A Taylor-like Expansion of a Commutator with a Function of Self-Adjoint, Pairwise Commuting Operators

TL;DR

This paper develops a multidimensional Taylor-like expansion for the commutator of a bounded operator with a function of multiple commuting self-adjoint operators, extending previous one-dimensional results using advanced functional calculus techniques.

Contribution

It generalizes the Taylor expansion of commutators to multiple dimensions for functions of commuting self-adjoint operators, employing almost analytic extensions and Helffer-Sj"ostrand formula.

Findings

Established a multidimensional commutator expansion.

Extended the Helffer-Sj"ostrand formula to higher dimensions.

Provided a framework for analyzing functions of commuting operators.

Abstract

Let $A$ be a $\nu$-vector of self-adjoint, pairwise commuting operators and $B$ a bounded operator of class $C^{n_0}(A)$. We prove a Taylor-like expansion of the commutator $[B,f(A)]$ for a large class of functions $f\colon\mathbm{R}^\nu \to \mathbm{R}$, generalising the one-dimensional result where $A$ is just a self-adjoint operator. This is done using almost analytic extensions and the higher-dimensional Helffer-Sj\"ostrand formula.