Multiple operator integrals, pseudodifferential calculus, and asymptotic expansions

TL;DR

This paper extends multiple operator integrals to unbounded operators using pseudodifferential calculus, enabling new expansions in noncommutative geometry and spectral theory.

Contribution

It introduces a framework for operator integrals with unbounded operators via pseudodifferential calculus, facilitating spectral action expansions in noncommutative geometry.

Findings

Provided a perturbative expansion of the spectral action for spectral triples.

Derived an asymptotic expansion of the trace of heat operators as time approaches zero.

Extended the theory of multiple operator integrals to unbounded operators in a noncommutative setting.

Abstract

We push the definition of multiple operator integrals (MOIs) into the realm of unbounded operators, using the pseudodifferential calculus from the works of Connes and Moscovici, Higson, and Guillemin. This in particular provides a natural language for operator integrals in noncommutative geometry. For this purpose, we develop a functional calculus for these pseudodifferential operators. To illustrate the power of this framework, we provide a pertubative expansion of the spectral action for regular $s$-summable spectral triples $(\mathcal{A}, \mathcal{H}, D)$, and an asymptotic expansion of $\mathrm{Tr}(P e^{-t(D+V)^2})$ as $t \downarrow 0$, where $P$ and $V$ belong to the algebra generated by $\mathcal{A}$ and $D$, and $V$ is bounded and self-adjoint.