Hölder functions are operator-Hölder
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L. N. Nikol′skaya and Yu. B. Farforovskaya
Translated by: S. Kislyakov - St. Petersburg Math. J. 22 (2011), 657-668
- DOI: https://doi.org/10.1090/S1061-0022-2011-01161-6
- Published electronically: May 3, 2011
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Abstract:
Let $A$ and $B$ be selfadjoint operators in a separable Hilbert space such that $A-B$ is bounded. If a function $f$ satisfies the Hölder condition of order $\alpha$, $0<\alpha <1$, i.e., $|f(x)-f(y)|\leq L|x-y|^{\alpha }$, then $\|f(A)-f(B)\|\leq CL{\|A-B\|}^\alpha$, where $C$ is a constant, specifically, $C=2^{1-\alpha }+2\pi \sqrt {8}\frac {1}{{(1-2^{\alpha -1})}^2}$. This result is a consequence of a general inequality in which the norm of $f(A)-f(B)$ is controlled in terms of the continuity modulus of $f$. Similar results are true for the quasicommutators $f(A)K-Kf(B)$, where $K$ is a bounded operator.References
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Bibliographic Information
- L. N. Nikol′skaya
- Affiliation: Institut Mathématique de Bordeaux, Université Bordeaux-1, 351 Libération, 33405 Talence, France
- Email: andreeva@math.u-bordeaux1.fr
- Yu. B. Farforovskaya
- Affiliation: Division of Mathematics, M. A. Bonch-Bruevich St. Petersburg University of Telecommunication, Naberezhnaya Moĭki 61, St. Petersburg 191186, Russia
- Email: rabk@sut.ru
- Received by editor(s): November 15, 2010
- Published electronically: May 3, 2011
- © Copyright 2011 American Mathematical Society
- Journal: St. Petersburg Math. J. 22 (2011), 657-668
- MSC (2010): Primary 47A56, 47A60, 47B25
- DOI: https://doi.org/10.1090/S1061-0022-2011-01161-6
- MathSciNet review: 2768964