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Lipschitz-Type Bounds for Functions of Operators with Noncompact Perturbations

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Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 290))

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Abstract

This article is dedicated to Lipschitzness of operator functions in the setting of noncompact perturbations that arise in problems of mathematical physics and noncommutative geometry. We survey known results on the subject and state several new results.

Research supported in part by NSF grant DMS-1554456.

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References

  1. A.B. Aleksandrov, V.V. Peller, Operator lipschitz functions. Uspekhi Mat. Nauk 71(4) (430), 3–106 (2016) (Russian). Translation: Russian Math. Surveys 71(4), 605–702 (2016)

    Google Scholar 

  2. P.J. Ayre, M.G. Cowling, F.A. Sukochev, Operator Lipschitz estimates in the unitary setting. Proc. Am. Math. Soc. 144(3), 1053–1057 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Caspers, S. Montgomery-Smith, D. Potapov, F. Sukochev, The best constants for operator Lipschitz functions on Schatten classes. J. Funct. Anal. 267(10), 3557–3579 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Caspers, D. Potapov, F. Sukochev, D. Zanin, Weak type commutator and Lipschitz estimates: resolution of the Nazarov-Peller conjecture. Am. J. Math. 141(3), 593–610 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Y.B. Farforovskaya, An example of a Lipschitz function of selfadjoint operators that yields a non-nuclear increase under a nuclear perturbation. J. Soviet. Math. 4, 426–433 (1975) (Russian).

    Article  Google Scholar 

  6. R.L. Frank, A. Pushnitski, Schatten class conditions for functions of Schrödinger operators. Ann. Henri Poincaré 20(11), 3543–3562 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. R.L. Frank, A. Pushnitski, Kato smoothness and functions of perturbed self-adjoint operators. Adv. Math. 351, 343–387 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. I.C. Gohberg, M.G. Krein, in Introduction to the Theory of Linear Nonselfadjoint Operators. Translations of Mathematical Monographs, vol. 18 (American Mathematical Society, Providence, 1969)

    Google Scholar 

  9. P.D. Hislop, C.A. Marx, Dependence of the density of states on the probability distribution. Part II: Schrödinger operators on \({\mathbb {R}}^d\) and non-compactly supported probability measures. Ann. Henri Poincaré 21, 539–570 (2020)

    Google Scholar 

  10. A. McIntosh, Counterexample to a question on commutators. Proc. Am. Math. Soc. 29, 337–340 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  11. V.V. Peller, Hankel operators in the theory of perturbations of unitary and selfadjoint operators. Funktsional. Anal. i Prilozhen. 19(2), 37–51 (1985) (Russian). Translation: Funct. Anal. Appl. 19, 111–123 (1985)

    Google Scholar 

  12. D. Potapov, F. Sukochev, Operator-Lipschitz functions in Schatten-von Neumann classes. Acta Math. 207(2), 375–389 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Simon, in Trace Ideals and Their Applications. Mathematical Surveys and Monographs, vol. 120, 2nd edn. (American Mathematical Society, Providence, 2005)

    Google Scholar 

  14. A. Skripka, Untangling noncommutativity with operator integrals. Not. Am. Math. Soc. 67(1), 45–55 (2020)

    MathSciNet  MATH  Google Scholar 

  15. A. Skripka, Lipschitz estimates for functions of Dirac and Schrödinger operators. J. Math. Phys. 62(1), 013506 (2021)

    Google Scholar 

  16. A. Skripka, A. Tomskova, in Multilinear Operator Integrals: Theory and Applications. Lecture Notes in Mathematics, vol. 2250 (Springer International Publishing, New York, 2019), XI+192 pp.

    Google Scholar 

  17. W. van Ackooij, B. de Pagter, F.A. Sukochev, Domains of infinitesimal generators of automorphism flows. J. Funct. Anal. 218(2), 409–424 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. T. van Nuland, A. Skripka, Spectral shift for relative Schatten class perturbations. J. Spectr. Theory (in press)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, USA

    Anna Skripka

Authors
  1. Anna Skripka
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Corresponding author

Correspondence to Anna Skripka .

Editor information

Editors and Affiliations

  1. Schmid College of Science & Technology, Chapman University, Orange, CA, USA

    Daniel Alpay

  2. Institut für Angewandte Mathematik, Technische Universität Graz, Graz, Steiermark, Austria

    Jussi Behrndt

  3. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Fabrizio Colombo

  4. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Irene Sabadini

  5. The Donald Bren Distinguished Presidential Chair in Mathematics, Chapman University, Orange, CA, USA

    Daniele C. Struppa

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Skripka, A. (2023). Lipschitz-Type Bounds for Functions of Operators with Noncompact Perturbations. In: Alpay, D., Behrndt, J., Colombo, F., Sabadini, I., Struppa, D.C. (eds) Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis. Operator Theory: Advances and Applications, vol 290. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-21460-8_9

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