Trace Expansions and Equivariant Traces on an Algebra of Fourier Integral Operators on $\mathbb R^n$

TL;DR

This paper develops trace expansions and equivariant traces for an algebra of Fourier integral operators on b^n, extending classical residue concepts and utilizing the Shubin calculus to analyze operator properties.

Contribution

It introduces a noncommutative residue and localized equivariant traces for an algebra generated by pseudodifferential, Heisenberg-Weyl, and metaplectic operators, extending trace theory.

Findings

Derived trace expansions using an auxiliary Shubin operator.

Defined a generalized noncommutative residue for the algebra.

Constructed localized equivariant traces on the operator algebra.

Abstract

We consider the operator algebra $\mathscr A$ on $\mathscr S(\mathbb R^n)$ generated by the Shubin type pseudodifferential operators, the Heisenberg-Weyl operators and the lifts of the unitary operators on $\mathbb C^n$ to metaplectic operators. With the help of an auxiliary operator in the Shubin calculus, we find trace expansions for these operators in the spirit of Grubb and Seeley. Moreover, we can define a noncommutative residue generalizing that for the Shubin pseudodifferential operators and obtain a class of localized equivariant traces on the algebra.