Locally definite normal operators in Krein spaces

TL;DR

This paper introduces spectral points of positive type for bounded normal operators in Krein spaces and demonstrates the existence of local spectral functions and positivity properties of spectral subspaces.

Contribution

It establishes the concept of spectral points of two-sided positive type and shows that certain spectral subspaces are uniformly positive and normal in a Hilbert space.

Findings

Normal operators have local spectral functions on two-sided positive type sets.

Spectral subspaces corresponding to positive type spectral sets are uniformly positive.

Restrictions of these operators to such subspaces are normal in a Hilbert space.

Abstract

We introduce the spectral points of two-sided positive type of bounded normal operators in Krein spaces. It is shown that a normal operator has a local spectral function on sets which are of two-sided positive type. In addition, we prove that the Riesz-Dunford spectral subspace corresponding to a spectral set which is only of positive type is uniformly positive. The restriction of the operator to this subspace is then normal in a Hilbert space.