Local spectral theory for normal operators in Krein spaces

TL;DR

This paper extends local spectral theory to normal operators in Krein spaces, showing conditions under which these operators have spectral functions and are similar to normal operators in Hilbert spaces, with applications to quadratic operator polynomials.

Contribution

It introduces a new spectral analysis framework for normal operators in Krein spaces, linking sign type spectra to spectral functions and similarity to Hilbert space normal operators.

Findings

Normal operators with real spectra parts have local spectral functions.

Restrictions of such operators are normal in some Hilbert space.

Operators with entirely positive/negative spectra are similar to Hilbert space normal operators.

Abstract

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space. We use this result to show the existence of operator roots of a class of quadratic operator polynomials with normal coefficients.