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Zapiski Nauchnykh Seminarov POMI, 2019, Volume 480, Pages 26–47
(Mi znsl6763)
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This article is cited in 1 scientific paper (total in 1 paper)
Some remarks concerning operator Lipschitz functions
A. B. Aleksandrov St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
Abstract:
We consider examples of operator Lipschitz functions $f$ for which the operator Lipschitz seminorm $\|f\|_{\mathrm{OL}(\mathbb{R})}$ coincides with the Lipschitz seminorm $\|f\|_{\mathrm{Lip}(\mathbb{R})}$. In particular, we consider the operator Lipschitz functions such that $f'(0)=\|f\|_{\mathrm{OL}(\mathbb{R})}$. It is well known that every function $f$ whose the derivative $f'$ is positive definite has this property. In the paper it is proved that there are other functions having this property. It is also shown that the identity $|f'(t_0)|=\|f\|_{\mathrm{OL}(\mathbb{R})}$ implies that the derivative of $f$ is continuous at $t_0$. In fact, a more general statement is established concerning commutator Lipschitz functions on a closed subset of the complex plane.
Key words and phrases:
operator Lipschitz functions.
Received: 26.08.2019
Citation:
A. B. Aleksandrov, “Some remarks concerning operator Lipschitz functions”, Investigations on linear operators and function theory. Part 47, Zap. Nauchn. Sem. POMI, 480, ПОМИ, СПб., 2019, 26–47
Linking options:
https://www.mathnet.ru/eng/znsl6763 https://www.mathnet.ru/eng/znsl/v480/p26
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Abstract page: | 160 | Full-text PDF : | 35 | References: | 29 |
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