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Spectral mapping theorems for evolution semigroups on spaces of almost periodic functions. (English) Zbl 1059.47048
This paper deals with the spectral mapping theorem for evolution semigroups on spaces of periodic or almost periodic functions. The authors prove that the weak spectral mapping theorem holds on the space of periodic functions. They also prove that the spectral mapping theorem holds on the space of (weakly asymptotically) almost periodic functions. Finally, in the hyperbolic case, the result is applied to establish a unique periodic or (weakly asymptotically) almost periodic solution for some nonautonomous differential equations in a Banach space.

MSC:
47D06 One-parameter semigroups and linear evolution equations
34G10 Linear differential equations in abstract spaces
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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