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Algebra i Analiz, 2010, Volume 22, Issue 4, Pages 198–213 (Mi aa1200)  

This article is cited in 8 scientific papers (total in 8 papers)

Research Papers

Hölder functions are operator-Hölder

L. N. Nikol'skayaa, Yu. B. Farforovskayab

a Institut de Mathématiques de Bordeaux, Université Bordeaux, Talence, France
b St. Petersburg State University of Telecommunications, St. Petersburg, Russia

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, $C=2^{1-\alpha}+2\pi\sqrt 8\frac1{(1-2^{\alpha-1})^2}$. 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.

Keywords: operator-Hölder functions, Adamar–Schur multipliers.

Full text: PDF file (280 kB)
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English version:
St. Petersburg Mathematical Journal, 2011, 22:4, 657–668

Bibliographic databases:

Received: 01.07.2009

Citation: L. N. Nikol'skaya, Yu. B. Farforovskaya, “Hölder functions are operator-Hölder”, Algebra i Analiz, 22:4 (2010), 198–213; St. Petersburg Math. J., 22:4 (2011), 657–668

Citation in format AMSBIB
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\by L.~N.~Nikol'skaya, Yu.~B.~Farforovskaya
\paper H\"older functions are operator-H\"older
\jour Algebra i Analiz
\yr 2010
\vol 22
\issue 4
\pages 198--213
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\zmath{https://zbmath.org/?q=an:1228.47020}
\transl
\jour St. Petersburg Math. J.
\yr 2011
\vol 22
\issue 4
\pages 657--668
\crossref{https://doi.org/10.1090/S1061-0022-2011-01161-6}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Aleksandrov A.B., Peller V.V., “Functions of perturbed unbounded self-adjoint operators. Operator Bernstein type inequalities”, Indiana Univ. Math. J., 59:4 (2010), 1451–1490  crossref  mathscinet  zmath  isi  elib  scopus
    2. A. B. Aleksandrov, V. V. Peller, “Functions of perturbed dissipative operators”, St. Petersburg Math. J., 23:2 (2012), 209–238  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    3. Aleksandrov A.B., Peller V.V., Potapov D.S., Sukochev F.A., “Functions of normal operators under perturbations”, Adv. Math., 226:6 (2011), 5216–5251  crossref  mathscinet  zmath  isi  elib  scopus
    4. Aleksandrov A.B., Peller V.V., “Estimates of operator moduli of continuity”, J. Funct. Anal., 261:10 (2011), 2741–2796  crossref  mathscinet  zmath  isi  elib  scopus
    5. Nazarov F.L., Peller V.V., “Functions of Perturbed N-Tuples of Commuting Self-Adjoint Operators”, J. Funct. Anal., 266:8 (2014), 5398–5428  crossref  mathscinet  zmath  isi  elib  scopus
    6. A. B. Aleksandrov, V. V. Peller, “Operator Lipschitz functions”, Russian Math. Surveys, 71:4 (2016), 605–702  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. Aleksandrov A.B., Nazarov F.L., Peller V.V., “Functions of noncommuting self-adjoint operators under perturbation and estimates of triple operator integrals”, Adv. Math., 295 (2016), 1–52  crossref  mathscinet  zmath  isi  elib  scopus
    8. Peller V.V., “Multiple operator integrals in perturbation theory”, Bull. Math. Sci., 6:1 (2016), 15–88  crossref  mathscinet  zmath  isi  elib  scopus
  • Алгебра и анализ St. Petersburg Mathematical Journal
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