Spectral conditions for admissibility of evolution equations in Hilbert space

Abstract

We study properties of solutions of the evolution equation u ′ ( t ) = ( B u ) ( 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 ˆ ( λ ) of closed operators on H such that B ( e i λ t x ) = e i λ t B ˆ ( λ ) x (whenever e i λ t 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 ˆ ( λ ) − i λ ] is invertible for all λ ∈ Λ and sup λ ∈ Λ ‖ [ B ˆ ( λ ) − i λ ] −1 ‖ < ∞ .