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

 

Modulus of continuity of operator functions


Authors: Yu. B. Farforovskaya and L. Nikolskaya
Original publication: Algebra i Analiz, tom 20 (2008), nomer 3.
Journal: St. Petersburg Math. J. 20 (2009), 493-506
MSC (2000): Primary 47B15
Published electronically: April 8, 2009
MathSciNet review: 2454458
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Abstract: Let $ A$ and $ B$ be bounded selfadjoint operators on a separable Hilbert space, and let $ f$ be a continuous function defined on an interval $ [a,b]$ containing the spectra of $ A$ and $ B$. If $ \omega _f$ denotes the modulus of continuity of $ f$, then

$\displaystyle \Vert f(A)-f(B)\Vert \leq 4\Big[\log\Big(\frac{b-a}{\Vert A-B\Vert}+1\Big)+1\Big]^2 \cdot \omega _f(\Vert A-B\Vert).$

A similar result is true for unbounded selfadjoint operators, under some natural assumptions on the growth of $ f$.


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Additional Information

Yu. B. Farforovskaya
Affiliation: Mathematics Department, State University of Telecommunication, St. Petersburg, Russia
Email: rabk@sut.ru

L. Nikolskaya
Affiliation: Institut de Mathématiques de Bordeaux, Université Bordeaux-1, 351 Cours de la Libération, 33405 Talence, France
Email: andreeva@math.u-bordeaux.fr

DOI: http://dx.doi.org/10.1090/S1061-0022-09-01058-9
PII: S 1061-0022(09)01058-9
Keywords: Selfadjoint operator, operator function, modulus of continuity
Received by editor(s): June 14, 2007
Published electronically: April 8, 2009
Article copyright: © Copyright 2009 American Mathematical Society