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

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 $B_{\text{be},1}^1({\mathbb{R}}^2)$, then we have the following Lipschitz type estimate in the Schatten–von Neumann norm ${\text{bS}}_p$, $1\le p\le 2$ norm: $\mathrm{\parallel }f(A_1,B_1)-f(A_2,B_2){\mathrm{\parallel }}_{{\text{bS}}_p}\le \text{const}(\mathrm{\parallel }{A}_1-A_2{\mathrm{\parallel }}_{{\text{bS}}_p}+\mathrm{\parallel }{B}_1-B_2{\mathrm{\parallel }}_{{\text{bS}}_p})$. However, the condition $f\in B_{\text{be},1}^1({\mathbb{R}}^2)$ does not imply the Lipschitz type estimate in ${\text{bS}}_p$ with $p>2$. The main tool is Schatten–von Neumann norm estimates for triple operator integrals.