Variation of discrete spectra of non-negative operators in Krein spaces

TL;DR

This paper investigates how the discrete spectrum of a non-negative operator in a Krein space changes under Schatten class perturbations, extending a classical Hilbert space result to Krein spaces.

Contribution

It generalizes Kato's theorem on spectral variation from Hilbert spaces to Krein spaces for non-negative operators under Schatten class perturbations.

Findings

Discrete eigenvalues' differences form an lp-sequence

Extended enumerations of eigenvalues are used

Results extend classical Hilbert space theorems to Krein spaces

Abstract

We study the variation of the discrete spectrum of a bounded non-negative operator in a Krein space under a non-negative Schatten class perturbation of order $p$. It turns out that there exist so-called extended enumerations of discrete eigenvalues of the unperturbed and the perturbed operator, respectively, whose difference is an $\ell^p$-sequence. This result is a Krein space version of a theorem by T.Kato for bounded selfadjoint operators in Hilbert spaces.