Real trace expansions

TL;DR

This paper establishes trace expansions for certain pseudo-differential operators on compact manifolds as a parameter approaches zero, linking the coefficients to non-commutative residues and canonical traces.

Contribution

It introduces new trace expansion results for operators with smooth, compactly supported functions, extending previous heat kernel and zeta function analyses.

Findings

Trace of $A\,\eta(t\mathcal{L})$ admits an expansion as $t \to 0^+$

Coefficients relate to non-commutative residue and canonical trace

No meromorphic properties unlike heat or zeta functions

Abstract

In this paper, we show that the trace of the operators $A\eta(t\mathcal{L})$ where $A$ and $\mathcal {L}$ are classical pseudo-differential operators on a compact manifold $M$ and $\mathcal {L}$ is elliptic and self-adjoint admits an expansion in powers of $t\to 0^+$. The functions $\eta$ being smooth and compactly supported on $\mathbb R$ have no meromorphic properties, unlike in the case of the heat trace or zeta functions. We also show that the constant coefficients in our expansions are related to the non-commutative residue and the canonical trace of $A$.