Dixmier traces for discrete pseudo-differential operators

In this paper we provide sharp results for the Dixmier traceability of discrete pseudo-differential operators on ${\mathrm{\ell }}^2({\mathbb{Z}}^n)$. 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.