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

We show that if the Boyd indices of a symmetrically normed ideal $J$ of $B\left(H\right)$ are nontrivial (differ from 1 and $\infty$), then for any Lipschitz function $f$ on $C$, the map $N\mapsto f\left(N\right)$ is Lipschitz on the set of normal operators with respect to the norm ${\parallel ·\parallel }_J$. 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 $f(N+X)-f\left(N\right)\in J$. 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.