On Compact Perturbations of Locally Definitizable Selfadjoint Relations in Krein Spaces

Abstract.

The aim of this paper is to prove two perturbation results for a selfadjoint operator A in a Krein space \( \mathcal{H} \) which can roughly be described as follows: (1) If Δ is an open subset of \( {\ifmmode\expandafter\overline\else\expandafter\=\fi{\mathbb{R}}} \) and all spectral subspaces for A corresponding to compact subsets of Δ have finite rank of negativity, the same is true for a selfadjoint operator B in \( \mathcal{H} \) for which the difference of the resolvents of A and B is compact. (2) The property that there exists some neighbourhood Δ_∞ of ∞ such that the restriction of A to a spectral subspace for A corresponding to Δ_∞ is a nonnegative operator in \( \mathcal{H}, \) is preserved under relative \( \mathfrak{G}_p \) perturbations in form sense if the resulting operator is again selfadjoint. The assertion (1) is proved for selfadjoint relations A and B. (1) and (2) generalize some known results.