Boundary values of resolvents of selfadjoint operators in Krein spaces

Abstract

We prove in this paper resolvent estimates for the boundary values of resolvents of selfadjoint operators on a Krein space: if H is a selfadjoint operator on a Krein space H, equipped with the Krein scalar product 〈⋅|⋅〉, A is the generator of a C0-group on H and I⊂R is an interval such that:1)H admits a Borel functional calculus on I,2)the spectral projection 1I(H) is positive in the Krein sense,3)the following positive commutator estimate holds:Re〈u|[H,iA]u〉⩾c〈u|u〉,u∈Ran1I(H),c>0. then assuming some smoothness of H with respect to the group eitA, the following resolvent estimates hold:supz∈I±i]0,ν]‖〈A〉−s(H−z)−1〈A〉−s‖<∞,s>12. As an application we consider abstract Klein–Gordon equations∂t2ϕ(t)−2ikϕ(t)+hϕ(t)=0, and obtain resolvent estimates for their generators in charge spaces of Cauchy data.