Local Definitizability of $T^{[*]}T$ and $TT^{[*]}$

TL;DR

This paper investigates the local spectral properties of the operators $T^{[*]} T$ and $TT^{[*]}$ in Krein spaces, establishing conditions under which their local definitizability is equivalent and comparing their critical points.

Contribution

It provides new criteria for local definitizability of $T^{[*]} T$ and $TT^{[*]}$ in Krein spaces, linking their spectral properties under specific resolvent set conditions.

Findings

$T^{[*]} T$ is locally definitizable iff $TT^{[*]}$ is, given certain resolvent conditions.

The critical points of $T^{[*]} T$ and $TT^{[*]}$ are compared.

Spectral properties of operator products are analyzed in the context of Krein spaces.

Abstract

The spectral properties of two products $AB$ and $BA$ of possibly unbounded operators $A$ and $B$ in a Banach space are considered. The results are applied in the comparison of local spectral properties of the operators $T^{[*]} T$ and $TT^{[*]} $ in a Krein space. It is shown that under the assumption that both operators $T^{[*]} T$ and $T^{[*]} $ have non-empty resolvent sets, the operator $T^{[*]} T$ is locally definitizable if and only if $TT^{[*]} $ is. In this context the critical points of both operators are compared.