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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
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