Global functional calculus for operators on compact Lie groups

TL;DR

This paper develops a global functional calculus for elliptic operators on compact Lie groups, providing symbolic descriptions of complex powers and a constructive proof of G {a}rding inequality without classical pseudo-differential assumptions.

Contribution

It introduces a new functional calculus framework for elliptic operators on compact Lie groups, extending beyond classical pseudo-differential operators.

Findings

Symbolic descriptions of complex powers of elliptic operators

Constructive symbolic proof of G {a}rding inequality

Extension of functional calculus to broader operator classes

Abstract

In this paper we develop the functional calculus for elliptic operators on compact Lie groups without the assumption that the operator is a classical pseudo-differential operator. Consequently, we provide a symbolic descriptions of complex powers of such operators. As an application, we give a constructive symbolic proof of the G\r{a}rding inequality for operators in $(\rho,\delta)$-classes in the setting of compact Lie groups.