Spectral conditions for admissibility of evolution equations in Hilbert space

AbstractWe 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 Bˆ(λ) of closed operators on H such that B(eiλtx)=eiλtBˆ(λ)x (whenever eiλtx∈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) [Bˆ(λ)−iλ] is invertible for all λ∈Λ and supλ∈Λ‖[Bˆ(λ)−iλ]−1‖<∞