Dixmier traces for discrete pseudo-differential operators

TL;DR

This paper establishes criteria for when discrete pseudo-differential operators on integer lattices are Dixmier traceable, introduces a class of symbols for such operators, and provides a formula linking the Dixmier trace to the Wodzicki residue.

Contribution

It introduces a class of classical symbols for discrete pseudo-differential operators and derives a formula for their Dixmier trace using Connes' equivalence.

Findings

Characterization of Dixmier traceability for discrete pseudo-differential operators

Introduction of a symbol class ensuring traceability

Explicit formula for the Dixmier trace via the Wodzicki residue

Abstract

In this paper we provide sharp results for the Dixmier traceability of discrete pseudo-differential operators on $\ell^2(\mathbb{Z}^n)$. In this setting, we introduce a suitable notion of a class of classical symbols which provide a class of Dixmier traceable discrete pseudo-differential operators. We also present a formula for the Dixmier trace of a Dixmier traceable discrete pseudo-differential operator by using the Connes equivalence between the Wodzicki residue and the Dixmier trace.