The behavior of functions of operators under perturbations

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 ${\text{bS}}_p$ 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.