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This article is cited in 2 scientific papers (total in 2 papers)
Brief communications
Lipschitz Functions, Schatten Ideals, and Unbounded Derivations
È. V. Kissina, D. S. Potapovb, F. A. Sukochevb, V. S. Shulmanc a STORM Research Center, London Metropolitan University
b University of New South Wales, Australia
c Vologda State Technical University
Abstract:
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.
Keywords:
functions of operators, operator Lipschitz functions, Schatten classes, unbounded derivations.
Received: 10.04.2010
Citation:
È. V. Kissin, D. S. Potapov, F. A. Sukochev, V. S. Shulman, “Lipschitz Functions, Schatten Ideals, and Unbounded Derivations”, Funktsional. Anal. i Prilozhen., 45:2 (2011), 93–96; Funct. Anal. Appl., 45:2 (2011), 157–159
Linking options:
https://www.mathnet.ru/eng/faa3021https://doi.org/10.4213/faa3021 https://www.mathnet.ru/eng/faa/v45/i2/p93
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