Functions of perturbed noncommuting unbounded selfadjoint operators

Abstract

Let f f be a function on R 2 \mathbb {R}^2 in the inhomogeneous Besov space {\text \textit {\Russian {B}}}_{\infty ,1}^{1}(\mathbb {R}^2). For a pair ( A , B ) (A,B) of not necessarily bounded and not necessarily commuting self-adjoint operators, the function f ( A , B ) f(A,B) of A A and B B is introduced as a densely defined linear operator. It is shown that if 1 ≤ p ≤ 2 1\le p\le 2 , ( A 1 , B 1 ) (A_1,B_1) and ( A 2 , B 2 ) (A_2,B_2) are pairs of not necessarily bounded and not necessarily commuting selfadjoint operators such that both A 1 − A 2 A_1-A_2 and B 1 − B 2 B_1-B_2 belong to the Schatten–von Neumann class S p {\boldsymbol {S}}_p and f\in {\text \textit {\Russian {B}}} _{\infty ,1}^{1}(\mathbb {R}^2), then the following Lipschitz type estimate holds: \begin{equation*} \|f(A_1,B_1)-f(A_2,B_2)\|_{{\boldsymbol {S}}_p} \le \operatorname {const}\|f\|_{\text \textit {\Russian {B}}_{\infty ,1}^{1}}\max \big \{\|A_1-A_2\|_{{\boldsymbol {S}}_p},\|B_1-B_2\|_{{\boldsymbol {S}}_p}\big \}. \end{equation*}