Gohberg lemma, compactness, and essential spectrum of operators on compact Lie groups

TL;DR

This paper extends the Gohberg lemma to compact Lie groups, providing estimates for operator distances to compact operators and criteria for compactness based on matrix-valued symbols, impacting spectral analysis.

Contribution

It introduces a version of the Gohberg lemma for compact Lie groups and establishes new bounds for the essential spectrum using matrix-valued symbols.

Findings

Derived a lower estimate for the distance to compact operators on compact Lie groups.

Provided criteria for operator compactness based on matrix-valued symbols.

Established bounds for the essential spectrum of operators.

Abstract

In this paper we prove a version of the Gohberg lemma on compact Lie groups giving an estimate from below for the distance from a given operator to the set of compact operators on compact Lie groups. As a consequence, we prove several results on bounds for the essential spectrum and a criterion for an operator to be compact. The conditions are given in terms of the matrix-valued symbols of operators.