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

Abstract

We consider an algebra $\mathscr A$ of Fourier integral operators on $\mathbb R^n$. It consists of all operators $D: \mathscr S(\mathbb R^n)\to \mathscr S(\mathbb R^n)$ on the Schwartz space $\mathscr S(\mathbb R^n)$ that can be written as finite sums $$ D= \sum R_gT_w A, $$ 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 $D \in \mathscr A$ and a suitable auxiliary Shubin pseudodifferential operator $H$ we establish expansions for $\mathop{\mathrm {Tr}}(D(H-\lambda)^{-K})$ as $|\lambda| \to \infty$ in a sector of $\mathbb C$ for sufficiently large $K$ and of $\mathop{\mathrm {Tr}}(De^{-tH})$ as $t\to 0^+$. We also obtain the singularity structure of the meromorphic extension of $z\mapsto \mathop{\mathrm{Tr}}(DH^{-z})$ 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.