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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

Hölder functions are operator-Hölder


Authors: L. N. Nikol′skaya and Yu. B. Farforovskaya
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 22 (2010), nomer 4.
Journal: St. Petersburg Math. J. 22 (2011), 657-668
MSC (2010): Primary 47A56, 47A60, 47B25
Published electronically: May 3, 2011
MathSciNet review: 2768964
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Abstract | References | Similar Articles | Additional Information

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., $ \vert f(x)-f(y)\vert\leq L\vert x-y\vert^{\alpha }$, then $ \Vert f(A)-f(B)\Vert\leq CL{\Vert A-B\Vert}^\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.


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Additional 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

DOI: http://dx.doi.org/10.1090/S1061-0022-2011-01161-6
PII: S 1061-0022(2011)01161-6
Keywords: Operator-Hölder functions, Adamar–Schur multipliers
Received by editor(s): November 15, 2010
Published electronically: May 3, 2011
Article copyright: © Copyright 2011 American Mathematical Society