A new proof and generalizations of Gearhart’s theorem

Abstract

Abstract. Let H be a Hilbert space, let AP (R, H) be the space of almost periodic functions from R to H, and let A be a closed densely defined linear operator on H. For a closed subset Λ ⊂ R, let M (Λ) be the subspace of AP (R, H) consisting of functions with spectrum contained in Λ. We prove that the following properties are equivalent: (i) for every function f ∈ M (Λ) there exists a unique mild solution..} yields a new proof of the well-known Gearhart's spectral mapping theorem.