Functions of perturbed noncommuting unbounded self-adjoint operators

TL;DR

This paper establishes Lipschitz estimates for functions of noncommuting self-adjoint operators perturbed within Schatten classes, extending functional calculus to a broader class of functions in Besov spaces.

Contribution

It introduces a framework for defining functions of noncommuting unbounded self-adjoint operators and proves Lipschitz bounds in Schatten norms for perturbations.

Findings

Lipschitz estimates hold for functions in inhomogeneous Besov space $B_{ infty,1}^1$.

Perturbations in Schatten class $oldsymbol{S}_p$ lead to controlled differences in operator functions.

The results extend functional calculus to noncommuting unbounded operators with perturbations.

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: $$ \|f(A_1,B_1)-f(A_2,B_2)\|_{\boldsymbol{S}_p} \le\operatorname{const}\max\big\{\|A_1-A_2\|_{\boldsymbol{S}_p},\|B_1-B_2\|_{\boldsymbol{S}_p}\big\}. $$