The Wodzicki residue for pseudo-differential operators on compact Lie groups

TL;DR

This paper computes the Wodzicki residue for pseudo-differential operators on compact Lie groups using analytic continuation of traces, expressing it via the matrix-valued symbol on the non-commutative phase space.

Contribution

It introduces a method to compute the Wodzicki residue for classical pseudo-differential operators on compact Lie groups without requiring ellipticity, extending previous results.

Findings

Derived explicit formula for Wodzicki residue in terms of matrix symbols.

Extended the class of operators for which the residue can be computed.

Removed ellipticity condition in the residue calculation for Hörmander classes.

Abstract

Let $G$ be an arbitrary compact Lie group. In this work we apply the method of the analytic continuation of traces in order to compute the Wodzicki residue for a classical pseudo-differential operator on $G$ in terms of its matrix-valued symbol (which is globally defined on the non-commutative phase space $G\times \widehat{G},$ with $\widehat{G}$ being the unitary dual of $G$). Our main theorem is complementary to the results in [2], where we remove the ellipticity hypothesis when the operators belong to the H\"ormander classes on $G$ defined by local coordinate systems.