Essential Spectrum and Fredholm Properties for Operators on Locally Compact Groups

TL;DR

This paper investigates the essential spectrum and Fredholm properties of integral and pseudodifferential operators on locally compact groups, extending spectral analysis techniques using crossed product C*-algebras.

Contribution

It introduces new spectral analysis methods for operators affiliated with crossed products on non-commutative groups, especially for unimodular, type I groups.

Findings

Extended the structure of the essential spectrum to new classes of operators.

Analyzed the role of representations in spectral theory.

Applied to operators with operator-valued symbols involving the unitary dual of G.

Abstract

We study the essential spectrum and Fredholm properties of integral and pseudodiferential operators associated to (maybe non-commutative) locally compact groups G. The techniques involve crossed product C*-algebras. We extend previous results on the structure of the essential spectrum to operators belonging (or affiliated) to the Schr\"odinger representation of certain crossed products. When the group G is unimodular and type I, we cover a new class of pseudo-differential differential operators with operator-valued symbols involving the unitary dual of G. We use recent results on the role of families of representations in spectral theory and the notion of quasi-regular dynamical system.