
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 where and are classical pseudo-differential operators on a compact manifold and is elliptic and self-adjoint admits an expansion in powers of . The functions being smooth and compactly supported on 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 .
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