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Russian Mathematical Surveys, 2016, Volume 71, Issue 4, Pages 605–702
DOI: https://doi.org/10.1070/RM9729 (Mi rm9729) 		 
This article is cited in 57 scientific papers (total in 57 papers)

Operator Lipschitz functions

A. B. Aleksandrova, V. V. Pellerb

a St. Petersburg Department of the Steklov Mathematical Institute of the Russian Academy of Sciences
b Michigan State University, East Lansing, Michigan, USA
English version PDF (1257 kB) Citations (57)
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DOI: https://doi.org/10.1070/RM9729
Abstract: The goal of this survey is a comprehensive study of operator Lipschitz functions. A continuous function $f$ on the real line $\mathbb{R}$ is said to be operator Lipschitz if $\|f(A)-f(B)\|\leqslant\mathrm{const}\|A-B\|$ for arbitrary self-adjoint operators $A$ and $B$. Sufficient conditions and necessary conditions are given for operator Lipschitzness. The class of operator differentiable functions on $\mathbb{R}$ is also studied. Further, operator Lipschitz functions on closed subsets of the plane are considered, and the class of commutator Lipschitz functions on such subsets is introduced. An important role for the study of such classes of functions is played by double operator integrals and Schur multipliers.
Bibliography: 77 titles.
Keywords: functions of operators, operator Lipschitz functions, operator differentiable functions, self-adjoint operators, normal operators, divided differences, double operator integrals, Schur multipliers, linear-fractional transformations, Besov classes, Carleson measures.
Funding agency Grant number
Russian Foundation for Basic Research 14-01-00198
National Science Foundation DMS 130092
The first author was supported by the Russian Foundation for Basic Research (grant no. 14-01-00198), and the second author by the National Science Foundation (grant no. DMS-130092).
Received: 02.05.2016
Russian version:
Uspekhi Matematicheskikh Nauk, 2016, Volume 71, Issue 4(430), Pages 3–106
DOI: https://doi.org/10.4213/rm9729
Bibliographic databases:
Document Type: Article
UDC: 517.983.28+517.984.4+517.983.24
MSC: Primary 26A16, 47A56; Secondary 47B15
Language: English
Original paper language: Russian
Citation: A. B. Aleksandrov, V. V. Peller, “Operator Lipschitz functions”, Uspekhi Mat. Nauk, 71:4(430) (2016), 3–106; Russian Math. Surveys, 71:4 (2016), 605–702
Citation in format AMSBIB
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    • This publication is cited in the following 57 articles:
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