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

Abstract: 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 ${\Bbb R}^2$ that belongs to the inhomogeneous Besov class $B_{\infty,1}^1({\Bbb R}^2)$, 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 $\boldsymbol{S}p$ with $p\in[1,2]$ and $f\in B{\infty,1}^1({\Bbb R}^2)$, then the following Lipschitz type estimate holds: [ |f(A_1,B_1)-f(A_2,B_2)|{\boldsymbol{S}_p} \le\operatorname{const}|f|{B_{\infty,1}^1}\max\big{|A_1-A_2|{\boldsymbol{S}_p},|B_1-B_2|{\boldsymbol{S}_p}\big}. ]