Operator-Lipschitz functions on symmetrically normed ideals with non-trivial Boyd indices

Abstract

We show that if the Boyd indices of a symmetrically normed ideal J of B(H) are non-trivial (differ from 1 and oo) then for any Lipschitz function f on C, the map N --> f(N) is Lipschitz on the set of normal operators with respect to the norm ||.||_J . In particular, we study ideals for which some enhanced forms of the Fuglede Theorem hold; this is necessary for the work with functions of normal operators. We also consider functions that have the property of J-stability with respect to an ideal J: if a normal operator N is perturbed by an operator X in J in such a way that the operator N +X is normal, then f(N +X) - f(N) belongs to J. As applications we present various results on Gateaux and Frechet J-differentiability of functions, and on the action of Lipschitz functions on the domains of derivations