Operator Lipschitz functions (English translation)

Alexei Aleksandrov, Vladimir Peller

Published 2016-11-05Version 1

The purpose of this survey is a comprehensive study of operator Lip\-schitz functions. A continuous function $f$ on the real line ${\Bbb R}$ os called operator Lipschitz if $\|f(A)-f(B)\|\le\operatorname{const}\|A-B\|$ 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 ${\Bbb R}$ . Next, we consider operator Lipschitz functions on closed subsets of the plane and introduce the class of commutator Lipschitz functions on such subsets. An important role for the study of such classes of functions is played by double operator integrals and Schur multipliers.