On the structure of the essential spectrum of elliptic operators on metric spaces

TL;DR

This paper characterizes the essential spectrum of a broad class of elliptic-like operators on metric measure spaces by analyzing their behavior at infinity, focusing on the structure of the associated $C^*$-algebra.

Contribution

It provides a novel description of the essential spectrum for elliptic operators on metric spaces, extending classical Euclidean results to more general settings.

Findings

Description of the essential spectrum via localizations at infinity

Analysis of the ideal structure of the $C^*$-algebra generated by these operators

Extension of Euclidean elliptic operator theory to metric measure spaces

Abstract

We give a description of the essential spectrum of a large class of operators on metric measure spaces in terms of their localizations at infinity. These operators are analogues of the elliptic operators on Euclidean spaces and our main result concerns the ideal structure of the $C^*$-algebra generated by them.