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Pré-Publication, Document De Travail Année : 2013

Boundary values of resolvents of self-adjoint operators in Krein spaces

Vladimir Georgescu
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Christian Gérard
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Dietrich Häfner

Résumé

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.
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Dates et versions

hal-00748181 , version 1 (05-11-2012)
hal-00748181 , version 2 (06-11-2012)
hal-00748181 , version 3 (07-01-2013)
hal-00748181 , version 4 (31-07-2013)

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Vladimir Georgescu, Christian Gérard, Dietrich Häfner. Boundary values of resolvents of self-adjoint operators in Krein spaces. 2013. ⟨hal-00748181v4⟩
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