Noncommutative Residues, Equivariant Traces, and Trace Expansions for an Operator Algebra on ${\mathbb{R}}^n$

Abstract

We consider an algebra $\mathcal{A}$ of Fourier integral operators on ${\mathbb{R}}^n$. It consists of all operators $D:\mathcal{S}({\mathbb{R}}^n)\to \mathcal{S}({\mathbb{R}}^n)$ on the Schwartz space $\mathcal{S}({\mathbb{R}}^n)$ that can be written as finite sums $$D=\sum R_g{T}_{w}A,$$ with Shubin type pseudodifferential operators $A$, Heisenberg-Weyl operators $T_w$, $w\in {\mathbb{C}}^n$, and lifts $R_g$, $g\in \mathrm{U}(n)$, of unitary matrices $g$ on ${\mathbb{C}}^n$ to operators $R_g$ in the complex metaplectic group. For $D\in \mathcal{A}$ and a suitable auxiliary Shubin pseudodifferential operator $H$ we establish expansions for $\mathrm{Tr}(D(H-\lambda {)}^{-K})$ as $|\lambda |\to \infty$ in a sector of $\mathbb{C}$ for sufficiently large $K$ and of $\mathrm{Tr}(D{e}^{-tH})$ as $t\to 0^{+}$. We also obtain the singularity structure of the meromorphic extension of $z\mapsto \mathrm{Tr}(D{H}^{-z})$ to $\mathbb{C}$. Moreover, we find a noncommutative residue as a suitable coefficient in these expansions and construct from it a family of localized equivariant traces on the algebra.

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