Lipschitz Functions, Schatten Ideals, and Unbounded Derivations

It is proved that, for any Lipschitz function $f(t_1,\dots,t_n)$ of $n$ variables, the corresponding map $f_{op}\colon(A_1,\dots,A_n)\mapsto f(A_1,\dots,A_n)$ on the set of all commutative $n$-tuples of Hermitian operators on a Hilbert space is Lipschitz with respect to the norm of each Schatten ideal $\mathcal{S}^p$, $p\in(1,\infty)$. This result is applied to the functional calculus of normal operators and contractions. It is shown that Lipschitz functions of one variable preserve domains of closed derivations with values in $\mathcal{S}^p$. It is also proved that the map $f_{op}$ is Fréchet differentiable in the norm of $\mathcal{S}^p$ if $f$ is continuously differentiable.