Spectral properties of weakly asymptotically almost periodic semigroups in the sense of Stepanov (Q1390734)

Let \(E\) be a complex Banach space, and \(f:R_+\to E\) a map, \(f\in L^p_{\text{loc}}(R_+,E)\). One denotes by \(\widetilde f:R_+\to L^p([0,1];E)\) the map defined by \(\widetilde f(t)=\{f(t+s),s\in[0,1]\}\). The map \(f\) is called \(S^p\)-asymptotically almost periodic if \(\widetilde f\in L^p([0,1];E)\), and the function \(\widetilde f\) defined above is asymptotically almost periodic in the classical sense (Fréchet). One of the basic results established in this paper is concerned with the generalization of the classical representation of asymptotically almost periodic function as the sum of an almost periodic function and another continuous function vanishing at infinity. Most of the investigation relates to the theory of semigroups, including reference to the spectral properties. The starting point for the research appears to be some work due to \textit{E. Vesentini} [``Introduction to continuous semigroups''; Scuola Normale Superiore, Pisa (1996); Adv. Math. 128, 217-241 (1997; Zbl 0882.22002)].