Almost commuting functions of almost commuting self-adjoint operators

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.