# The Dixmier trace and the noncommutative residue for multipliers on compact manifolds

## TL;DR

This paper provides explicit formulas for the Dixmier trace and noncommutative residue of pseudo-differential operators on compact manifolds, utilizing global symbol calculus and Fourier analysis, applicable to manifolds with or without boundary.

## Contribution

It introduces new formulas for these traces using global symbols and extends analysis to manifolds with boundary, incorporating representation theory and symbolic calculus.

## Key findings

- Formulas for Dixmier trace and noncommutative residue derived
- Application to invariant pseudo-differential operators on compact Lie groups
- Extension of analysis to manifolds with boundary

## Abstract

In this paper we give formulae for the Dixmier trace and the noncommutative residue (also called Wodzicki's residue) of pseudo-differential operators by using the notion of global symbol. We consider both cases, compact manifolds with or without boundary. Our analysis on the Dixmier trace of invariant pseudo-differential operators on closed manifolds will be based on the Fourier analysis associated to every elliptic and positive operator and the quantization process developed by Delgado and Ruzhansky. In particular, for compact Lie groups this can be done by using the representation theory of the group in view of the Peter-Weyl theorem and the Ruzhansky-Turunen symbolic calculus. The analysis of invariant pseudo-differential operators on compact manifolds with boundary will be based on the global calculus of pseudo-differential operators developed by Ruzhansky and Tokmagambetov.