The spectrum and the Weyr characteristics of operator pencils and linear relations

TL;DR

This paper explores the relationship between the spectra and Weyr characteristics of operator pencils with unbounded coefficients and associated linear relations, revealing their equivalence and analyzing perturbation effects.

Contribution

It establishes the equivalence of spectra and Weyr characteristics for operator pencils and linear relations, and studies their behavior under one-dimensional perturbations.

Findings

Spectra of operator pencils and linear relations coincide.

Weyr characteristics are the same for both operator pencils and linear relations.

One-dimensional perturbations affect Weyr characteristics in a quantifiable way.

Abstract

The relation between the spectra of operator pencils with unbounded coefficients and of associated linear relations is investigated. It turns out that various types of spectrum coincide and the same is true for the Weyr characteristics. This characteristic describes how many independent Jordan chains up to a certain length exist. Furthermore, the change of this characteristic subject to one-dimensional perturbations is investigated.