Almost commuting functions of almost commuting self-adjoint operators

TL;DR

This paper develops a functional calculus for almost commuting self-adjoint operators using Besov class functions, ensuring the operators almost commute and satisfy the Helton–Howe trace formula, with triple operator integrals as a key tool.

Contribution

It introduces a new functional calculus for almost commuting self-adjoint operators based on Besov class functions, extending previous frameworks.

Findings

Constructed a linear functional calculus for almost commuting operators.

Operators $(A,B)$ and $'(A,B)$ almost commute for functions in the Besov class.

Helton–Howe trace formula holds for the constructed calculus.

Abstract

Let $A$ and $B$ be almost commuting (i.e, $AB-BA\in\bS_1$) self-adjoint operators. We construct a functional calculus $\f\mapsto\f(A,B)$ for $\f$ in the Besov class $B_{\be,1}^1(\R^2)$. This functional calculus is linear, the operators $\f(A,B)$ and $\psi(A,B)$ almost commute for $\f,\,\psi\in B_{\be,1}^1(\R^2)$, $\f(A,B)=u(A)v(B)$ whenever $\f(s,t)=u(s)v(t)$, and the Helton--Howe trace formula holds. The main tool is triple operator integrals.