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 $\cH$, equipped with the Krein scalar product $\langle \cdot| \cdot \rangle$, $A$ is the generator of a $C_{ 0}-$group on $\cH$ and $I\subset \rr$ is an interval such that: \begin{ itemize} \item[]1) $H$ admits a Borel functional calculus on $I$, \item[]2) the spectral projection $\one_{ I}(H)$ is positive in the Krein sense, \item[]3) the following { \em positive commutator estimate} holds: \[ \Re \langle u| [H, \i A]u\rangle\geq c \langle u| u\rangle, \ u \in { \rm Ran}\one_{ I}(H), \ c>0. \] \end{ itemize} then assuming some smoothness of $H$ with respect to the group $\e^{ \i t A}$, the following resolvent estimates hold: \[ \sup_{ z\in I\pm \i]0, \nu]}\| \langle A\rangle ^{ -s}(H-z)^{ -1}\langle A\rangle^{ -s}\| <\infty, \ s>\12. \] As an application we consider abstract Klein-Gordon equations \[ \p_{ t}^{ 2}\phi(t)- 2 \i k \phi(t)+ h\phi(t)=0, \] and obtain resolvent estimates for their generators in { \em charge spaces} of Cauchy data.
Categories: math-ph math.AP math.MP math.SP