Boundary values of resolvents of self-adjoint operators in Krein spaces
arXiv:1211.0791
Abstract
We prove in this paper resolvent estimates for the boundary values of resolvents of selfadjoint operators on a Krein space: if is a selfadjoint operator on a Krein space $\cH$, equipped with the Krein scalar product , is the generator of a group on $\cH$ and $I\subset \rr$ is an interval such that: \begin{itemize} \item[]1) admits a Borel functional calculus on , \item[]2) the spectral projection $\one_{I}(H)$ is positive in the Krein sense, \item[]3) the following {\em positive commutator estimate} holds: \[ \Re \langle u| [H, ıA]u\rangle\geq c \langle u| u\rangle, \ u \in {\rm Ran}\one_{I}(H), \ c>0. \] \end{itemize} then assuming some smoothness of with respect to the group $\e^{ıt A}$, the following resolvent estimates hold: \[ \sup_{z\in I\pm ı]0, ν]}\| \langle A\rangle ^{-s}(H-z)^{-1}\langle A\rangle^{-s}\| <\infty, \ s>\12. \] As an application we consider abstract Klein-Gordon equations \[ \p_{t}^{2}ϕ(t)- 2 ık ϕ(t)+ hϕ(t)=0, \] and obtain resolvent estimates for their generators in {\em charge spaces} of Cauchy data.