Spectral conditions for admissibility of evolution equations in Hilbert space

We study properties of solutions of the evolution equation u (t) = (Bu)(t) + f (t) (∗), where B is a closable operator on the space AP(R,H) of almost periodic functions with values in a Hilbert space H such that B commutes with translations. The operator B generates a family B(λ) of closed operators on H such that B(eiλt x) = eiλt B(λ)x (whenever eiλt x ∈ D(B)). For a closed subset Λ ⊂ R, we prove that the following properties (i) and (ii) are equivalent: (i) for every function f ∈ AP(R,H) such that σ(f ) ⊆ Λ, there exists a unique mild solution u ∈ AP(R,H) of Eq. (∗) such that σ(u) ⊆ Λ; (ii) [ B(λ)−iλ] is invertible for all λ ∈ Λ and supλ∈Λ [ B(λ) −iλ]−1 <∞. © 2006 Elsevier Inc. All rights reserved