generates a pre-semigroup S(.) (with S(0) = (Al - A)R(0; B)) that commutes with A. PROOF. Suppose S(.) is a pre-semigroup with S(0) as stated, commuting with A and generated by an extension A' of A. First, S(t)D(A) = S(t)R(A; A)X = R(A; A)S(t)X C D(A) for all t. For x = S(O)y with y E D(A) C D(A'), we have by Theorem 2.3 A'[S(.)y] =
i.e., [S(.)S(0)-1x]' = A[S(.)S(0)-1x] and of course [S(.)S(0)-1x](0) = x, that is, S(.)S(0)-1x solves ACP for x E S(0)D(A) = D(B), since S(0)D(A) = S(O)R(A; A)X = R(A; A)S(0)X = R(0; B)X = D(B).
If v : [0, oo) -+ D(A) is any solution of ACP with x = S(0)y E S(0)D(A), then
de[S(t-s)v(s)] = -S(t-s)A'v(s)+S(t-s)Av(s) = 0 since v(s) E D(A). Equating therefore the values of the constant s-function S(t - s)v(s) at s = 0 and s = t, we get S(O)v(t) = S(t)S(0)y, hence v(t) = S(t)y = S(t)S(O)-'x, meaning that ACP has a unique solution for each x E S(0)D(A). 78
Conversely, assume ACP with initial value x E D(B) has the unique C1- solution u(.; x) on [0, oo). If v := R(A; A)u(.; x), then
v' = R(A; A)u(.; x)' = R(A; A)Au(.; x) = Av, and
v(O) = R(A; A)x = R(A; A)R(O; B)y = R(0; B)R(A; A)y E D(B).
By the uniqueness assumption,
(x E D(B)).
R(A; A)u(.; x) = u(.; R(A; A)x)
(1)
We define now for all x E X
S(.)x := (AI - A)u(.; R(0; B)x) = Au(.; R(0; B)x) - u'(.; R(0; B)x).
(2)
Since R(0; B)x E D(B), and u(.; y) has values in D(A) for any y E D(B), the operator S(t) is everywhere defined on X, and is linear by the uniqueness hypothesis (for each t > 0). By (2), S(.)x is continuous for each x E X. By (1) with R(0; B)x(E D(B)) replacing x,
R(A; A)S(t)x = u(t; R(0; B)x) _ (Al - A)R(A; A)u(t; R(0; B)x) _ (AI - A)u(t; R(A; A)R(0; B)x) = (AI - A)u(t; R(0; B)R(A; A)x) = S(t)R(A; A)x,
and it follows that S(t) commutes with A for all t. Consider now U(.)x := u(.; R(0; B)x). The operator U(.) : X -, C1([0, b]; X) into the Banach space of all X-valued C'-functions on [0, b] with the usual norm, is shown to be closed. Indeed, if x -> x in X, and U(. )x --' v in C' ([0, b]; X), then for each t E [0, b], Au(t; R(0;
[U(.)xn]'(t)
[u(.; R(0;
v'(t).
Since A is closed, v(t) E D(A) and Av(t) = v'(t). Also v(0) = lim U(0)x = lim u(0; R(0; B)x,,)
= lim R(0; B)x = R(0; B)x. n
By uniqueness, it follows that v = u(.; R(0; B)x) = U(.)x, that is U(.) is closed, hence bounded, by the Closed Graph Theorem. Let M denote its norm. Then for
0
< (A + 1)Mllxl I
(x E X).
Since b is arbitrary, this shows that S(.) is B(X)-valued. We saw that it satisfies Property 1. with S(O) = (XI - A)R(O; B)x clearly injective, and it commutes with A. For x E D(A), write x = R(A; A)y; then by (1), S(.)x = (Al - A)u(.; R(0; B)R(A; A)y) = u(.; R(0; B)y) solves ACP (with the initial value S(O)x = R(0; B)y).
By Theorem 2.4, we conclude that S(.) is a pre-semigroup generated by an extension of A.
I
I
I
I
Taking B = -(Al - A)"+1 (for some non-negative integer n) with domain D(B) = D(A"+1) C D(A), the pre-semigroup S(.) generated by an extension of A satisfies S(O) = (,\I - A)(AI - A)-"-1 = R(A; A)". We thus have the following 2.6. COROLLARY. Let A E p(A). Then ACP for A has a unique C1-solution on [0, oo) for each x E D(An+1) if and only if an extension of A generates a presemigroup S(.) with S(0) = R(A; A)", which commutes with A. We consider next an exponentially tamed pre-semigroup S(.), that is, all three properties 1.,2.,3. are satisfied. By the injectivity of S(0) and the Uniform Boundedness Theorem, 0 < IIS(0)II < M := supe-atIIS(t)II < oo. t>o
2.7. DEFINITION. Let S(.) be a pre-semigroup with Property 3. Set
Y = {x E X;
S(0)-1 e-"tS(t)x
E Cb([0, oo); X)},
where Cb(...) denotes the Banach space of all X-valued bounded uniformly continuous functions on [0, oo), normed by 11f I Iu = sups>o Ilf(t)I1.
For x E Y, set IIxI1Y :=
11S(0)-'e-°tS(t)xllu.
Clearly 11.11y ? 11.11 on Y, and Y:= (Y, II.IIY) is a normed space.
Finally, we denote by [S(0)X] the Banach space S(0)X with the norm MIIS(0)-'xll.
IIxIIo := 80
2.8. THEOREM. Let A generate an exponentially tamed pre-semigroup S(.), and let Y be the space defined in 2.7. Then Y is a Banach subspace of X containing
[S(O)X] as a Banach subspace, and Ay (the part of A in Y) generates a C,semigroup T(.) in Y satisfying IIT(t)IIB(y) < eat. PROOF. Let
be Cauchy in Y. Since II.IIy > 11.1 1, it is Cauchy in X. Let then {S(0)-'e-a"S(t)x,,}
x = limx in X. By definition of the Y-norm, the sequence is Cauchy in Cb := C6([0, oo); X ). Let U E C6 be its Cb-limit. The bound-
edness of S(O) implies that e-atS(t)x = S(0)u(t) E S(O)X for all t, so that S(0)-le-atS(t)x = u(t) E C6, i.e., x E Y, and clearly II x,, - xfl y - 0. Thus Y is indeed a Banach subspace of X.
If x = S(O)y E S(O)X, then S(0)-1e-a"S(t)x = e-"S(t)y E C6 by Property 3., so that S(0)X C Y. Also Ilxlly < IIe-atS(t)yllu <- MIIyII = 11X110-
Hence [S(0)X] is a Banach subspace of Y. If X E Y, then for each fixed s > 0, S(0)-1 e-a1S(t)S(s)x = e-atS(t + s)x E Cb,
(as a function of t), so that S(s)Y C Y. We may then define T(.) := S(0)-1S(.) on Y. Then for x E Y, IIT(t)xlly = sup
IIS(0)-'e-asS(s)S(0)-'S(t)xiI
9
at supl]S(0)-1e-a('+t)S(s+t)x{I < eatllxlIY d
This shows that T(.) is B(Y)-valued and IIT(t)IIB(Y) <- eat. The semigroup property of T(.) follows trivially from (1'). S(0)-le-atS(t)x The C0-property of T(.) follows from the uniform continuity of for x E Y. Indeed, given e > 0, there exists 6 > 0 such that for 0 < h < b, IIS(0)-le-a(t+h)S(t
+ h)x -
s(O)-le-ats(t)xllu
< C.
Therefore
IIS(0)-1e-atS(t)[T(h)x - x]II < S(0)-le-atS(t)xll eahllS(0)-1e-a(t+h)S(t+ h)x eahe + (eah - 1)IIxIIY
+(eah -1)IIS(0)-le-atS(t)xII 5
Taking the supremum over all t > 0, and letting then h -+ 0, we obtain
Iimsup IIT(h)x - xlly < e, h-o+ 81
and the arbitrariness of a gives the C0-property of T(.) in Y. Let then A' be the generator of T(.) in Y. For x E D(A') C Y and h > 0, h-1 [S(h)x - S(0)xJ =
S(0)h-1
[T(h)x - x]
S(0)A'x E s(o)x,
where the limit (as h -+ 0 is in Y, hence in X). Therefore x E D(A) and Ax = A'x E Y, by definition. This shows that x E D(Ay) and Ayx = A'x, i.e., A' C Ay. The Laplace transform L(a)x of S(.)x is well-defined for A > a, and calculations identical with those in the proof of Theorem 1.15 show that
L(A)(AI - A)x = S(0)x
(x E D(A)).
If (AI - A)x = 0, it follows that S(0)x = 0, hence x = 0, i.e., (AI - A) is injective on D(A), and therefore (Al - Ay) is injective (for A > a). Also, by Theorem 1.15, R(A; A') E B(Y), so that in particular AI - A' is surjective (for those A), and we saw above that Al - A' C AI - Ay. It follows that D(A') = D(Ay).IIII
82
B. SEMI-SIMPLICITY MANIFOLD (real spectrum case)
In this section, we generalize Theorem 1.49 to operators A with real spectrum, for which iA is not assumed to generate a Ca-group. Consider the Poissonian of A,
Let 11. 11 1 denote the L'(R)-norm (with respect to the Lebesgue measure).
2.9. DEFINITION. The semi-simplicity manifold for A is the set of all x E X such that R(t + iu; A)x = 0 for all t E R; and (2) sup,>0 I Ix*P(., s)xlll < oo for all x* E X*. (1) limb, 1 .
Note that Condition (1) is valid for all x E X when iA generates a C0-group; indeed, since R(t + iu; A) = iR(-u + it; iA), we have in that case II R(t + iu; A)I I = O(1/lul) for 0 # u E R. 2.10. LEMMA. If x E X satifies Condition (2), then s > 0, IIx*II = 1} < 00-
IIXIIA :=
PROOF. First, for s > 0 and x E X fixed, assume that x*P(.,s)x E V(R) for all x* E X*, and consider then the linear map V, : x* -+ x*P(.,s)x
of X* into L'(R). If xn -+ x* in X* and V,x,*, -+ f in L',, then by Fatou's lemma,
J
If (t) - x* P(t, s)xl dt =
jlirninflf(t) - x* P(t, s)xldt
r
If(t)-V,xnldt=0, 83
i.e., V,x* = f, so that V, is closed, hence bounded, by the Closed Graph Theorem. When Condition (2) is satisfied by x, the family of bounded operators V, satisfies supIIV,x*II1 < 00
(x* E X*).
8>0
By the Uniform Boundedness Theorem, sup,>0 I I V, I I < oo, which means precisely
that
IIxIIA < 00. IIII
2.11. THEOREM. Let A be an operator with real spectrum acting in the reflexive Banach space X, and let Z be its semi-simplicity manifold, normed by Then Z is a Banach subspace of X, invariant for any U E B(X) commuting with A, and there exists a spectral measure on Z, E(.), such that 1. for each b E 8(R), E(b) commutes with every U E B(X) which commutes with A; 2. D(Az) _ {x E Z; f,uE(ddu)x exists and belongs to Z}, and
Ax = fuE(du)x
(x E D(Az));
3. for all non-real ( E C and x E Z,
R((; A)x =
E(du)x.
Moreover, Z is "maximal-unique" in the following sense: if W is a linear manifold
in X and F(.) is a spectral measure on W with Property 3., then W C Z and F(b) = E(b)/W for all b E 8(R). Note that the "existence" of the integral in 2. is in the sense of Section 1.48, i.e., as the strong limit in X b r uE(du)x := lima,b 1. uE(du)x.
J
PROOF. For fixed x E Z and. x* E X*, the function x*P(t, s)x is an analytic function of t + is in C+, hence (complex) harmonic there, and satisfies sup IIx*P(., s)xlll <_ IIxIIAIIx*ll < 00 s>0
(by Lemma 2.10). Therefore there exists a unique regular complex Borel measure µ(.; x, x*) on 8(R) such that (1) IIxIIAIIx*II 84
and
x*P(t, s)x = fp(t - u, s),u(du; x, x*) (t E R, s > 0),
(2)
where p(t, s) _
a +e is the Poisson kernel for the upper half-plane (cf. [SW, pp. 49-53]). The uniqueness of the Poisson integral representation implies that for each b E 8(R), µ(b;.,.) is a bilinear form; hence, by (1) and the reflexivity of X, there exists a unique linear transformation
E(b):Z-4X such that
µ(b; x, x*) = x*E(5)x,
(x E Z, x* E X*)
(3)
and
(x E Z).
IIE(b)xII < IIxIIA
(4)
It follows from (3) and Pettis' theorem that E(.)x is a regular countably additive X-valued measure on 8(R), and we may rewrite (2) in the form
[R(t - is; A) - R(t + is; A)]x =
[
fit
1
t - is - u
- t +is - u ]E(du)x 1
(5)
for all t E R,s > 0, and x E Z. For X E Z fixed, consider the function F(C)x
_E(du)x
(C E C - R).
By (4), F(.)x is well-defined, analytic in C - R, and it follows from (1) that
IIF(()xll < III
CI
In particular, F(t + is)x -+ 0 as Isl -+ oo. By (5),
F(t - is)x - F(t + is)x = R(t - is; A)x - R(t + is; A)x
(6)
for all t E R,s > 0, and x E Z. Set G(.)x = F(.)x - R(.; A)x. This is an analytic function in C - IR, satisfying G(()x = G((- )x in its domain. Therefore, for each x* E X*, the functions x*G(()x and [x*G(()x]-(= [x*G((-)x]-) are both analytic in C - R. Hence x*G(.)x is constant there, and since it vanishes as I`a,C1 -4 oo (by (1) in Definition 2.9 , and by
our previous observation about F(.)x), it follows that G(.)x = 0 for all x E Z, i.e.,
R((; A)x = l C
1
u E(du)x
(7) 85
for all (EC-R andxEZ. We shall verify now that (7) (i.e., Statement 3. of our theorem) imply all the other statements of Theorem 2.11. Let U E B(X) commute with A. Then U commutes with R((; A) for all non-real (. If X E Z, then R(t + iu; A)Ux = UR(t + iu; A)x -+ 0 as lul -r oo, for all t E R. Also, with notations as in Lemma 2.10, we have for fixed x E Z,
IIx*P(.,S)Uxlll = Ilx*UP(.,s)xlli =
I
I
II
I
(x E Z).
I
(8)
Thus Z is U-invariant, and by (7),
(-
U
E(du)Ux = R((; A)Ux = UR((; A)x UE(du)x
for all non-real ( and x E Z. By the uniqueness property of the Stieltjes transform (cf. [WI), it follows that E(S)Ux = UE(6)x for all x E Z and S E 8(R) (which proves Statement (1) of the theorem). In particular, taking U = R(A; A) for \ E p(A), we obtain
R(A; A)E(R)x = E(R)R(A; A)x
£hm o
I (- u-E(du)R(,\; A)x
= lim (R((; A)R(A; A)x = lim
((x [R(A; A) - R((; A& = R(A; A)x,
since x E Z (cf. Condition (1)). Hence
E(R)x = x
(x E Z).
(9)
By (4), this shows in particular that Ilxll <- IIXIIA 86
(x E Z).
(10)
Therefore, if {xn} C Z is it is also 11.11-Cauchy (and of course, II.IIAbounded, say by the constant K). Let x = limn xn in X. Then by (7) and (1), II
IIR(t+iu;A)xnII s and therefore IIR(t + iu; A)xII < 2.9.
.
IUT
IuIIIA <
IuI,
Thus x satisfies Condition (1) in Definition
For each x* E X* and s > 0, x*P(., s)xn
x*P(., s)x pointwise, so that by
Fatou's lemma,
Ilx*P(.,s)xlll
n
and we conclude that x E Z. Also, given e > 0, let no be such that I Ixn - xm I IA < e for all n, m > no. Then I l x*P(., s)(xn - x,n)I II < e for all unit vectors x* E X*, ,q > 0, and n, m > n0. Letting m - oo, Fatou's lemma implies that IIxn - XIIA < e for all n > no, i.e., xn -, x in the We then conclude that (Z, II.IIA) is a Banach subspace of X.
-
ForxEZandAEp(A), 1
E(du)R(A; A)x = R((; A)R(A; A)x U
1
[R(A; A) - R((; A)]x =
_
1
[
1
1
J[
u
-
1
u
]E(du)x
E(du)x],
for all non-real (. By the uniqueness property of the Stieltjes transform,
E(b)R(A;A)x = 12
X
1 uX6(u)E(du)x,
(11)
for all x E Z, A E p(A), and b E 8(R) (where X6 denotes the characteristic function of b).
Since R(A; A) commutes with E(b), it follows from (11) that for x E Z,
R(A;A)E(b)x = f
1 uE(du)x -+ 0
when IAI -> oo. 87
Also for all unit vectors x * E X*, we have by (11)
IIx*P(.,s)E(b)xlll =
8 j Ix* Jb (t _U)2 + s2 E(du)xldt
7r
x*Exl(du) = Ix*ExI (b) <- Il Is < J6I (t - )2 + s2 Hence E(6)Z C Z and
x, x*)II <_ IIxIIA.
I
(x E Z, b E 13(R)).
II E(b)xII A <- IIxIIA
(12)
This shows that E(b) E T(Z); in fact E(b) E B(Z, II.IIA), with operator norm < 1. By (11) and (7), since E(b)x E Z for X E Z,
f
1 ux6(u)E(du)x = R(A;A)E(b)x = f
The uniqueness property of the
E(r)E(b)x =
1 uE(du)E(b)x.
Stransform implies that
f xv(u)X6(u)E(du)x = E(a n b)x
for all o, b E 13(R) and x E Z. We have thus shown that E is a spectral measure on Z. We prove now Statement 2. in the theorem. The argument yielding (4) in the proof of Theorem 1.49 shows that
D(Az) = R(A;A)Z
(13)
for anyAEC - R. Let x E D(Az). Write then x = R(A; A)y for a fixed non-real A and a suitable
yEZ. For-oo
JE(d)x =
IuE(du)R(A; A)y = f 6
L A_
A
u u E(du)y
uE(du)y
as a -r -oo and b -> oo. Thus f1 uE(du)x exists. Writing Auu = relation shows that uE(du)x = A it 88
rffi
1
u E(du)y
- E(R)y
A
A-u
-1, the last
= AR(a; A)y - y = AR(A; A)y = Ax E Z
(14)
(since x E D(Az)). Thus D(Az) C Z1, where Z1 denotes the set on the right of Statement 2. of the theorem. On the other hand, if x E Z1, denote z = f! uE(du)x; we have z E Z, and for non-real A, we obtain from (11) b
R(A; A)z = lim
(a,b)
rb
lim
(a,b) Ja
_J
J
uR(A; A)E(du)x
u
A-u E(du)x 1
U
= 12
A-u
E(du)x
u E(du)x - x = AR(A; A)x - x.
Hence
x = R(A; A)[Ax - zI E R(A; A)Z = D(Az),
so that D(AZ) = Z1. By (14), Ax = fauE(du)x for all x E D(Az). Finally, let W and F be as in the statement of the theorem. For x E W, the representation 3. (with F) of the resolvent implies (1) in Definition 2.9. Also, for all x* E X*, IIx*P(., s)xIIl = s
-s
f
I x*
dt
(t - u)2 + sz
J (t - u1
+ S2
F(du)xjdt
I x*Fxl (du) = I x*FxI (R),
so that (2) in Definition 2.9 is satisfied as well, i.e., x E Z. Thus W C Z, and the uniqueness property of the Stieltjes transform implies that F(.)x = E(.)x for xEW.IIII The discussion preceding Corollary 1.50 yields the following
2.12. COROLLARY. Let A be an operator with real spectrum, acting in the reflexive Banach space X, and let Z be its semi-simplicity manifold. Then Z = X if and only if A is a scalar-type spectral operator. When this is the case, E is the resolution of the identity for A.
We consider now the operational calculus r induced by E(.), the spectral measure on Z, as defined in Section 1.48. In the following, the space Z is normed by 11-11A89
2.13. THEOREM. The operational calculus r is a norm-decreasing algebra homomorphism of B(R) into B(Z). Moreover, for each h E B(R), r(h) maps D(AZ) into itself, and
Ar(h)x = r(h)Ax = juh(u)E(du)x
(x E D(Az)).
(1)
Note that the "improper" integral appearing in (1) is r(uh(u)), defined as usual for functions in Bio,(R).
PROOF. For x E Z and h E B(R), we have by (11) (in the proof of Theorem 2.11) R(A;A)r(h)x = f h(u)R(A; A)E(du)x = fm (u) E(du)x.
Therefore
IIR(A; A)r(h)xll <
IIhIII.IIIIA I
(2)
(A EC-R).
In particular, r(h)x satisfies 2.9 (1). By (2), for all x* E X*, II x*P(., s)r(h)xII i = 11x*
<
j(. - u, s)h(u)E(du)xI I1
f f lh(u)Ip(t - u, s)Ix*Exl(du)dt < IIhIIo. f f p(t - u, s)dtlx*ExI(du)
= IIhII.Ix*Exl( ):5 IIhII.IIxIIAllx*II, where I x*Exl denotes the variation measure of x*Ex = µ(.; x, x*Thus ). r(h)x satisfies 2.9 (2) as well, i.e., it belongs to Z for all x E Z, and moreover Ilr(h)xIIA <- IIhIIoIIXIIA
(h E B(R),x E Z).
(3)
This establishes that r is norm-decreasing from B(R) into B(Z). Since E is a spectral measure on Z, it follows that r is multiplicative on the simple Borel functions, hence on B(R) as well. Next, let x E D(AZ). Then x = R(.1; A)y for A non-real and y E Z. Therefore, for any h E B(R),
r(h)x = R(A; A)r(h)y E R(A; A)Z = D(AZ),
i.e., r(h)D(AZ) C D(Az). 90
In particular, for h, x, y as before, by multiplicativity of r on B(R), the limit b
lim f uE(du)r(h)x = lim
(a,b) a
J
b
uh(u)E(du)x
(a,b) a
exists in X and belongs to Z, and equals AT(h)x (by Theorem 2.11). Finally, we observe that the bounded operator AR(A; A) = .\R(A; A) - I commutes with E, hence with r(h), and therefore Ar(h)x = AR(7; A)r(h)y = r(h)AR(A; A)y = -r(h)Ax.III I
Taking in particular the functions ht (u) = eitu (t, u E R), let
T(t) = r(ht)
(t E R).
Then T(.) is a group of contractions in Z, which map D(AZ) into itself. It is continuous with respect to the X-norm (as follows at once by dominated convergence), but not necessarily with respect to the Z-norm. Consider the continuous functions on R eitu
kt(u)
-1
itu
(t,u 36 0),
and kt(0) = 1 (for t 54 0). We have jkt(u)I _< 1 and kt(u) --+ 1 as t -+ 0 (for all u E R). For x E D(Az), we apply (1) (in Theorem 2.13) and the Dominated Convergence Theorem for vector measures:
t-1 [T(t)x - x] = i i
J
J
ukt(u)E(du)x = ir(kt)Ax
kt(u)E(du)Ax -+t_o iAx.
IL
(limit in X). Also s-1 [T(t
+ s)x - T(t)x] = s-1 [T(s)T(t)x - T(t)x] -+,-o iAT(t)x
(limit in X), since T(t)x E D(AZ) for x E D(Az), by Theorem 2.13. We formalize the above discussion in
2.14. COROLLARY. T(.) is a group of contractions in Z, continuous in the X-topology on Z, and leaving D(AZ) invariant. Moreover, in that topology, the 91
generator of T(.) coincides with iA on D(AZ), and u := T(.)x solves the ACP on R
u' = iAu,
u(0) = x
for x E D(Az). The basic properties of the operational calculus r on Bloc (R) are collected in the following
2.15. THEOREM. (i) T(Ah) = Ar(h) (0 0 A E C, h E Bloc( (ii) D(r(h) + T(g)) = D(7-(h + g)) fl D(r(g)), and
);
T(h + g)x = r(h)x + r(g)x for all x E D(r(h) + r(g)) and h, g E 1Bioc(R);
(iii) E(b)D(r(h)) C D(r(h)), and
r(h)E(b)x = E(b)r(h)x = r(hXb)x
(x E D(r(h)),
for all compact b C R and h E 1Btoc(R);
(iv) D(r(h)r(g)) = D(T(hg)) fl D(7-(g)), and r(h)r(g)x = r(hg)x
for all x E D(r(h)r(g)) and h, g E 3toc(R). PROOF. (i) is trivial. In the following, h, g will denote arbitrary functions in Bloc (1R ).
Proof of (ii). Let x E D(r(h) + r(g)) := D(r(h)) fl D(r(g)). Then rb
lim a,b
Ja
h(u)E(du)x
exists in X and belongs to Z, and similarly for g. Therefore lima,b of the sum of the two integrals (i.e., of fa (h + g)(u)E(du)x) exists and belongs to Z. Thus x E D(r(h+g)) and T(h+g)x = r(h)x+r(g)x. On the other hand, if x E D(r(h+ g)) fl D(r(g)), then writing fa h(u)E(du)x = fa (h + g)(u)E(du)x - fa g(u)E(du)x, we see that x E D(r(h)), so we have the wanted equality of domains. (iii) Let b C R be compact, and x E D(r(h)). In particular, x E Z, and therefore E(6)x E Z. Since r is multiplicative on B(R), we have b
IL 92
h(u)E(du)E(b)x = r(hx[a,b])r(Xb)x = r(hX[a,b]nb)
= f h(u)X[a,b]nb(u)E(du)x.
In the last integral, the integrand is majorized by the bounded function Ihj, and converges pointwise to hX6 when a --+ -oo and b -+ oo. By dominated convergence
for vector measures, it follows that lima,b of that integral exists in X and equals T(hX6)x E Z (since hX6 E 3(R)). Hence E(b)x E D(T(h)) and T(h)E(b)x = r(hXb)x.
We also have r(h)x E Z, because x E D(T(h)). Therefore, by (11) in the proof of Theorem 2.11, we have for A E C - R 1
u E(du)T(h)x = R(A; A)T(h)x
A
= R(A; A) lim Jab h(u)E(du)x =1a m a,b
b
= lim a,b
h(u)
J '\ -u E(du)x
= J
/a
6
h(u)E(du)R(,\; A)x
h(u)
1A-u
E(du)x.
Hence E(b)T(h)x = f h(u)X6(u)E(du)x = 7-(hX6)x,
by the uniqueness property of the Stieltjes transform. This completes the proof of (iii).
(iv) Let X E D(T(h)T(g)), i.e., x E D(T(g)) and r(g)x E D(T(h)). By the multiplicativity of r on 3(R) and by (iii), 6
h(u)g(u)E(du)x = a
= T(hX[a,6])T(9X[a,b])x = T(hX[a,b])E([a, b])T(9)x
= T(hX[a,b])T(9)x -i r(h)T(g)x
when a --+ -oo and b --+ oo, since r(g)x E D(T(h)). Hence X E D(7-(hg)) and T(hg)x = T(h)T(g)x. In particular, D(T(h)T(g)) C D(T(hg)) fl D(T(g)). On the other hand, if x belongs to the right side of the last relation, we have in particular T(g)x E Z. Therefore, by the multiplicativity of r on 3(R) and by (iii), we have T(hX[a,b])7-(9)x = 7(hX[a,b])T(9)x
= T(hX[a,b])E([a, b])T(9)x = T(hx[a,b])T(9X[a,b])x 93
= r(hgX[a,b])x -'a,b r(hg)x E Z,
because x E D(r(hg)). Hence r(g)x E D(r(h)).IIII We show next that r operates in the desired way on polynomials. If p(u) = Eocrkuk with n > 1 and a # 0, we define as usual
p(Az) := EocrkAz = EoakAk restricted to
D(p(Az)) = D(A')
{x E D(AZ 1 ); AZ 1x E D(Az)}.
2.16. THEOREM. 1. D(p(Az)) = nk=1 D(r(uk)); 2. p(Az)x = r(p)x for all x E D(p(Az)). PROOF. We first prove the following
LEMMA. For n = 1, 2,... and any A E p(A),
(i)D(AZ)={xED(A");AkxEZ,k=0,1,...,n}; (ii)D(AZ) = R(A; A)" Z.
PROOF (of Lemma). (i) is easily verified by induction. The validity of (ii) for n = 1 was observed before. Assume (ii) for n - 1 (where n > 2). Since Z is R(A; A)-invariant,
R(A; A)"Z C R(A; A)"-1Z = D(Az 1), by the induction hypothesis. Let x E R(A; A)" Z; then x E D(AZ 1), and writing x = R(A; A)"y with y E Z, we have
A"-1x = [AR(A;
A)]"-1 R(X; A)y
= [AR(X; A) - I]"-1R(X; A)y
=-R(A; A)[AR(A; A) - I]"-1y E D(A) f1 Z.
Hence x E D(A") and A"x = [AR(A; A)]"y = [AR(A; A) - I]"y E Z.
By (i), this shows that x E D(AZ).
On the other hand, if x E D(AZ), then by (i), x E D(An) and Akx E Z for (,\I - A)"x E Z, and x = R(A; A)"y E R(A; A)"Z (for
k = 0,..., n. Therefore y
A E p(A)), and (ii) follows for n.IIII 94
Back to the proof of the theorem, let x E D(A") and fix A E p(A). By the lemma, write x = R(A; A)"y with y E Z. Applying (11) in the proof of Theorem 2.11 repeatedly, we obtain E(du)x = E(du)R(A; A)k [R(A; A)"-k y] = (A
- u)-kE(du)[R(A;
A)"-ky]
for k = 1, ..., n, since R(A; A)" -k y E Z.
Hence, for -oo < a
J
u
ukE(du)x -ia,b
A
u)kE(du)[R(A;
u
A)"-ky]
A)"-ky],
,u)kE(du)[R(',i (1) J since [u/(A - u)]k is a bounded function of u on R. Thus f1 ukE(du)x exists (in X) and equals the integral in (1), which belongs to Z for k = 1, ..., n, by Theorem 2.13. This proves the inclusion c in Statement 1. of the theorem. (A
Next, let x belong to the set on the right of 1. For each k = 0,..., n, denote zk =
J
ukE(du)x (E Z).
By (11) in the proof of Theorem 2.11, we have for k = 1,...,n, R(A; A)zk = lima,b fQ ukR(A; A)E(du)x u E(du)x = lima & f.' uk-I A-u = lima b fa uk-i(j - 1)E(du)x = lima,b[AR(A; A) - I) fa uk-'E(du)x
= [AR(A; A) - I]zk-,. Therefore
zk-1 = R(A; A)(Azk-l - Zk) E D(A) n Z, and
Azk-i = [.AR(A; A) - I](Azk-1 - zk) = \R(.1; A)zk - [.AR(.1; A) - I]zk = zk,
for k = 1, ..., n. Since zo = x, it follows from the above recursion that x E D(A") and Akx = Zk E Z for k < n, i.e., x E D(AZ) by Part (i) of the lemma. This proves Statement 1. of the theorem, and also the relation Akx =
(x E D(Akz ,
k = 1, 2,...),
which clearly implies Statement 2.1111 95
C. SEMI-SIMPLICITY MANIFOLD (Case R+ C p(-A))
We generalize the construction of the semi-simplicity manifold to operators -A with spectrum in a half-plane, say in the closed left half-plane, to fix the ideas. Actually, all we need for our construction is that Ig+ := (0, oo) be contained in the resolvent set of -A. Let then
R(t) := R(t; -A)
(t > 0),
and
S:= AR(I - AR). The function S(t) = tR(t) [I - tR(t)] is a well-defined B(X)-valued function on R+, and for all k = 1, 2,..., the powers Sk are of class C°°. In the following discussion, the Ll (lR
,
tt )- norm is denoted by I
l
IIi
The Beta function is
B(s,t) :=
r(s)r(t) r(s + t)
(s, t E l[8+).
2.17. DEFINITION. Let -A be an operator with (0, oo) C p(-A), and let S be the operator function defined above. The "semi-simplicity manifold" for -A is the set Z of all x E X such that sup kEN
Ilx*Skxlli
B(k,k)
< oo
for all x* E X*. Using the Closed Graph Theorem, Fatou's lemma, and the Uniform Boundedness Theorem as in the proof of Lemma 2.10, we obtain
2.18. LEMMA. For all x E Z, Ilxllz := sup{ 96
l1B(k
k
klll
,
llxll; k E N, llx*II = 1) < oo.
2.19. LEMMA. The space Z :_ (Z, is a Banach subspace of X, invariant for any U E B(X) commuting with A. and IIUIIB(z) IIUIIB(x) PROOF. The proof is analogous to the one we gave for the real-spectrum case (see proof of Theorem 2.11).
2.20. THEOREM. Let -A be an operator in the reflexive Banach space X, whose resolvent set contains the axis R+, and let Z be its semi-simplicity manifold. Then there exists a spectral measure on Z,
E(.) : B(R') -+ T(Z), such that 1. for each b E B(R+), E(5) commutes with every U E B(X) which commutes with A; 2. (i) D(Az) = {x E Z; limb-0 f o'6 sE(ds)x exists in X and belongs to Z}, and (ii) Ax = f°° sE(ds)x (x E D(Az)), where the last integral is defined as the limit in (i); 3. R(t)x = f0'00 1-+'-;E(ds)x (x E Z,t > 0). Moreover, Z is maximal-unique relative to Property 3., in the sense of Theorem 2.11.
PROOF. Let Lk be the Widder formal differential operators
ckMk-ID2k-lMk
Lk
(k E N),
where M : f(t) -. tf(t);
D : f -> f'
are respectively the "multiplication" and the differentiation operators acting on functions of t E R+. The constants Ck are given by cl = 1 and Ck
__
(k > 2)
.
r(k - 1)r(k +T)
By Leibnitz' rule, j)-'
Lk =
ci = 1 and ck =
(-1)k-1 B(k
7where
- 1, k + 1)-1 for k > 2. Since
Dk+i-1(x*Rx) = (-1)k+i-1r(k+ j)x*Rk+ix, 97
we have
Lkx*R(t)x = c',r't-lx*(tR)I 3 ..0
(k) (-tR)'x = ckt-lx*Sk(t)x,
where cs = 1 and ck = B(k - 1, k + 1)-1 for k > 2. Therefore, for x E Z and x* E X*, 1000 ILk(x*Rx)ldt = CkIIx*Skxlll
IIxIIZIIx*II,
(1)
trivially for k = 1, and because B B(k+1 = kkl < 1 for k > 1. We now rely on the following complex version of Widder's theorem (cf. [W; Theorem 16, p. 361]): Let f be a C°° complex function on R+, such that
kEN
0
t f (t) exists, and there exists a unique complex regular Then the limit c = Borel measure p on R+ such that Ilkll < 2K + lcl and
f(t) = f
0
)
(t E
t(+s)
Taking f = x*Rx with x E Z and x* EX*fixed, wehave K
Let MZ := 2IIxilz + Hx. By Widder's theorem, there exists a unique complex regular Borel measure p(.; x, x*) such that Mzllx*II and
x*R(t)x = f o
µ(ds; x, x*)
t+s
(t E II8+)
'
forallxEZandx* EX*. This implies in particular that
lltR(t)xll < M.
(t E R+).
(2')
The uniqueness of the Stieltjes transform implies that for each fixed 6 E B(R+) and x E Z, µ(6; x,.) is a continuous linear functional on X*, so that, by reflexivity of 98
X, there exists a unique function E(.)x : 13(R+) - X (for each fixed x E Z) such that p(.;x,x`) = x*E(.)x (x" E X*). Necessarily, E(b) is a linear operator with domain Z, and
(b E B(R ),x E Z).
IIE(b)xHI : M.
By Pettis' theorem, E(.)x is a strongly countably additive vector measure, and
R(t)x = °° E(ds)x
(t > O' X E Z).
t+s
This is Property 3., which corresponds to (7) in the proof of Theorem 2.11. As in the latter case, we shall see that it implies that E is a spectral measure on Z satisfying Properties 1. and 2. of our theorem. Property 1. is an immediate consequence of the uniqueness property of the Stieltjes transform. Taking then, in particular, U = R(u) for u > 0 fixed, we obtain for x E Z
R(u)E(R+)x
t
E(R+)R(u)x
= limtR(t)R(u)x = lim
l!'0010M t
t uR(u)x - lim
sE(ds)R(u)x
tR(t)x = R(u)x,
t by the resolvent equation and (2'). Since R(u) is one-to-one, it follows that E(1R+)
I/Z. For t, u > 0, t # u, and x E Z, we have by Property 3., the resolvent equation, and the fact that R(u)x E Z, t+s
E(ds)R(u)x = R(t)R(u)x
luu
t+s[u--sE(ds)xJ.
t+sJE(ds)x=J
Fs t By the uniqueness of the Stieltjes transform,
E(ds)R(u)x =
u+s
E(ds)x,
(3)
and inductively,
E(ds)R(u)kx =
s) k
(u
+
E(ds)x, 99
for all k E N, u > 0, and x E Z. Therefore
E(ds)p(R(u))x = p(u
+
s)E(ds)x
for all polynomials p. In particular,
E(ds)Sk(u)x = (u + s)k [1 -
U s]kE(ds)x u+
(us)k
E(ds)x, (u + s)2A
for all u > 0, k E N, and x E Z. Property 1. for U = Sk(u) implies then that
x*Sk(u)E(b)x = f
(
k (+ ))2k x*E(ds)x.
u
By Tonelli's theorem, IIx*SkE(b)xIIi
JaJo
(
16 /
(+ )kzk du
B(k,k)Ix*ExI(6)
(1 +t)2k
< B(k,k)MxIIx*II,
for all x* E X*, k E N, b E 13(R+), and x E Z. Therefore E(b)x E Z for x E Z (i.e.,
E(b) E T(Z)), and (x E Z,b E B(R+).
IIE(b)xIIz <_ M.
(4)
It now follows from Property 3. with the vector E(b)x E Z (whenever x E Z), Property 1. (with U = R(u) for any u > 0), and (3): R(u)E(b)x = 10"0 u + E(ds)E(b)x s
=EbRux= () ( )' J
6
()sEdsx ( )
,
therefore, by the uniqueness of the Stieltjes transform,
E(u)E(b)x = f 100
(s)6(s)E(ds)x = E(a n b)x
for all a, b E 13(R+). In conclusion, E is a spectral measure on Z.
Since D(AZ) = R(t)Z for any t > 0, write any given x E D(Az) as x = R(t)y for a fixed t > 0 and a suitable y E Z. Then b
sE(ds)x = Job
i
t+
sE(ds)y
00
--4b-00 0
=
E(ds)y
t+s
f[i - t+s t ]E(ds)y [I - tR(t))y(E Z) = AR(t)y = Ax.
If Z1 denotes the set on the right of Property 2(i), we obtained that D(Az) C Zi ;uid Property 2(ii) is valid on D(AZ). On the other hand, if x E Z1, denote the limit in Property 2(i) by z E Z. Then for any t > 0, R(t)z = slim
sR(t)E(ds)x =
J
bli/
0
0
J'
t s s
t + s E(ds)x
E(ds)x = x - tR(t)x.
Therefore x = R(t)[z + tx] E R(t)Z = D(Az), so that D(AZ) = Zi. Suppose now that W is a linear manifold in X and F is a spectral measure on lT with Property 3. of E. Fix x E W. Differentiating repeatedly, we obtain
Rk(t)r =
J0
(I t
s )kF(ds)x
for all k = 1, 2, ... and t > 0. Therefore (5)
for all polynomials p and t > 0. In particular, Sk(t)x = {tR(t)[1
- tR(t)))k = J{J___(1 -
t
=Jo' (ts)k2k (t + s) P(d,),. 101
Therefore, for all x* E X* and k E N, we have by Tonnelli's theorem I IB(k, k)1 < B(k,
B(k, k)-1
k)-1
00
/
F
=
(00 (t(+ s)2k Ix*FxI(ds)dt
(1 + u)2k uu I x*FxI (ds)
IIx*FxII < oo.
Hence x E Z, i.e., W C Z. Also, for all x E W C Z, we have
R(t)x =
t
J
1
s F(ds)x
= or t + s E(ds)x
(t > 0),
and therefore F(b)x = E(b)x for all b E B(R+), by the uniqueness property of the Stieltjes transform. I
I
II
As before, the important special case Z = X gives the following result.
2.21. THEOREM. Let -A be an operator with R+ C p(-A), acting in the reflexive Banach space X, and let Z be its semi-simplicity manifold. Then the following statements are equivalent: (a) Z = X. (b) K := sup11 11=1 IIxIIz < oo. (c) A is spectral of scalar type, with spectrum in [0,00). PROOF. Since I I I < I I Iz, the equivalence of (a) and (b) follows from the Closed Graph Theorem. Assume now (a). By (2) and the Uniform Boundedness Theorem, I
I
H := sup IItR(t)II < oo, o
and so M. < MIIxII
(x E X),
where M := 2K + H. It now follows from (4) that E(b) E B(X) (actually, II E(b)II s(x) < M) for all n E B(R+). Therefore E is a spectral measure in the usual sense, and Property 2. of Theorem 2.20 just states that A is a scalar-type spectral operator with resolution of the identity E, and then necessarily a(A) C [0, 00) (cf. [DS,III]). We show finally that (c) implies (a) (even without the reflexivity hypothesis). Let E be the resolution of the identity of the scalar-type operator A with spectrum 102
in (0, oo). Then E is a spectral measure on X satisfying Property 3. of the theorem (on X). By the maximality property of Z, we have necessarily X = Z.I The operational calculus results contained in Theorems 2.13, 2.15, and 2.16, and in Corollary 2.14 (with the obvious modification), are generalized in a routine way to the present situation (i.e., with the assumption R+ C p(-A)). Observe that Theorem 2.20 applies in particular to the case where -A generates a C,,-semigroup of contractions, T(.). In that case, by (5) (for the spectral measure on Z, E(.)), we have
[t R(t
)]' x
f
t i 1 s]"E(ds)x = J
° [E(dt)x n
ao
-In-oo
Jo
e-t'E(ds)x
for all x E Z and t > 0, by the Lebesgue Dominated Convergence Theorem for vector measures. By Theorem 1.36, it follows that
T(t)x =
j&t8E(ds)x
(x E Z, t _> 0),
that is, T(.)x is the Laplace-Stieltjes transform of the vector measure E(.)x for all
xEZ.
We may consider the more general question of constructing a maximal Banach subspace on which an arbitrary family of closed operators is the Laplace-Stieltjes transform of a vector measure. This will be done in the next section.
103
D. LAPLACE-STIELTJES SPACE
Denote by C the Laplace transform, (t > 0),
(,C0)(t) :=
acting on a space of functions to be specified as we proceed. We may choose for example the space C-:= C°°(1[8+) of all complex Coo-functions with compact support in R+. Let K(X) denote the set of all closed operators acting on X.
2.22. DEFINITION. Let F : [0, oo) --> K(X) be such that F(O) = I. The "Laplace-Stieltjes space" for F is the set W of all x in the "common domain" of F,
D:= n D(F(s)), s>0
such that F(.)x is strongly continuous on [0, oo), and IIxIIw := sup{II
J0
0(s)F(s)xdsll; 0 E C°°, II,COII. = 1)
is finite.
2.23. THEOREM. Let W be the Laplace-Stieltjes space for F, as in Definition 2.22, normed by 11.11w. Then W is a Banach subspace of X, and in case X is reflexive, there exits a uniquely determined function E on B([0, oo)) into the closed unit ball B(W, X), of B(W, X), such that (i) for each x E W, E(.)x is a regular countably additive X-valued measure, and
(ii) F(t)x = fo e-'*E(ds)x for all t > 0 and x E W. (iii) If T E B(X) leaves the common domain D invariant and commutes with F(s)ID for all s > 0, then T E B(W) (with IITIIB(w) S IITIIB(x)) and TE(b) _ E(S)T on W, for all 6 E B([0, oo)). Moreover,the pair (W, E) is maximal-unique in the following sense: if (Y, E') is
a pair with the properties of (W, E) (not including (iii)), then (Y, E') C (W, E), 104
meaning that Y is continuously imbedded in W and E'(b) = E(b)ly for all 5 E a([0, oo)).
The proof depends on a general criterion for belonging to the range of the adjoint T* of a densely defined operator T.
2.24. LEMMA. Let E, .F be normed spaces, and let T : £ -+ F be a densely defined linear operator. Let u* E 6* and M > 0 be given. Then there exists v* E D(T*) with lIv*II < M such that u* = T*v* if and only if lu*ul < MllTull
(u E D(T)).
(*)
PROOF. If u* = T*v* with v* E D(T*) such that IIv*ll < M, then for all u E D(T), lu*ul = I(T*v*)(u)I = Iv*(Tu)I <_ IIv*II.IITuII <_ MllTull
Conversely, if (*) is satisfied, define
7r:ran (T)-+ C by
ir(Tu) = u*u
(u E D(T)).
If u, u' E D(T) are such that Tu = Tu', then by (*),
lu*u - u*u'l = Iu*(u - u')I < MIIT(u - u')II = 0, so that 7r is well-defined. It is linear and bounded on ran(T), with norm < M (by (*)). By the Hahn-Banach Theorem, there exists v* E F* such that IIv*II < M and 'U
ran (T) = 7r
Thus
v*(Tu) = u*u
(u E D(T)).
This shows that v* E D(T*) and T*v* = u*.IIII
Note that for T E B(£,.F), Condition (*) needs to be required only for all u in a dense subset of E. We apply the lemma to the Laplace transform: 105
2.25. LEMMA. A function h : [0, oo) is the Laplace-Stieltjes transform h(t) = f000 e_tsy(ds) of a regular complex Borel measure p on [0, oo) with total variation norm IIuII < M if and only if it is continuous and
If'
h(t)O(t)dtl < MIIL011.
for all 0 E C'°(R+)). PROOF. If h is a Laplace-Stieltjes transform, it is certainly continuous, so that the integrals f000 h(t)cb(t)dt make sense for all 0 E C°°, and I
f'
h(t)O(t)dtI = I
j(L4')(s)z(ds)i
0
< ull.IIL011- < MIIL511.. For the converse, apply Lemma 2.24 to the operator L L1([0,oo)) -' Co([0,oo), where Co([0, oo)) denotes the space of all complex continuous functions on [0, oo) vanishing at oo. Its adjoint space is the space M([0, oo)) of all regular complex Borel measures on [0, oo) with the total variation norm, and L* M([0, oo)) -' LO°([0, oo))
is the Laplace-Stieltjes transform (by a simple application of Fubini's theorem). If the given continuous function h satisfies our lemma's condition, then since 111411oo <_ 110111:=110IIL'c[o,oo)),
we have necessarily Ilhll,,. < M, i.e., h E (L1)*, and by Lemma 2.24, there exists
p E M([0,oo)) with IIuII < M such that h = £ p (everywhere, by continuity of both sides).II11
2.26. LEMMA. The Laplace-Stieltjes space W for F is a Banach subspace of X, and if T E B(X) leaves V invariant and commutes with each F(s)1D, then T E B(W) (with B(W) -norm < IITII). PROOF. Clearly, W is a linear manifold in X, and 11.11w is a semi-norm on W. If x E W, then
III' q5(t)F(t)xdtll < IIxIIwII C0IIo(<- IIxIIwIl41I1) 106
(1)
for all ¢ E C"°, hence necessarily sup
(x E W).
IIF(t)xII S IIxIIw
(2)
In particular, since x = F(O)x, IIxll 5 IIxIIw, and therefore W :_ (W, II.I1w) is a normed subspace of X. We prove its completeness. Let {xn} be Cauchy in W (hence in X), and let x be its X-limit. For e > 0 given, let no E N be such that
Ilxn - xmIIW < efor all n,m>n,. Then for all qE C°° andn,m>n0, II f 0(t)F(t)(xn - xm)dtll <- e11,C011. <- 6!10111.
(3)
0
Hence F(t)(xn - xm )I I C e
II
(n, m > n,; t > 0),
that is, {F(t)xn} is uniformly Cauchy in X on [0, oo). Let then g(t) := limn F(t)xn
(limit in X, uniformly in t E [0, oo)). Since xn E D(F(t)) and xn --+ x in X, it follows that x E D(F(t)) and F(t)x = g(t) for each t > 0, because F(t) is a closed operator. Thus F(.)xn -* F(.)x uniformly on [0, oo), so that F(.)x is continuous on (0, oo) and (0 E C°°)
f 0r 0(t)F(t)xndt -+ f0 q(t)F(t)xdt strongly in X. Letting n - oo in (3), we obtain
III' Hence
(m >
- xm)dtll <<
o;
E C°°).
x - xm I l w < e for all m > n,; therefore x - xm E W (and so x =
(x - xm) + xm E W), and I l x - xm I I w --+ 0 when m -+ oo. Thus W is complete.
If T is as in the statement of the lemma, then for each x E W, we have Tx E D, F(.)Tx = T[F(.)x) is continuous, and for all ¢ E C°°,
'
it f ¢(t)F(t)Txdtll = IITJ0 0(t)F(t)xdtlI 0
<_ 117'IIB(x)IIxIIwII1C0II-,
so that TW C W and IITIIB(w) <_
PROOF OF THEOREM 2.23. For each x E W and x* E X*, x*F(.)x is a
' if
complex continuous function on (0, oo) satisfying ¢(t)[x*F(t)x)dtI <- IIxIIwIIff-0II.IIx*Il
(0 E C,°). 107
By Lemma 2.25, there exists a unique p = µ(.; x, x*) E M([0, oo)) such that
II,t(.;x,x*)II < IlxIIwIIx*II
(4)
x*F(t)x = 1 M e-t'µ(ds; x, x*)
(5)
and 0
for all t > 0, x E W, and x* E X *. The uniqueness of the representation (5) implies the bilinearity of µ(b; ., .) for each fixed b E 8([0, oo)), and since X is reflexive, it follows from (4) that there exists a unique E(b) E B(W, X), such that µ(b; x, x*) = x*E(b)x
(6)
for allxE W, X* EX*, and6EB([0,o0)). Statements (i),(ii),(iii) of the theorem follow now from (6), Pettis' theorem, (5), and Lemma 2.26.
Let (Y, E') be as in the statement of the theorem. Property (ii) for Y contains implicitly the fact that Y is contained in the common domain V of F(.). Also if x E Y, F(.)x is X-continuous on [0, oo) (as the Laplace-Stieltjes transform of the vector measure E'(.)x), and by Lemma 2.25,
< sup{IIx*E'(.)xII; IIx*II = 1} := Ky < oo, that is, x E W. Since Kx < K I Ix I I Y for a suitable finite constant K, the inclusion Y C W is topological. The fact E'(.) = E(.)Iy follows from the uniqueness property of the Laplace-Stieltjes transform of regular measures. IIII
We shall apply Theorem 2.23 to semigroups of closed operators
2.27. DEFINITION. The family {T(t); t > 0} of closed operators is called a semigroup of closed operators if T(0) = I, and T(s)T(t)x = T(s + t)x for all x in the common domain D of T(.).
2.28. THEOREM. Let T(.) be a semigroup of closed operators on the reflexive Banach space X, and let W be its Laplace- Stieltjes space. Then there exists a uniquely determined spectral measure on W,
E : B([O,oo)) -4B(W)I, such that
00
T(t)x 108
fe8E(ds)x
(t >_ 0; x E W).
Moreover, E(.) commutes with every operator U E B(X) such that UD C V and UT(.)x = T(.)Ux for all x E D. Also, if
T : h --r f r h(s)E(ds)x, 0
then T is a norm-decreasing algebra homomorphism of B([0, oo)) into B(W).
PROOF. Let (W.E) be associated with the family F(.) = T(.) as in Theorem 2.23. Let X E W. Then for each s > 0, T(.)T(s)x = T(.-fs)x is strongly continuous on [0, oo), and for all 0 E C°° := C°°(R+), denoting 0,(u) :_ ¢(u - s), we have II
f
q5(t)T(t)[T(s)x]dtll = II
0
f
b,(u)T(u)xdull
llX1lWIIC'1II. < IIx11w114IIoo,
since (Cq,)(t) = e-t'(C)(t), so that llC0,1I, < II,C011OO. Thus T(s)x E W, and IIT(s)xllw < llxllw This shows that T(.) is a semigroup of contractions on the Banach subspace W of X (continuous with respect to the X-norm!). By Theorem 2.23, for x E W and t > 0, T(t)
J0
e-"`E(du)x = T(t)T(s)x 00
= T(t + s)x =
e-t"e-"E(du)x.
0
By linearity, this shows that
T(t)r(h)x =
J0
°O e-t°h(u)E(du)x,
(1)
for all x E W and h(u) = E C and sj > 0. These finite linear combinations are dense in Cb := Cb([0, oo)), the space of all bounded continuous complex functions on [0, oo). If h E Cb, pick a sequence hk of such combinations such that hk -+ h uniformly on (0,oo). Then for each t > 0, r(hk)x E D(T(t)), T(hk)x --4k T(h)x, and by (1), T(t)T(hk)x = f000 e-t"hk(u)E(du)x --*k f0,30 e-t"h(u)E(du)x. Since T(t) is closed, it follows that r(h)x E D(T(t)) and (1) is valid for all h E C6. This is easily extended to h E 16([0, oo)). Indeed, since the vector measure E(.)x is regular (for each x E W), there exists a sequence {hk} C C6 such that llhkll°o = llhllOO and hk -F h pointwise almost everywhere with respect to E(.)x. By the Lebesgue Dominated Convergence Theorem for vector measures
(cf.[DS-I, p. 328]), T(hk)x -+ r(h)x in X, r(hk)x E D(T(t)) (for each t > 0), and 109
by (1) for hk, T(t)T(hk)x ---p f °O e-'uh(u)E(du)x. Since T(t) is closed, it follows that -r(h)x E D(T(t)) and (1) is valid for h. Thus r(h)x E D, and by (1), we have for all E C°°, 11
f
0(t)T(t)[T(h)x]dtII =11 f 0(t) f e-tuh(u)E(du)xdt11
0
0
= 11 fr(G0)(u)h(u)E(du)x11
0
S 11h11o11x11w11C011oo.
0
(cf. Theorem 2.23). Therefore 117-(h)x11w S 11h11.11x11w
(x E W,h E B([0,oo)),
(2)
i.e., T is a norm-decreasing (linear) map of B([0, oo) into B(W). Taking h = X6 , we have E(b) E B(W)1, and by (1), for x E W, etc...,
f °° e-tuE(du)[E(b)x] = T(t)[E(b)x] _ 0
f
By the uniqueness property of the Laplace-Stieltjes transform of regular measures, it follows that E(du)E(b)x = X6(u)E(du)x, and therefore
E(o,)E(b)x = E(o, n b)x
for all a,b E B([0,oo)) and x E W. Thus E is a spectral measure on W, and r is necessarily multiplicative on B([0, oo)), since it is multiplicative on the simple Borel functions, and satisfies (2) on B([0, oo)). In view of Theorem 2.23, this completes the proof of Theorem 2.28.1111
In the special case of a C,,-semigroup of contractions T(.), with generator -A, since R+ C p(-A), the semi-simplicity manifold Z for -A is well-defined, as well as the Laplace-Stieltjes space W for T(.). As expected, we have
2.29. THEOREM. Let -A generate the Co-semigroup of contractions T(.) on the reflexive Banach space X. Let Z and W be the semi-simplicity manifold for A and the Laplace-Stieltjes space for T(.), respectively. Then Z = W, topologically. PROOF. The observations preceding Definition 2.22 and the maximality of W show that Z C W. 110
On the other hand, if x E W, then
R(t)x =
J0
oo
w e-t'T(s)xds =
'
e-t' f 00 e-"`E(du)xds
J0
0
e-(t+u)'dsE(du)x
E(du)x,
= Jo Jo = fo 0 t + u where the change of integration order is easily justified by the Tonnelli and Fubini theorems. By the multiplicativity of the map r induced by E(.), the spectral measure on W, we get for all k E N,
Sk(t)x := {tR(t)[1- tR(t)]}kx =
= JOB{ (t + u)2 }kE(du)x = Therefore, for all unit vectors x* E X*,
J0
{ T+ u
[1
j00+
t+
u]}kE(du)x
)-2kE(du)x.
oo
11x*Skxlli <_
J
j00 (t)k(1 + t)-2kdt Ix*E()xl (du)
00
ds Ix*E()xl (du) = J0 Jo sk(1 + s)-2k s
=B(k,k)IIx*E(.)xll
Ilxllz := Sup{IIxlI, llB(kkk)l' ; k E N,
IIx*II =1} < IIxIIw < oo,
that is, W C Z. We thus proved that Z = W, and since both are Banach spaces and 1 I.1 I z < I1.11 w, it follows that the norms are equivalent (by a well-known theorem of Banach). 1111
We consider now a "variation" of the construction of the Laplace-Stieltjes space for a family F(.) as given in Definition 2.22. In that construction, the constraint on E C°° := C° °(R+) was Il IIoo = 1. Since ¢ vanishes in a neighborhood of 0, (R+, Lo E L' dt) (with the usual Lebesgue measure dt), and has L1(R+, dt)-norm IIL0Il1 < II0IIL1(I+,ds/s)-
It then makes sense to replace the norm 11.11 by the norm 11.111 in the constraint on 0 (in Definition 2.22). 111
2.30. DEFINITION. Let {F(t); t > 0) be a family of closed operators with common domain V. The "Integrated Laplace space" for F(.) is the set Y of all x E V such that F(.)x is X-continuous on R+, and IlxIIy
suP{IIxII, II f 0(t)F(t)xdtII; 0 E
IILOII1 =11
0
is finite.
2.31. THEOREM. Let Y be the Integrated Laplace space for the family F(.) of closed operators on the arbitrary Banach space X. Then Y is a Banach subspace
of X, invariant for every T E B(X) commuting with F(t)Ip for all t > 0 (and IITIIB(Y) <_ IITIIB(x)), and there exists a uniquely determined map
S(.) : [0, oo) - B(Y, X) with the following properties: (1) S(0) = 0; (2) I IS(t)x - S(u)xII < It - uI IIxI IY (t, u > O; x E Y); (3) F(t)x = t f o' e-t"S(u)xdu (t > O; x E Y). Moreover, the pair (Y,S) is "maximal-unique" in the usual sense. PROOF. The basic properties of Y are verified in precisely the same way as the corresponding properties of the Laplace-Stieltjes space W. Denote by Lip, the space of all complex functions f on [0, oo) such that f (0) = 0 and (t, u > 0). I f(t) - f(u)I <_ MIt - uI
The smallest possible constant M above is called the Lipshitz constant for the "Lipshitz function" f. The remainder of the proof depends on the following
LEMMA. Let h : R+ --+ C and K > 0 be given. Then there exists f E Lip, with Lipshitz constant < K such that h(t)lt is the Laplace transform of f on R+ if and only if h is continuous and I
f
q(t)h(t)dtI < KIIC0II1
(*)
0
for all ¢ E C,"0.
PROOF (of Lemma). If h(t)/t = (,C f)(t) for all t > 0, where f E Lip, has Lipshitz constant < K, then h is clearly continuous, and f is locally absolutely continuous, its derivative (which exists a.e.) satisfies IIf'II0o < K, and
f(t) = (,f')(t) := f 0f'(s)ds t 112
(t > 0).
For any 0 E C°°, integration by parts and Fubini's theorem give 00
¢(t)h(t)dt =
J
F m te-t°(Jf')(u)du¢(t)dt J0 J0
0J J ef'(u)duO(t)dt = j(Jcb)(u)f'(u)du. 0
Thus
for all 0 E C°°. Conversely, suppose h is continuous on R+ and satisfies (*). For any 0 E C°° := C°°(R+), 11,4111 < J OW fo"O te-t'duJ4)(t)Idt/t < IIOIILI(IL+,dt/t). 0
,C : L'(dt/t) := L'(R+,dt/t) -+ L'(dt) := L'(R+,dt)
(1)
is a contraction. We identify [L'(dt/t)]* with the space of all complex measurable functions h on R+ such that th(t) E L°° := L°°(R+,dt), normed by the essential supremum norm of th(t), with the duality given by
F F < O, h >= f c6(t)h(t)dt = f 4)(t)[th(t)](dt/t). 0
0
By Fubini's theorem, for all 0 E L'(dt/t) and 0 E L°O, < ,C¢,V' >= f r(GO)(s)V,(s)ds =
j(Jb)(t)4)(t)dt.
The use of Fubini's theorem is justified because
J
f
e-8LI4)(t)I.I0(s)Idtds < II0II0II)1IL1(dt/t) < 00-
This shows.that the operator C defined in (1) has the adjoint .C* =,C : L°°(dt) --t [L'(dt/t)]*. 113
By (*) for h, I
J
'[th(t)][4(t)/t]dtl < KII,C.I11 < K IIcIIL1(dt1t)
for all 0 E C°°, and therefore IIth(t)II0 < K. This means that h E [L1(dtlt)]*, and (*) is precisely Condition (*) in Lemma 2.24 for the operator T = L. There exists therefore 0 E L°°(dt) with IItkIIo° < K, such that h = LVY (everywhere on 1R+, by continuity of both sides). Now f := JV, E Lip,,, with Lipshitz constant II IIoo < K, and an integration by parts shows that h(t) = t f °O e-te f (s)ds. i I
I I
PROOF OF THEOREM 2.31. Fix x E Y and x* E X*. The function h := x*F(.)x satisfies the criterion in the lemma with K = IIxIIyIIx*lI. There exists therefore a unique function f = f(.; x, x*) E Lip,,, such that (1) x*F(t)x = t fo e-t"f(u;x,x*)du (t > 0); and
(2) If(t;x,x*)-f(u;x,x*)I <- It u
(t > 0;x E Y, X* E X*). The uniqueness of the representation (1) implies that f(t;.,.) is a bounded bilinear form (for each fixed t), and there exists therefore a uniquely determined operator S(t) E B(Y, X **) such that (3) f (t; x, x*) = [S(t)x](x*) for all x E Y and x* E X *, and by (2), (4) II S(t)x - S(u)xll x** 5 It - uI.I IxIIy for all t, u > 0 and x E Y. For t = 0, the left side of (3) vanishes for all x, x*, and therefore S(0) = 0. By (4), the integral fo e-t"S(u)xdu (with t > 0 and x E Y) converges strongly in X**, and we may then rewrite (1) in the form rc[F(t)x] = t
J0
M e-t"S(u)xdu,
where rc denotes the canonical imbedding of X into X**. Let 7r denote the canonical homomorphism it : X** -+ X**/kX. Since 7r is continuous and 7rrc = 0, we obtain
0=lrrcFtx 114
t
te e-
?rSuxdu
t>0.
The uniqueness of the Laplace transforms implies that ir[S(u)x] = 0, i.e., S(u)x E nX for all u > 0 and x E Y. Identifying as usual r .X with X, we may restate the above relations as the statements (1)-(3) of the theorem. The maximal-uniqueness is an immediate consequence of the necessity part of the lemma and the uniqueness of the Laplace transform. I I I I
Note that Statement (2) in Theorem 2.31 means that S(.) is of class Lipo,l as a B(Y, X)-valued function (where the index 1 indicates that the Lipshitz constant is < 1), that is, (t, u > 0). II S(t) - S(u)II B(y,x) 5 It - uI In particular, the Laplace transform CS exists in the B(Y, X)-norm on (0, oo), and by Statement (3) of the theorem, F(.) is B(Y,X)-valued and F(t) = t(GS)(t)
for allt>0.
We express this relation between F(.) and S(.) by saying that F(.) is the integrated Laplace transform of the B(Y, X)- valued Lipo,l-function S(.). 2.32. COROLLARY. Let F(.) be a family of closed operators on (0, oo), operating on the arbitrary Banach space X, and let Y :_ (Y, II.IIy) be its Integrated Laplace space. Then the following statements are equivalent.
(1) Y = X.
(2) K:=
IIxIIy < oo.
(3) F(.) is the integrated Laplace transform of a B(X)-valued Lipo,K-function. In view of the characterization of semigroups generators given in Lemma 5 (in the proof of Theorem 1.38), it is interesting to consider the special family F(t) = R(t; A) for a given operator A.
2.33. DEFINITION. The operator A on the Banach space X, with (0, oo) C p(A)
is said to generate an integrated semigroup of bounded type < K if R(.; A) is the integrated Laplace transform of a B(X)-valued Lipo,K-function S(.) on [0, oo).
The (uniquely determined) function S(.) is called the integrated semigroup generated by A. By Corollary 2.32, we have
2.34. COROLLARY. An operator A with (0, oo) C p(A), acting in a Banach space X, is the generator of an integrated semigroup of bounded type < K if and only if II
J0
q5(t)R(t; A)dtII < KIIC III
for all 0 E C° °(R+). Recall that I I.I I i denotes the Ll (R
,
dt)-norm. 115
The more general case where (a,oo) C p(A) for some a > 0 and S(.) is of exponential type < a is easily reduced to the case above by translation. We omit the details. If n E N is given, the operator A (with (0, oo) C p(A)) generates an n-times
integrated semigroup of bounded type < K if R(t; A) = tn(LS)(t) on (0, oo), for S as in Definition 2.33, i.e., if t-(n-')R(t; A) is the integrated Laplace transform of S. These objects have been studied recently, and have been found to be useful in the analysis of the Abstract Cauchy Problem. We refer to the bibliography for additional information. Let us only state the following immediate consequence of Corollary 2.32 and of the above observation:
2.35. COROLLARY. The operator A with (0, oo) C p(A) generates an n-times integrated semigroup of bounded type < K if and only if 0(t)R(t;A)td-111
for all 0 E C°°(R+).
There is a simple relation between exponentially bounded n-times integrated semigroups and (exponentially bounded) pre-semigroups. In order to simplify the
statement, we consider only the case n = 1 and we assume that [0, oo) C p(A) (making a translation as needed). 2.36. THEOREM. Let A be such that [0, oo) C p(A). Then A generates the integrated semigroup S(.) (of bounded type < K) if and only if it generates the pre-semigroup A-' + S(.) (of class LipA-1,K) PROOF. The characterization in Lemma 5 (in the proof of Theorem 1.38) carries over to pre-semigroups in the following form: The operator A (with [0, oo) C p(A)) generates the pre-semigroup W(.) of class LipA-I if and only if
A-'R(t; A)x =
jeW(u)xdu
for allt>0,xEX. Writing briefly.CW for the above Laplace transform (understood in the strong operator topology), since LA-' = t-'A-1, the above condition is equivalent to (t > 0), A-'[-t-1 + R(t; A)] =,C(-A-1 + W)(t)
A-' [-I + tR(t; A)] = tL(-A-1 + W)(t), R(t; A) = t(LS)(t)
where S:= -A-' + W.IIfl 116
(t > 0),
E. SEMIGROUPS OF UNBOUNDED SYMMETRIC OPERATORS
In this section, Stone's theorem is generalized to semigroups of unbounded symmetric operators. Let A = [0, c] (c > 0), and let {T(t); t E Al be a family of unbounded operators
acting in a Hilbert space X, with T(0) =I and D(T(t)) := Dt (t E A). We assume that D. C Dt for s > t, and that the linear manifold
D:=U{Dt;0
on
Dt+,.
The (weak) continuity condition takes the following form: for each 0 < s E A, (T(.)x, x) is continuous on [0, s], for all x E D,. Briefly, the above family of operators will be called a local semigroup (on A).
It is symmetric if each T(t) is a symmetric operator (t E A), that is (T(t)x, y) = (x, T(t)y)
(X, y E Dt, t E A).
2.37. THEOREM. Let T(.) be a symmetric local semigroup on A. Then there exists a unique selfadjoint operator H such that for all 0 < s E A, D. C D(e-tH) and T(t) = e-tH on D, for all t E [0, s/2].
PROOF. Fix 0 < s E A and x E D and consider the continuous function f (t) := (T(t)x, x)
(t E D,).
For any n E N, if tl, ..., t E [0, s] are such that ti + tj E [0, s], then x E Dt;, T(ti)x E D,_t; C Dt, (since s - t, > tj), and T(tj)T(ti)x = T(ti + tj)x. Therefore, by the symmetry hypothesis,
f(ti +tj) = (T(ti)x,T(t;)x). 117
If now ei i ..., cn E C, then
f(ti +ti) = Ei,icic (T(ti)x,T(ti)x) = IIEiciT(ti)xll2 > 0. By a theorem of Widder [W1], this positivity property of the continuous function f on [0, s] implies the existence of a unique regular positive Borel measure a = a(.; x) on R such that a-tu E L1(R,a) and
f(t) = J e-tua(du)
(t E [0,s]).
(1)
For n, m E N, Cl, ..., Cn; dl, ..., dnt complex, and s1, ..., sn; tl, ..., t,,,, real such that .tit +ti E [0, s], the preceding calculation shows that (Et 1c T(si)x, Ej"1diT(ti)x) = Ei,.icid.i (T(si)x,T(ti)x)
= Ei,icidj-.f(si + ti) = Ei,icidi f e-(9++ti)ua(du) =
f(Eicie-9`u)(Eidie-tiu)-a(du).
(2)
Let Y be the closed span of {T(t)x;t E [0,s/21}, and let U(T(t)x) := e-"(E L'(a), since e-"u E L1(a) for t E [O,s/2]). If g E L2(a) is orthogonal to all the functions a-tu with t E [0, s/2], then the function G(z) := f]R e-zug-(u)a(du), which is analytic in the strip S := {z E C;Rez E (0,s/2)} and continuous in its closure S-, must vanish identically. Hence f1 e-irug(u)-a(du) = 0 for all real r. By the uniqueness property of the Fourier-Stieltjes transform, it follows that g(u)-a(du) = 0, and therefore f,gg-da = 0, i.e., g is the zero element of L2(a). Thus {e-tu;t E [0,s/2]} is fundamental in L2(a), and it follows from (2) that U extends linearly to a unitary operator from Y onto L2(a).
For each z E S-, the function hz(u) = e-:u is in L2(a). The L2(a)-valued function z -* hz is continuous in S-. Indeed, let O(u) = 1 for u > 0 and O(u) = e-'u for it < 0. Let z, w E S-, and denote Rez = t, Rew = r. Then
Ie-zu - e-wu12 < (e-tu + e-ru)2
= e-2tu + e-2ru + 2e-(t+r)u < 40(u) E Ll (a), since 2t, 2r, t + r < s. It then follows by dominated convergence that HHhz
when w --f z. 118
- h.1122(a) - 0
For each g E L2(a), (h ,z, g) = fA e--g(u)-a(du) is the Laplace-Stieltjes transform of the measure g-dce; it converges absolutely in S- since a-'u E L2(a) for t E [0, s/2], and is therefore analytic in S. Hence the L2(a)-valued function hr is analytic in S. Define
x(z) := U-lhz(E Y)
(z E S-).
(3)
Then, as a Y-valued function, x(.) is continuous in S-, analytic in S, and
x(t) = T(t)x
(t E [0, s/2]).
(4)
Now, given x c D, there exists 0 < s C A such that x E D,. Let x(.) be the function constructed above in the strip S-, and define
V(r)x := x(ir)
(r E R).
(5)
By (4) and the analyticity of x(.), V(.) is well- defined on D. For r, r' E R, we have
(V(r)x, V (r' )x) = (x(ir), x(ir')) = (hir, hir' )
= f Re-i(r-r)ua(du)
(6)
Thus, by (1), IIV(r)x112 = a(R) = f(0) =
IIxII2
(7)
for all r E R and x E D, i.e., each V(r) is an isometry from D to X. We verify its linearity as follows. If x, y E D and A, ,u E C, there exists 0 < s E 0 such that x, y E D,. Then by (4), for all t E [0, s/2],
(.fix + py)(t) = T(t)(Ax +µy) = AT(t)x +,uT(t)y = Ax(t) + µy(t),
and therefore, by analyticity of x(.) and y(.) in S and their continuity in S-, the same relation is valid with t replaced by ir, i.e., V (r)(.1x +µy) = AV (r)x + µV (r)y for all real r. Since V(r) is isometric on the dense linear manifold D, it extends as a linear isometry on X.
The function r --} V(r)x = x(ir) is continuous for each x E D (as observed above), and V(r) is isometric on X; therefore V(.)x is continuous on R for all
xEX.
Let X E D. and t, t' E 0 such that t+t' E A. The semigroup property T(t+t')x = T(t)T(t')x implies, by uniqueness of the analytic continuation onto the imaginary
axis, that V (r)V (r' )x = V (r + r')x for all r, r' E R and x E D hence for all x E X, by density of D. Thus V(.) is a group of operators on R; in particular, the isometries V(r) are onto, i.e., V(.) is a strongly continuous unitary group. 119
By Stone's theorem, we have V(r) = e-irH with H selfadjoint. Let E be the resolution of the identity for H. For all r E R and x E D we have by (6)
Je_(E(du)x,x)
= (V(r)x,x) = je_1"o(du;x),
and therefore (E(.)x, x) = a(.; x), by the uniqueness property of the FourierStieltjes transform.
Since e-z" E L2(a) for z E S-, we have x E D(e-,H). The vector functions e-zHx and x(z) are both analytic in S and continuous in S-; on the imaginary axis, we have a-irHx = V(r)x = x(ir), so that a-zHx = x(z) for all z E S In particular, by (4),
T(t)x = e-tHx
(t E [0, s/2], x E D,).
(8)
If also H' is a selfadjoint operator satisfying (8), then the analytic continuation employed in the construction gives a-irHx = e-irH'x for all real r and all x E D, hence for all x E X, and therefore H = H' by the uniqueness in Stone's theorem.I I I I
We apply Theorem 2.37 to "analytic vectors".
2.38. DEFINITION. Let A be an (unbounded) operator on X. An analytic vector for A is a vector x E D°°(A) such that, for some t > 0 (depending on x),
0, let Dt = {x E D°°(A);
Et IIA"xII < co). W,
These are linear manifolds such that D, C Dt for s > t and U,>0 Dt = D is dense, by hypothesis. Define T(0) = I and R
T(t)x=E'0-A"x for x E Dt, t > 0 (the series converges in X for such x). 120
If x E Dt+,, EmEnmin,IIAm+nXII =
r-k(1/k!)Em_0(k)tmSk-mUUAkxll
k's)k
= Ek (t
IIAkxlI < oo.
(9)
Thus En n, I IAn(A'nx)II < oo, that is, Atmx E D. for all m = 0, 1, 2,..., and since A is closed, one verifies by induction that T(s)x E D(Am), and
T(s)Amx = AmT(s)x = En
sn
An+n`x.
(10)
W!
Therefore Em mM II AmT(S)xII = E. tM +IEn n Am+nxll < 00
by (9), i.e., T(s)x E Dt, and by (10) and absolute convergence, Sn T(t)T(s)x = ,En n An+mx m! En-t"`
W!
= Ek (t
k's)k
Akx = T(t + s)x
(for all t,s>0and xEDt+,).
foralltE[0,s] n
(T(t)x,x) = Ennt, (Anx,x), where the series converges absolutely; in particular, (T(.)x, x) is continuous on [0, s]. By symmetry of A, we clearly have (T(t)x, y) = (x, T(t)y) for all x, y E Dt.
We conclude that {T(t); t > 0} is a symmetric local semigroup. Let H be the sel f adj oint operator associated with it as in Theorem 2.37. If x E D, then x E D. for some s > 0, and in e-eHx = T(t)x := En Anx (11)
t' n!
for all t E [0, s/2].
In particular a-tHx E D,_t C D (for t E [0,.5/2]), and it follows that D is invariant for the C,,-(semi)group a-1H. Since D is also dense in X by hypothesis, Theorem 1.7 implies that D is a core for the generator H, if D C D(H). However for x E D, we have by (11)
Hx := lim t -1 t-.o+
[e-tHx
- x] = Ax, 121
i.e., x E D(H) and Hx = Ax. Thus indeed D C D(H), Hx = Ax for all x E D, and D is a core for H. Since A is closed, we have
H = (HID)- = (AID)- C A- = A, and since A is symmetric and H is selfadjoint, it follows that
AcA*cH`=H, i.e., A = H, so that A is indeed selfadjoint.I I II
If A is not assumed to be closed, the theorem applies to its closure A-, which is closed and symmetric, and every analytic vector for A is certainly an analytic vector for A-. We then have 2.40. COROLLARY. Let A be a symmetric operator with a dense set of analytic vectors. Then A is essentially selfadjoint.
122
F. LOCAL COSINE FAMILIES OF SYMMETRIC OPERATORS
Semigroups of operators are associated with the ACP
U'= Au
u(0) = X.
The second order ACP
u"=Au
u(0)=x, u'(0)=0
in Banach space is associated in a similar way to so-called "cosine operator functions"(cf. [G]).
2.41. DEFINITION. A cosine operator function on the Banach space X is a function C(.) : R , B(X) such that C(0) = I and C(t + s) + C(t - s) = 2C(t)C(s)
(t, s E
The concept that parallels that of a local semigroup is the following
2.42. DEFINITION. Let D be a dense linear manifold in X. A local cosine family (of operators) on D is a family {C(t); t E R} of operators on the Banach space X, such that for each x E D there exists e = e(x) > 0 such that (i) x E D(C(t)) and C(.)x is strongly continuous for ItI < e; (ii) C(0)x = x, and for ItI, Ist, It + si, It - sl < e, C(s)x E D(C(t)) and C(t + s)x + C(t - s)x = 2C(t)C(s)x. A result parallel to Theorem 2.37 for local cosine families of symmetric operators
in Hilbert space is stated below, first for the special case when all the operators C(t) are bounded below, that is,
(C(t)x,x) > iIxli
(x E D(C(t)),t E R).
(1)
Condition (1) implies in particular that all the operators C(t) are symmetric. The general case of a symmetric local cosine family is dealt with in Theorem 2.45. Since no parallel to Widder's theorem [W1] is known for the cosine transform, the proof will proceed differently. 123
2.43. THEOREM. Let D be a dense linear manifold in the (complex) Hilbert space X, and let C(.) be a local cosine family of bounded below operators on D. Then there exists a unique positive selfadjoint operator A such that C(t)x = cosh(tA'/2)x
for all x E D and ItI < e(x).
Note that the family {cosh(tA1/2);t E R} is a cosine family of bounded below selfadjoint operators that extends the local family C(.). PROOF. Since C(t) is symmetric for each t, it is closable, and its closure C(t)clearly satisfies (i), (ii), and (1). We may then assume that C(.) is a local cosine family of closed bounded below operators on D, replacing C(t) by C(t)- if needed (by (i), the conclusion of the theorem remains unchanged). Fix a sequence {hn) of non-negative C°°- functions on R, such that hn(t) = 0 for ItI > 1/n and ff hn(t)dt = 1. Let x E D, and fix n(x) > 1/e(x). Denote
xn = J hn(s)C(s)xds
(n > n(x)),
(2)
where the integral is a well-defined strong integral, by (i) in Definition 2.42. Clearly xn --+ x strongly. Since D is dense by hypothesis, it follows that the set
Do :_ {xn;x E D,n > n(x)} is dense in X.
Fix n > n(x). If ItI < en(x) := e(x) - 1/n (note that 1/n < 1/n(x) < e(x)), Condition (ii) implies that C(s)x E D(C(t)) for all s with Isi < 1/n. Also C(t)C(s)x = (1/2)[C(t + s)x + C(t - s)x] is strongly continuous for Isi < 1/n (because It + sI, It - sI < e(x)). Since C(t) is closed, it follows from Theorem 3.3.2 in [HP] that xn E D(C(t)) and
C(t)xn = j hn(s)C(t)C(s)xds
(3)
(ItI < en(x)). Let u > 0 be such that It + UI, It - uI < en(x) (for a given t such that ItI < En(x)) By (3) [C(t + u) + C(t - u) - 2C(t)]xn
_
hn(s)[C(t + u) + C(t - u) - 2C(t)]C(s)xds s
124
j hn(s)C(s)[...]xds =
_
J
hn(s)C(s)[2C(t)C(u) - 2C(t)]xds
hn(s)C(t)[2C(s)C(u) - 2C(s)]xds
r
= J hn(s)C(t)[C(s + u) + C(s - u) - 2C(s)]xds
_
[hn(v - u) + hn(v + u) - 2hn(v)]C(t)C(v)xdv.
In the last integral, integration extends over an interval where Iti, Ivi, Jt+vj, It-vj < e(x), so that Conditions (i),(ii) imply that C(t)C(v)x is strongly continuous there (as a function of v), and therefore
u-2[C(t + u) + C(t - u) - 2C(t)]xn --,u-o
JI
h'(v)C(t)C(v)xdv
(4)
strongly (for Itl < en(x)). Let
D1 = {C(t)xn; X E D, n > n(x), 0 < t < en(x)}.
Since Do C D1, D1 is dense in X, and as before, if y E D1, there exists e'(y) > 0 such that y E D(C(t)) for Iti < e'(y). By (1), 2u-2(C(u)y - y, y) ? 0
(iu! < e'(y)).
(5)
Writing y = C(t)xn for some n > n(x) and some t E [0, en(x)), we have
2u-2[C(u)y - y] = u-2[2C(u)C(t)xn - 2C(t)xn] = u-2[C(t + u) + C(t - u) - 2C(t)]xn.
(6)
By (4), the last expression has a (strong) limit as u -+ 0, which we denote Aoy. The operator Ao is linear on the dense domain D1, and positive (by (5) and (6)), i.e.,
(Aoy, y) ? 0
(y E D1).
Let A be the Friedrichs selfadjoint extension of Ao (cf. Theorem XII.5.2 in [DS,II]),
and let E be its resolution of the identity. Denote E,n = E([0, m]) and An = E,,,A = fo sE(ds) for m E N. Note that An, is a bounded positive (selfadjoint) operator. Let xn E Do; for It! < en(x),
d EmC(t)xn = EmAoxn = Amxn 125
by (4) and the definition of Ao. From the spectral representation, cosh(tA,;2)E,nxn is also a solution of v'1 = Am v,
v(O) = Emxn,
v'(O) = 0.
By the uniqueness of the solution, we have E,,,.C(t)xn, = cosh(tA7z2)E,,,xn = for Iti < en(x).
When m -+ oo, E,,,C(t)xn -+ C(t)xn (for each Itl < en(x)); in particular, E,nxn -+ xn. Also
cosh(tA/2)E,nxn = E,,,C(t)xn -+ C(t)x,, (when m -, oo). Since cosh(tA1/2) is closed, it follows that xn E D(cosh(tA1/2) and cosh(tA'12)xn = C(t)xn
(7)
for x,, E Do and Itl < en(x).
For n-ioo,xn-+x,and by (3), C(t)xn = (1/2) J hn(s)[C(t + s) + C(t - s)]xds -> C(t)x for Iti < e(x). Since cosh(tA1/2) is closed, it follows from (7) that x is in its domain, and
C(t)x = cosh(tA1/2)x
(8)
for all x E D and Itl < e(x). The uniqueness of A is proved as follows. If B is also a positive selfadjoint operator satisfying the identity in the theorem, and if E and F are the resolutions of the identity for A and B respectively, then (x E D, Iti < e(x)).
cosh(tA1/2)x = cosh(tB1/2)x
Since I cosh z < cosh( 2z), the above identity (written in term of the corresponding
spectral integrals) extends analytically to t complex in the strip J*tj < e(x). In particular for t E iR, we have cos(sA1/2)x = cos(sB1/2)x for all x E D, hence for all x E X by density (since the operators are bounded), and for all s E R. Thus J 00 cos(su1/2)E(du)x = 0
126
J
0
r cos(su1/2 )F(du)x
for all x E X and s E R. By the uniqueness property of the cosine transform, it follows that E = F, and therefore A = B.IIII We consider next the general case of a local cosine family of symmetric operators.
The following notation will be used. If A is a selfadjoint operator, and E is its resolution of the identity, we let A+ := o
"o uE(du);
A- :_ -
J
uE(du) 00
with the usual domains. { [x, y]; x, y E X } is considered as a Hilbert space The cartesian product X 2 with the inner product ([x, y], [x', y']) := (x, x') + (y, y'). If T is an operator on X with domain D(T), we let T [x, y] :_ [Tx, -Ty]
([x, y] E
D(T)2).
2.44. LEMMA. If T is symmetric, then T has a selfadjoint extension.
PROOF. Let J[x, y] := [y, -x]. Then J is unitary, J2 = I (the identity operator on X2), and JD(T) = D(T). One verifies that
T - iI = J(T + iI)J. Therefore
[ran (T - iI)]1 = J[ran (T + U)]-L. This implies that T (which is obviously symmetric) has equal deficiency indices (n_ = n+), and has therefore a selfadjoint extension (cf. [DS,II], Chapter XII).III]
2.45. THEOREM. Let D be a dense linear manifold in the (complex) Hilbert space X, andlet C(.) be a local cosine family of symmetric operators on D. Then there exists a selfadjoint operator A on X such that C(t)x = cosh[t(A+)112]x + cos[t(A-)112]x
for allxEDandlt]<e(x). PROOF. As in the proof of Theorem 2.43, we may assume that each C(t) is closed. With notation as in the proof of that theorem, we obtain the operator A0 defined on D1 (up to that point, the "bounded below" hypothesis was not used). Since
(C(u)y - y, y) = (y, C(u)y - y)
(y E D1, I uI < e '(y)), 127
it follows from (4) and (6) that A0, with the dense domain D1, is symmetric. Let A be a selfadjoint extension of the operator A0 associated with AO (cf. Lemma 2.44), and let E be its resolution of the identity. Consider the projections Em := E([0, co)) and the bounded positive selfadjoint operators Am := E,+nA for m E N. For xn E Do, let
nm(t) := E+[C(t)xn, 0]
(Iti < en(x))
Since [C(t)xn, 0] E Dl = D(Ao ), Ao C A, and the projection Em commutes with A, it follows from (4) and the definition of AO that d Cnm(t) = Em [AoC(t)xn, 0] = E,+nAo [C(t)xn, 0] = E,+nA[C(t)xn,0] = E+AEm[C(t)xn,0] = AmCnm(t)
for Itl < , (x). By uniqueness of the solution of the second order ACP, we then have for all Iti < en(x): m(t) = cosh[t(An)1/21C+ (0) cosh[ ...]E,+nCnm(0) = cosh[t(A+)1/2]Cnm(0).
(9)
When m -; oo,
6+ (t)
Sn (t)
E([O, oo))[C (t)xn, o]
for Iti < en(x). In particular, nm(0)
6,+, (0), and cosh[t(A+)1/2]6ri (0)
6n (t)
by (9). Since cosh[...] is closed (it is selfadjoint!), it follows that 6n (0) belongs to its domain, and C+(t)
(hti <
(x))
(10)
Similarly, letting Em = E([-m.0]), etc..., we find that cosh[t(-A-)1/2]Sn (0)
_ n (t)
(11)
(Iti < en(x)),
where en (t) := E((-oo, 0])[C(t)xn, 0]. Adding (10) and (11), we obtain [C(t)xn,0] = cosh[t(A+)1/2]Cn(0) +cos[t(A-)1121
for Itf < en(x).
n
(0)
(12)
Let P denote the orthogonal projection of X2 onto X (identified with its imbedding in X2 as { [x, 0]; x E X}). Then P commutes with Ao, hence with 128
A, and so A := PA is a selfadjoint operator on X with resolution of the identity PE(.) = E(.)P. We have cosh[t(A+)1/27tn Jb (0)
= J0
cosh(tu1/2)E(du)[x,,,01
/00
= J0
cosh(tu112)[E(du)P][xn, 0] = cosh[t(A+)112]xn,
and similarly for the second term in (12) (we have identified [xn,0] with xn). Thus C(t)xn = cosh[t(A+)1/2]xn + cos[t(A-)1/2]xn
(13)
for all Iti < 'En(X)-
Recall now that for each x E D, xn -+ x and C(t)xn -+ C(t)x when n -+ 00 (for Itl < e(x)). Since the operator on the right of (13) is closed, it follows that x belongs to its domain and the identity in Theorem 2.45 is verified. I I
II
The following concept corresponds to that of analytic vectors in the context of cosine families.
2.46. DEFINITION. A semianalytic vector for A is a vector x E DO°(A) such that
En
0 (2n)!
for some t > 0 (depending on x).
IIAnxII < 00
'
If A is positive, we may immitate the proof of Theorem 2.39, applying Theorem 2.43 to the local cosine family of bounded below operators t2n
C(t)x
E°_o
(2n)! Anx,
to obtain the following result:
2.47. THEOREM (Nussbaum's Semianalytic Vectors Theorem). Let A be a positive operator with a dense set of semianalytic vectors. Then A is selfadjoint. The details of the proof are omitted. Note that if A is not assumed to be closed, the conclusion is that A is essentially selfadjoint.
129
NOTES and REFERENCES
PART I. GENERAL THEORY.
The standard books on semigroups are [D, G, HP, P], with chapters in general texts like [DS I-III, Katl, RS].
The Hille-Yosida space. The terminology and Theorem 1.23 are from [K5].
Semigroup convergence. Theorem 1.32 goes back to [Tr].
Exponential formulas. The treatment here follows [D,P], and is based on work by [Kat3, C1, C2, Tr].
Perturbations. Theorem 1.38 is due to Hille-Phillips. The proof given here is basically the one in [DS1].
Groups. Theorem 1.40 is from [N]. Theorem 1.41 is the classical Stone theorem. Theorem 1.49 and the following analysis are from [K3].
Analyticity. Theorem 1.54 is from [Liu], but the short proof given here is new.
Non-commutative Taylor formula. The results of this section are from [K7].
PART II. GENERALIZATIONS.
Pre-semigroups. The concept appears in germinal form in [DP] (under the name of regularizable semigroups). In [DPg], the name "C-semigroup" is coined, and the detailed analysis of these families is started (see [DL1-DL3, Ml, M2, MT1MT3, T1, T21, as a partial list for this subject). Since a C-semigroup is not a semigroup (unless C = I), we prefered to call it here a pre-semigroup. Theorems 2.3-2.5 are from [DL1]. Theorem 2.8 is from [DL2] (but we coined the term "exponentially tamed" as a reference to Property 3.). 130
The Semi-simplicity manifold. The concept goes back to [K1] for a single bounded operator, with extensions to unbounded operators appearing in [K1, K2, KH2, KH31. Theorem 2.11 is from [KH2]. A variant of this theorem is found in [KH3]. Theorem 2.20 is from [K2] (see also [K4]). Lemma 2.24 is from [KH3] (see also [DLK]). The concepts of the Laplace-Stieltjes space and of the Integrated Laplace space for a family of closed operators were introduced and studied in [DLK]. Theorems 2.23, 2.28, 2.29, and 2.31 are from [DLK] (with some modification of the
proofs). Theorem 2.36 is a special case of the main result of [DL3]. Integrated semigroups were introduced in [Neu].
Semigroups of unbounded symmetric operators. First results on this subject were obtained in [De] and [Nus]. A general theory of semigroups of unbounded operators in Banach space was developed in [Hl, H2]. Theorem 2.37 is from [KL], as well as the proof of Theorem 2.39 (which appeared originally in [Nell, with a different proof). Another proof of Theorem 2.37 is found in [Fr), and serves a model for the proofs of Theorems 2.43 and 2.45 (first published in [KH3]). The concept of "semianalytic vector" is due to Nussbaum, as well as Theorem 2.47 (with a proof independent of the result on local cosine families; see [RS]). The results on local semigroups are generalized to a Banach space setting in [KH1] (see also [K4]). For cosine families of closed operators, a "semi-simplicity manifold" can be constructed as in [KH3] to provide a spectral integral representation, as Theorem 2.28 does it for semigroups of closed operators (cf. Theorem 4.2 in [KH3], with the obvious modifications needed in Definition 4.1 and in the proof of the theorem).
131
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ABOUT THIS VOLUME This Research Note presents some aspects of the theory of semigroups of operators, mostly from the point of view of its interaction with spectral theory. In order to make it self-contained, a concise description of the the basic theory of semigroups, with complete proofs, is included in Part I. Some of the author's recent results, such as the construction of the Hille-Yosida space for general operators. the semi-simplicity manifold, and a Taylor formula for semigroups as functions of their generator. are also included in Part 1. Part II describes recent generalizations (most of them in book form for the first time), including pre-semigroups, semi-simplicity manifolds in situations more general than that considered in Part I. semigroups of unbounded symmetric operators. and an analogous result on "local cosine families" and semi-analytic vectors. It is hoped that the book will inspire more research in this field.
Readership: Graduate students and researchers working in operator theory and its applications.
PITMAN RESEARCH NOTES IN MATHEMATICS SERIES The aim of this series is to disseminate important new material of a specialist nature in economic form. It ranges over the whole spectrum of mathematics and also reflects the changing momentum of dialogue between hitherto distinct areas of pure and applied parts of the discipline. The editorial board has been chosen accordingly and will from time to time be recomposed to represent the full diversity of mathematics as covered by Mathematical Reviews.
This is a rapid means of publication for current material whose style of exposition is that of a developing subject. Work that is in most respects final and definitive. but not yet refined into a formal monograph. will also be considered for a place in the series. Normally homogeneous material is required, even if written by more than one author. thus multi-author works will be included provided that there is a strong linking theme or editorial pattern. Proposals and manuscripts: See inside book.
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