Operator smoothness in Schatten norms for functions of several variables: Lipschitz conditions, differentiability and unbounded derivations

The paper studies the action of functions of several variables on Schatten–von Neumann ideals p , 1< p <∞, of compact operators on Hilbert spaces. It shows that a function on ℝ n is an p ‐Lipschitz function on families of n commuting selfadjoint operators if and only if it is a Lipschitz function on ℝ n in the usual sense. It is proved also that a function in the disc algebra is an p ‐Lipschitz function on the set of all contractions if and only if its derivative is bounded on the disc. Furthermore, a function f on ℝ is Gateaux (respectively, Frechet) p ‐differentiable on an open subset α of ℝ if and only if f is differentiable on α and has bounded derivative on all its compact subsets (respectively, if and only if f ∈ C 1 (α)). Finally, it is established that Lipschitz functions of one or several variables preserve the domains of all closed *‐derivations on p .