Triple operator integrals in Schatten--von Neumann norms and functions of perturbed noncommuting operators

TL;DR

This paper investigates how functions of noncommuting self-adjoint operators behave under perturbations, establishing Lipschitz estimates in Schatten--von Neumann norms for functions in a specific Besov class, using triple operator integrals.

Contribution

It provides new Lipschitz-type bounds for functions of noncommuting operators in Schatten norms, extending understanding of perturbation effects in operator theory.

Findings

Lipschitz estimates hold for p in [1,2] in Schatten norms.

Condition on function f is in Besov class B_{∞,1}^1(R^2).

Estimates do not extend to p > 2.

Abstract

We study perturbations of 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 Schatten--von Neumann norm $\bS_p$, $1\le p\le2$ norm: $\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_p}\le\const(\|A_1-A_2\|_{\bS_p}+\|B_1-B_2\|_{\bS_p})$. However, the condition $f\in B_{\be,1}^1(\R^2)$ does not imply the Lipschitz type estimate in $\bS_p$ with $p>2$. The main tool is Schatten--von Neumann norm estimates for triple operator integrals.