Advances in Mathematics AI1613 advances in mathematics 128, 217241 (1997) article no. AI971613 Spectral Properties of Weakly Asymptotically Almost Periodic Semigroups Edoardo Vesentini Scuola Normale Superiore, 56126 Pisa, Italy Received January 1, 1996; accepted December 30, 1996 The classical theory of continuous almost periodic functions on the real line was extended by Frechet in 1941 [14, 15], to the case of continuous almost periodic functions on half-lines. Since then, the theory of asymptoti- COREcally almost periodic functionsas they were called by Frechetbecame Metadata, citation and similar papers at core.ac.uk Providedthe by subject Elsevier - Publisher of several Connector investigations and of generalizations in various direc- tions: from scalar-valued functions on half-lines to scalar-valued functions defined on more general topological semigroups [10], and from scalar- valued or finite-dimensional vector-valued functions (already considered by Frechet) to functions with values in Banach spaces endowed with different topologies [4, 12]. As an extension of the notion of a periodic operator-valued group defined on the real line [8, 5], it is natural to investigate strongly con- tinuous groups and semigroups of continuous linear operators giving rise, in various ways, to almost periodic, or asymptotically almost periodic func- tions. Several investigations have been carried out in this context.1 It turns out that the existence of functions of this type introduces some constraints on the spectral structure of the groups and semigroups. Given a strongly continuous semigroup T, acting on a complex Banach space E, generated by a linear operator X, the main purpose of this paper is investigating some aspects of these constraints bearing on the spectrum of X. It will be shown, for example in Sections 6, 7, and 8, that, under fairly general conditions, the intersection of the imaginary axis of C with the union of the point spec- trum and the residual spectrum of X contains (up to the factor &- &1) the frequencies of all almost periodic or asymptotically almost periodic functions associated to T. Furthermore, the closure of this intersection is discrete. In view of this fact, stronger hypotheses on the point spectrum of X yield further constraints on the semigroup T, as examples of eventually differentiable semigroups will show in Section 7. The case in which E is 1 See, e.g., [13, 6, 17, 2224] and bibliographical references therein. 217 0001-8708Â97 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved. File: 607J 161301 . By:DS . Date:15:07:07 . Time:08:45 LOP8M. V8.0. Page 01:01 Codes: 3962 Signs: 2473 . Length: 50 pic 3 pts, 212 mm 218 EDOARDO VESENTINI weakly sequentially complete and the intersection of the imaginary axis of C with the point spectrum of X is a harmonious set [18] will be investigated in Section 8. It will be shown that, roughly speaking, T acts as a uniformly almost periodic group on the closure, for the graph norm, of the space spanned by all eigenvectors of X. In the final part of this paper (Sections 9 and 10), a class of semigroups of linear contractions of a complex Hilbert space, behaving in an ``asymptotically almost periodic mode,'' will be investigated. All these results depend on various applications of the ergodic theorem for semigroups, which will be discussed (Sections 25) in the first part of this article. 1 Let J be either R or R+ . A subset 4 of J is called relatively dense in J if there exists l>0 such that every interval of length l in J has a non-empty intersection with 4. Throughout this article, E denotes a complex Banach space, and E$ the topological dual of E. For x # E and * # E$, (x, *) will denote the value of * on x. If f: J Ä E, H( f ) is the set of all translates f( v +|)offby all | # J. If f is an E-valued function defined on R and =>0, then a real number { # R"[0], such that & f (t+{)&f(t)&= (1) for all t # R, is called an =-period of f. The function f is said to be almost periodic if, for every =>0, the set of all =-periods of f is relatively dense in R.Iffis almost periodic and continuous, then it is necessarily bounded on R. Let Cb (J, E) be the Banach space of all bounded continuous func- tions J Ä E, endowed with the uniform norm. Continuous almost periodic functions on R are characterized by the following classical theorem2. Theorem 1. A continuous function f: R Ä E is almost periodic if and only if H( f ) is relatively compact in Cb(R, E). If f # Cb (R, E) is almost periodic, the limit 1 a lim f (t) dt (2) a Ä + a |0 2 See [4] for a proof and for further bibliographical references. File: 607J 161302 . By:DS . Date:15:07:07 . Time:08:45 LOP8M. V8.0. Page 01:01 Codes: 2721 Signs: 1884 . Length: 45 pic 0 pts, 190 mm ALMOST PERIODIC SEMIGROUPS 219 exists in E [4]. If % # R, the function t [ e&i%tf (t) also is almost periodic. The corresponding limit 1 a lim e&i%t f (t) dt (3) a Ä + a |0 vanishes, with the possible exception of an at most countable set of values of % [4]. These values are called the frequencies of f, and the corresponding values of (3) are called the Fourier coefficients of f. If all of them vanish, then f=0. A function f # Cb(R+ , E) such that H( f ) is relatively compact in Cb(R+, E) is said to be asymptotically almost periodic. 3 Theorem 2. A function f # Cb(R+ , E) is asymptotically almost periodic if, and only if, any one of the following two equivalent conditions are fulfilled: there exist a unique almost periodic function g # Cb(R, E) and a unique function h # Cb(R+ , E), vanishing at +, such that f=h+g|R+ ;(4) for every =>0 there exist l>0 and k0 such that every interval of length l contains some { for which (1) holds whenever tk and t+{k. The functions g and h are called, respectively, the principal term and the correction term of f. If f is asymptotically almost periodic, the limit (3) exists for all % # R and vanishes for all values of % # R with the possible exception of an at most countable set. These real numbers and the corresponding values of (3) are called respectively the frequencies and the Fourier coefficients of f. If h is any element Cb(R+ , E) vanishing at +, then 1 a lim h(t) dt=0. a Ä + a |0 Hence, if an # E, n # Z, are the Fourier coefficients of f and if g is the principal part of f, then (4) and the inequality (see [4]) a 2 1 2 : &an& lim &g(t)& dt a Ä + a |0 3 This theorem is due to Frechet [14, 15] in the cases in which E=C (with a minor improvement by DeLeeuw and Glicksberg [10]) or dim E<, and to Ruess and Summers in the general case (see [23] also for further bibliographical references). File: 607J 161303 . By:DS . Date:15:07:07 . Time:08:45 LOP8M. V8.0. Page 01:01 Codes: 2778 Signs: 1783 . Length: 45 pic 0 pts, 190 mm 220 EDOARDO VESENTINI imply that a 2 1 2 : &an& lim & f (t)& dt. a Ä + a |0 A function f # Cb(J, E) is said to be Eberlein weakly almost periodic [12, 23] if H( f ) is weakly relatively compact in Cb(J, E). If f # Cb(R+ , E)is Eberlein weakly almost periodic, the limit (2) exists. 2 Throughout this paper T: R+ Ä L(E) will be a strongly continuous semi- group of continuous linear operators acting on E, and X: D(X)/EÄE will be its infinitesimal generator. In the following, r(X), _(X), p_(X), r_(X),4 and c_(X) will denote respectively the resolvent set, spectrum, point spectrum, residual spectrum, and continuous spectrum of the linear operator X. If T is uniformly bounded (&T(t)&M for some M1 and all t # R+), then, for all % # R, ker(X&i%I) & R(X&i%I)=[0] (where R(X&i%I) denotes the closure of the range R(X&i%I)ofX&i%I), and ker(X&i%I)ÄR(X&i%I), which is the set of all x # E for which the limit a 1 &i%t Pi% x := lim e T(t) xdt (5) a Ä + a |0 exists, is closed (see, e.g., [25]). According to the ergodic theorem, Pi% # L(ker(X&i%I)ÄR(X&i%I)) is a projector, with norm &Pi% &M, whose range is the eigenspace ker(X&i%I)ofX. Let X$: D(X$)/E$ÄE$ be the dual operator of X; let E+ be the closure of D(X$) in E$, and let X + be the part of X$inE+, i.e. the restriction of X$ to the linear space D(X +):=[* # D(X$): X$* # E+], 4 Since the terminology is not well established, note that, in this paper, r_(X) is the set of all ` # C for which X&`I is injective and R(X&`I){E. File: 607J 161304 . By:DS . Date:15:07:07 . Time:08:45 LOP8M. V8.0. Page 01:01 Codes: 2663 Signs: 1412 . Length: 45 pic 0 pts, 190 mm ALMOST PERIODIC SEMIGROUPS 221 + + + which is dense in E . Then T(t)$ E /E for all t # R+ , and, setting + + T (t):=T(t)$|E + , one defines a strongly continuous semigroup T : R+ Ä L(E+), which is generated by X + : the dual semigroup of T (see, e.g., [20, 25]).