Functions of perturbed noncommuting unbounded self-adjoint operators

Abstract

Let $f$ be a function on ${\mathbb{R}}^2$ in the inhomogeneous Besov space $B_{\infty ,1}^1({\mathbb{R}}^2)$. For a pair $(A,B)$ of not necessarily bounded and not necessarily commuting self-adjoint operators, we define the function $f(A,B)$ of $A$ and $B$ as a densely defined linear operator. We show that if $1\le p\le 2$, $(A_1,B_1)$ and $(A_2,B_2)$ are pairs of not necessarily bounded and not necessarily commuting self-adjoint operators such that both $A_1-A_2$ and $B_1-B_2$ belong to the Schatten--von Neumann class ${\mathbf{S}}_p$ and $f$ is in the above inhomogeneous Besov space, then the following Lipschitz type estimate holds: $$\Vert f(A_1,B_1)-f(A_2,B_2){\Vert }_{{\mathbf{S}}_p}\le \mathrm{const}\max \{\Vert A_1-A_2{\Vert }_{{\mathbf{S}}_p},\Vert B_1-B_2{\Vert }_{{\mathbf{S}}_p}\}.$$

18 pages. arXiv admin note: text overlap with arXiv:1505.07173