Functions of pairs of unbounded noncommuting self-adjoint operators under perturbation

For a pair $(A,B)$ of not necessarily bounded and not necessarily commuting self-adjoint operators and for a function $f$ on the Euclidean space $\mathbb{R}^2$ that belongs to the inhomogeneous Besov class $B^1_{\infty,1}(\mathbb{R}^2)$, we define the function $f(A,B)$ of these operators as a densely defined operator. We consider the problem of estimating the functions $f(A,B)$ under perturbations of the pair $(A,B)$. It is established that if $1\le p\le2$, and $(A_1,B_1)$ and $(A_2,B_2)$ are pairs of not necessarily bounded and not necessarily commuting self-adjoint operators such that the operators $A_1-A_2$ and $B_1-B_2$ belong to the Schatten–von Neumann class $\mathrm{S}_p$ with $p\in[1,2]$ and $f\in B^1_{\infty,1}(\mathbb{R}^2)$, then the following Lipschitz type estimate holds: $\|f(A_1,B_1)-f(A_2,B_2)\|_{\mathrm{S}_p}\le\operatorname{const}\|f\|_{B^1_{\infty,1}}\max\{\|A_1-A_2\|_{S_p},\|B_1-B_2\|_{\mathrm{S}_p}\}$.