Functions of pairs of unbounded noncommuting self-adjoint operators under perturbation

TL;DR

This paper develops Lipschitz estimates for functions of pairs of unbounded, noncommuting self-adjoint operators under perturbations, extending operator theory to a broader class of functions and operators.

Contribution

It introduces a method to estimate changes in functions of operator pairs under perturbations within Schatten classes, for noncommuting and unbounded operators.

Findings

Established Lipschitz type estimates in Schatten norms for operator functions.

Extended perturbation bounds to noncommuting, unbounded self-adjoint operator pairs.

Applied Besov class functions to operator perturbation analysis.

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 ${\Bbb R}^2$ that belongs to the inhomogeneous Besov class $B_{\infty,1}^1({\Bbb R}^2)$, 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 $1\le p\le2$, 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 $\boldsymbol{S}_p$ with $p\in[1,2]$ and $f\in B_{\infty,1}^1({\Bbb R}^2)$, then the following Lipschitz type estimate holds: \[ \|f(A_1,B_1)-f(A_2,B_2)\|_{\boldsymbol{S}_p} \le\operatorname{const}\|f\|_{B_{\infty,1}^1}\max\big\{\|A_1-A_2\|_{\boldsymbol{S}_p},\|B_1-B_2\|_{\boldsymbol{S}_p}\big\}. \]