## Real Trace Expansions

##### Doc. Math. 24, 2159-2202 (2019)
DOI: 10.25537/dm.2019v24.2159-2202

### Summary

In this paper, we investigate trace expansions of operators of the form $A\eta(t\mathcal{L})$ where $\eta:\mathbb{R}\rightarrow\mathbb{C}$ is a Schwartz function, $A$ and $\mathcal L$ are classical pseudo-differential operators on a compact manifold $M$ with $\mathcal L$ elliptic. In particular, we show that, under certain hypotheses, this trace admits an expansion in powers of $t\rightarrow 0^+$. We also relate the constant coefficient to the non-commutative residue and the canonical trace of $A$. Our main tool is the continuous inclusion of the functional calculus of $\mathcal{L}$ into the pseudo-differential calculus whose proof relies on the Helffer-Sjöstrand formula.

58J40, 58J42

### Keywords/Phrases

pseudodifferential operators on manifolds, non-commutative residues, canonical trace

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### Affiliation

Fischer, Véronique
Department of Mathematical Sciences, University of Bath, BA2 7AY Bath, UK