On the index of pseudo-differential operators on compact Lie groups

TL;DR

This paper investigates the analytical index of pseudo-differential operators on compact Lie groups using operator-valued symbols and extends the McKean-Singer index formula with a new functional calculus approach.

Contribution

It introduces a novel application of operator-valued symbols and develops an operator-valued functional calculus to analyze the index of pseudo-differential operators on compact Lie groups.

Findings

Extended the McKean-Singer index formula to operator-valued symbols.

Developed a new operator-valued functional calculus.

Provided a framework for computing the analytical index in this setting.

Abstract

In this note we study the analytical index of pseudo-differential operators by using the notion of (infinite dimensional) operator-valued symbols (in the sense of Ruzhansky and Turunen). Our main tools will be the McKean-Singer index formula together with the operator-valued functional calculus developed here.