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Operator smoothness in Schatten norms for functions of several variables: Lipschitz conditions, differentiability and unbounded derivations
Author(s) -
Kissin E.,
Potapov D.,
Shulman V.,
Sukochev F.
Publication year - 2012
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pds014
Subject(s) - mathematics , lipschitz continuity , differentiable function , bounded function , directional derivative , pure mathematics , smoothness , lipschitz domain , operator (biology) , function (biology) , open set , hilbert space , mathematical analysis , biochemistry , chemistry , repressor , transcription factor , gene , evolutionary biology , biology
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 .