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A spectral mapping theorem for evolution semigroups on asymptotically almost periodic functions defined on the half line. (English) Zbl 1010.47027
Let $$X$$ be a complex Banach space, $$AAP_0(\mathbb{R}_+,X)$$ be the space of all $$X$$-valued functions $$h$$ on $$\mathbb{R}_+$$ such that $$h(0)=0$$ and there exists an $$f \in C_0(\mathbb{R}_+,X)$$ and an almost periodic $$X$$-valued function $$g$$ such that $$h=f+g$$. It is proved that the evolution semigroup on $$AAP_0(\mathbb{R}_+,X)$$ is strongly continuous. Some properties of the generator of this evolution semigroup are established.

##### MSC:
 47G10 Integral operators 47D03 Groups and semigroups of linear operators
##### Keywords:
almost periodic functions; exponential stability
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