Spectral Points of Type $\pi_+$ and Type $\pi_-$ of Closed Operators in Indefinite Inner Product Spaces

TL;DR

This paper introduces spectral points of specific types for closed operators in indefinite inner product spaces, analyzes their stability under perturbations, and establishes properties of the resolvent and local spectral functions.

Contribution

It defines spectral points of type π+ and π− in indefinite inner product spaces and studies their stability, resolvent growth, and local spectral functions.

Findings

Spectral points of type π+ and π− are stable under compact perturbations.

The resolvent of symmetric operators has finite order growth near certain spectral points.

A local spectral function exists on intervals of type π+ or π−.

Abstract

We introduce the notion of spectral points of type $\pi_+$ and type $\pi_-$ of closed operators $A$ in a Hilbert space which is equipped with an indefinite inner product. It is shown that these points are stable under compact perturbations. In the second part of the paper we assume that $A$ is symmetric with respect to the indefinite inner product and prove that the growth of the resolvent of $A$ is of finite order in a neighborhood of a real spectral point of type $\pi_+$ or $\pi_-$ which is not in the interior of the spectrum of $A$. Finally, we prove that there exists a local spectral function on intervals of type $\pi_+$ or $\pi_-$.