The behavior of functions of operators under perturbations

TL;DR

This survey explores how functions of various classes of operators behave under perturbations, focusing on Lipschitz and differentiable functions, Schatten class perturbations, and trace formulas, including recent findings on operator H"older--Zygmund functions.

Contribution

It provides a comprehensive overview of the behavior of operator functions under perturbations, including recent novel results on operator H"older--Zygmund functions.

Findings

Analysis of operator Lipschitz and differentiable functions

Results on Schatten class perturbations and trace formulas

Recent discoveries on operator H"older--Zygmund functions

Abstract

This is a survey article. We consider different problems in connection with the behavior of functions of operators under perturbations of operators. We deal with three classes of operators: unitary operators, self-adjoint operators, and contractions. We study operator Lipschitz and operator differentiable functions. We also study the behavior of functions under perturbations of an operator by an operator of Schatten--von Neumann class $\bS_p$ and apply the results to the Livschits--Krein and Koplienko--Neidhardt trace formulae. We also include in this survey article recent unexpected results obtained in a joint paper with Aleksandrov on operator H\"older--Zygmund functions.