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

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