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 Uspekhi Mat. Nauk, 2016, Volume 71, Issue 4(430), Pages 3–106 (Mi umn9729)

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

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).

DOI: https://doi.org/10.4213/rm9729

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English version:
Russian Mathematical Surveys, 2016, 71:4, 605–702

Bibliographic databases:

UDC: 517.983.28+517.984.4+517.983.24
MSC: Primary 26A16, 47A56; Secondary 47B15

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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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. B. Aleksandrov, V. V. Peller, “Krein's trace formula for unitary operators and operator Lipschitz functions”, Funct. Anal. Appl., 50:3 (2016), 167–175
2. M. M. Malamud, H. Neidhardt, V. V. Peller, “Analytic operator Lipschitz functions in the disk and a trace formula for functions of contractions”, Funct. Anal. Appl., 51:3 (2017), 185–203
3. D. Potapov, A. Skripka, F. Sukochev, A. Tomskova, “Multilinear Schur multipliers and Schatten properties of operator Taylor remainders”, Adv. Math., 320 (2017), 1063–1098
4. A. B. Aleksandrov, V. V. Peller, “Multiple operator integrals, Haagerup and Haagerup-like tensor products, and operator ideals”, Bull. London Math. Soc., 49:3 (2017), 463–479
5. M. Malamuda, H. Neidhardt, V. Peller, “A trace formula for functions of contractions and analytic operator Lipschitz functions”, C. R. Math. Acad. Sci. Paris, 355:7 (2017), 806–811
6. V. V. Peller, “Functions of triples of noncommuting self-adjoint operators under perturbations of class $S_p$”, Proc. Amer. Math. Soc., 146:4 (2018), 1699–1711
7. A. R. Mirotin, “Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations, II”, Oper. Matrices, 12:2 (2018), 445–463
8. S. Minsker, “Sub-Gaussian estimators of the mean of a random matrix with heavy-tailed entries”, Ann. Statist., 46:6 (2018), 2871–2903
9. Malamud M.M., Neidhardt H., Peller V.V., “Absolute Continuity of Spectral Shift”, J. Funct. Anal., 276:5 (2019), 1575–1621
10. Aleksandrov A.B., Peller V.V., “Dissipative Operators and Operator Lipschitz Functions”, Proc. Amer. Math. Soc., 147:5 (2019), 2081–2093
11. Coine C., Le Merdy Ch., Skripka A., Sukochev F., “Higher Order S-2-Differentiability and Application to Koplienko Trace Formula”, J. Funct. Anal., 276:10 (2019), 3170–3204
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