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On Compact Perturbations of Locally Definitizable Selfadjoint Relations in Krein Spaces

Behrndt, Jussi; Jonas, Peter

Inst. Mathematik

The aim of this paper is to prove two perturbation results for a selfadjoint operator A in a Krein space H which can roughly be described as follows: (1) If Δ is an open subset of 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 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 H, is preserved under relative Sp 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.