Functions of almost commuting operators and an extension of the Helton-Howe trace formula

TL;DR

This paper develops a functional calculus for almost commuting self-adjoint operators using Besov class functions and extends the Helton-Howe trace formula to a broader class of functions.

Contribution

It introduces a new functional calculus for almost commuting operators in the Besov class and extends the Helton-Howe trace formula to these functions.

Findings

Constructed a linear functional calculus for operators in the Besov class

Operators almost commute under this calculus

Extended the Helton-Howe trace formula to all functions in the Besov class

Abstract

Let $A$ and $B$ be almost commuting (i.e., the commutator $AB-BA$ belongs to trace class) self-adjoint operators. We construct a functional calculus $\varphi\mapsto\varphi(A,B)$ for functions $\varphi$ in the Besov class $B_{\infty,1}^1({\Bbb R}^2)$. This functional calculus is linear, the operators $\varphi(A,B)$ and $\psi(A,B)$ almost commute for $\varphi,\,\psi\in B_{\infty,1}^1({\Bbb R}^2)$, and $\varphi(A,B)=u(A)v(B)$ whenever $\varphi(s,t)=u(s)v(t)$. We extend the Helton--Howe trace formula for arbitrary functions in $B_{\infty,1}^1({\Bbb R}^2)$. The main tool is triple operator integrals with integrands in Haagerup-like tensor products of $L^\infty$ spaces.