A local version of Gearhart’s theorem

Let {etA: t ≥ 0} be a C0—semigroup on the Hilbert space ℋ. If x0 ∈ ℋ is such that the local resolvent R(λ,A)x0 admits a bounded holomorphic extension to the open half plane {Reλ > 0}, then limt→∞‖etAR(λ0, A)x0‖ = 0 for each λ0 ∈ ρ(A). This resuit is used to find mild spectral conditions which ensure the decay at infmity to zero of solutions of higher order abstract Cauchy problems.