The classical theory of continuous almost periodic functions on the real line was extended by Fre chet in 1941 [14,15], to the case of continuous almost periodic functions on half-lines. Since then, the theory of asymptotically almost periodic functions as they were called by Fre chet became the subject of several investigations and of generalizations in various directions: from scalar-valued functions on half-lines to scalar-valued functions defined on more general topological semigroups [10], and from scalarvalued or finite-dimensional vector-valued functions (already considered by Fre chet) 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 continuous groups and semigroups of continuous linear operators giving rise, in various ways, to almost periodic, or asymptotically almost periodic functions. 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 spectrum 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 article no.