Functions of perturbed noncommuting unbounded self-adjoint operators (2109.02339v1)

Abstract: Let $f$ be a function on ${\Bbb R}^2$ in the inhomogeneous Besov space $B_{\infty,1}^1({\Bbb R}^2)$. 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 $1\le p\le2$, $(A_1,B_1)$ 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 $\boldsymbol{S}p$ and $f$ is in the above inhomogeneous Besov space, then the following Lipschitz type estimate holds: $$ |f(A_1,B_1)-f(A_2,B_2)|{\boldsymbol{S}p} \le\operatorname{const}\max\big{|A_1-A_2|{\boldsymbol{S}p},|B_1-B_2|{\boldsymbol{S}_p}\big}. $$