1211.0791 (Vladimir Georgescu et al.)

Boundary values of resolvents of self-adjoint operators in Krein spaces    [PDF]

Vladimir Georgescu, Christian Gérard, Dietrich Häfner

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 $<\cdot| \cdot>$, $A$ is the generator of a $C_{0}-$group on $\cH$ and $I\subset \rr$ is an interval such that: 1) $H$ admits a Borel functional calculus on $I$, 2) the spectral projection $\one_{I}(H)$ is positive in the Krein sense, 3) the following positive commutator estimate holds: [\Re \geq c , u \in {\rm Ran}\one_{I}(H), c>0.] 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]}| ^{-s}(H-z)^{-1}^{-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 charge spaces of Cauchy data.

View original: http://arxiv.org/abs/1211.0791