Spectral Mapping Theorems For Evolution Semigroups on Spaces of Almost Periodic Functions

Abstract

. We show that the weak spectral mapping theorem holds for evolution semigroups on spaces of periodic functions. Furthermore, we prove the spectral mapping theorem for evolution semigroups on spaces of (weakly asymptotically) almost periodic functions. The results are applied to the study of hyperbolic periodic and (weakly asymptotically) almost periodic evolution families U = {U (t, s) : t ≥ s}. We establish conditions on U which lead to unique periodic resp. (weakly asymptotically) almost periodic mild solutions u (t) = U (t, s) u (s) + f ^{t s} U (t, τ)f (τ) dτ, t ≥ s ∈ R, provided that f is periodic resp. (weakly asymptotically) almost periodic with relatively compact range.