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 $B_{\mathrm{\infty },1}^1({\mathbb{R}}^2)$, 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 $1\le p\le 2$, 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 ${\mathit{S}}_p$ with $p\in [1,2]$ and $f\in B_{\mathrm{\infty },1}^1({\mathbb{R}}^2)$, then the following Lipschitz type estimate holds: $$\mathrm{\parallel }f(A_1,B_1)-f(A_2,B_2){\mathrm{\parallel }}_{{\mathit{S}}_p}\le \mathrm{const}\mathrm{\parallel }f{\mathrm{\parallel }}_{B_{\mathrm{\infty },1}^1}\max \{\mathrm{\parallel }{A}_1-A_2{\mathrm{\parallel }}_{{\mathit{S}}_p},\mathrm{\parallel }{B}_1-B_2{\mathrm{\parallel }}_{{\mathit{S}}_p}\}\mathrm{.}$$