Essential Spectra of Linear Relations

TL;DR

This paper introduces five essential spectra for linear relations based on semi-Fredholm properties, explores their fundamental characteristics, and extends classical spectral theorems to this broader context.

Contribution

It generalizes Weyl's theorem and spectral mapping results to linear relations, expanding spectral theory beyond single-valued operators.

Findings

Defined five essential spectra for linear relations

Established basic properties of these spectra

Extended Weyl's theorem and proved a spectral mapping theorem

Abstract

Five essential spectra of linear relations are defined in terms of semi-Fredholm properties and the index. Basic properties of these sets are established and the perturbation theory for semi-Fredholm relations is then applied to verify a generalisation of Weyl's theorem for single-valued operators. We conclude with a spectral mapping theorem.