Triple operator integrals in Schatten--von Neumann norms and functions of perturbed noncommuting operators

Abstract: We study perturbations of functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the following Lipschitz type estimate in the Schatten--von Neumann norm $\bS_p$, $1\le p\le2$ norm: $|f(A_1,B_1)-f(A_2,B_2)|{\bS_p}\le\const(|A_1-A_2|{\bS_p}+|B_1-B_2|{\bS_p})$. However, the condition $f\in B{\be,1}^1(\R^2)$ does not imply the Lipschitz type estimate in $\bS_p$ with $p>2$. The main tool is Schatten--von Neumann norm estimates for triple operator integrals.