Operator–Lipschitz functions on symmetrically normed ideals with nontrivial Boyd indices

Abstract We show that if the Boyd indices of a symmetrically normed ideal of are nontrivial (differ from 1 and ), then for any Lipschitz function on , the map is Lipschitz on the set of normal operators with respect to the norm . 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 ‐stability with respect to an ideal : if a normal operator is perturbed by an operator in such a way that the operator is normal, then . As applications, we present various results on Gateaux and Frechet ‐differentiability of functions, and on the action of Lipschitz functions on the domains of derivations.