Spectral properties of weakly asymptotically almost periodic semigroups (Q1362599)

Let \(E\) be a Banach space. By a definition of M. Fréchet, a continuous and bounded function \(f\) mapping \(\mathbb{R}_+\) to \(E\) is said to be asymptotically almost periodic if the set of all translates of \(f\) by elements of \(\mathbb{R}_+\) is relatively compact in \(C_b(\mathbb{R}_+,E)\). This paper investigates strongly continuous groups and semigroups \(T\) of operators on \(E\) for which all the functions \(\lambda\mapsto f(T(\lambda))\), for \(f\in E'\), are almost periodic on \(\mathbb{R}\) or asymptotically almost periodic, respectively. Several theorems show that in this situation there are strong conditions on the spectrum of the generator of the semigroup. There are stronger results than in the general case if \(E\) is weakly sequentially complete. Finally, much stronger results are obtained for the case, where \(E\) is a Hilbert space.