Lipschitz Functions, Schatten Ideals, and Unbounded Derivations

È. V. Kissin ; D. S. Potapov ; F. A. Sukochev ; V. S. Shulman

Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 2, pp. 93-96

Résumé

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.

Détail

Keywords: functions of operators, operator Lipschitz functions, Schatten classes, unbounded derivations.