Functions of perturbed noncommuting self-adjoint operators

TL;DR

This paper investigates the behavior of functions of noncommuting self-adjoint operators, establishing Lipschitz estimates in the trace norm for functions in a specific Besov class, but not in the operator norm.

Contribution

It proves trace norm Lipschitz estimates for functions of noncommuting operators within a Besov class, highlighting limitations in operator norm estimates.

Findings

Lipschitz estimate holds in trace norm for functions in B_{∞,1}^1

No Lipschitz estimate in operator norm for the same class

Functions in the Besov class enable trace norm control of operator perturbations

Abstract

We consider functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the following Lipschitz type estimate in the trace norm: $\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_1}\le\const(\|A_1-A_2\|_{\bS_1}+\|B_1-B_2\|_{\bS_1})$. However, the condition $f\in B_{\be,1}^1(\R^2)$ does not imply the Lipschitz type estimate in the operator norm.