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 𝒮p, 1<p<∞, of compact operators on Hilbert spaces. It shows that a function on ℝn is an 𝒮p-Lipschitz function on families of n commuting selfadjoint operators if and only if it is a Lipschitz function on ℝn in the usual sense. It is proved also that a function in the disc algebra is an 𝒮p-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) 𝒮p -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∈C1(α)). Finally, it is established that Lipschitz functions of one or several variables preserve the domains of all closed *-derivations on 𝒮p.

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