Operator Lipschitz Functions

TL;DR

This survey comprehensively studies operator Lipschitz functions, providing conditions for their characterization, exploring operator differentiability, and examining their behavior on subsets of the plane using advanced tools like double operator integrals.

Contribution

It offers a detailed analysis of operator Lipschitz functions, including necessary and sufficient conditions, and extends the study to functions on subsets of the plane and commutator Lipschitz functions.

Findings

Characterization of operator Lipschitz functions through necessary and sufficient conditions

Analysis of operator differentiable functions on the real line

Application of double operator integrals and Schur multipliers

Abstract

The purpose of this survey article is a comprehensive study of operator Lipschitz functions. A continuous function $f$ on the real line ${\Bbb R}$ is called operator Lipschitz if $\|f(A)-f(B)\|\le{\rm const}\|A-B\|$ for arbitrary self-adjoint operators $A$ and $B$. We give sufficient conditions and necessary conditions for operator Lipschitzness. We also study the class of operator differentiable functions on ${\Bbb R}$. Then we consider operator Lipschitz functions on closed subsets of the plane as well as commutator Lipschitz functions on such subsets. Am important role is played by double operator integrals and Schur multipliers.