Real Trace Expansions

Abstract

In this paper, we investigate trace expansions of operators of the form $A\eta (t\mathcal{L})$ where $\eta :\mathbb{R}\to \mathbb{C}$ 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 $t\to 0^{+}$. 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.