Skip to main content
SpringerLink
Log in

Uniform Convergence on Subspaces in the von Neumann Ergodic Theorem with Discrete Time

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We consider the power-law uniform (in the operator norm) convergence on vector subspaces with their own norms in the von Neumann ergodic theorem with discrete time. All possible exponents of the considered power-law convergence are found; for each of these exponents, spectral criteria for such convergence are given and the complete description of all such subspaces is obtained. Uniform convergence on the whole space takes place only in the trivial cases, which explains the interest in uniform convergence precisely on subspaces. In addition, by the way, old estimates of the rates of convergence in the von Neumann ergodic theorem for measure-preserving mappings are generalized and refined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space (Nauka, Moscow, 1966) [in Russian].

    MATH  Google Scholar 

  2. N. K. Bari, Trigonometric Series (Fizmatgiz, Moscow, 1961) [in Russian].

    Google Scholar 

  3. I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Connected Quantities (Nauka, Moscow, 1965) [in Russian].

    Google Scholar 

  4. J. Ben-Artzi and B. Morisse, “Uniform convergence in von Neumann’s ergodic theorem in the absence of a spectral gap,” Ergodic Theory Dynam. Systems 41 (6), 1601–1611 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. G. Kachurovskii, I. V. Podvigin, and V. E. Todikov, “Uniform convergence on subspaces in von Neumann’s ergodic theorem with continuous time,” Siberian Electronic Math. Reports 20 (1), 183–206 (2023).

    MathSciNet  Google Scholar 

  6. A. G. Kachurovskii and I. V. Podvigin, “Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems,” Trans. Moscow Math. Soc. 77, 1–53 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. A. Prikhod’ko, “Ergodic automorphisms with simple spectrum and rapidly decreasing correlations,” Math. Notes 94 (6), 968–973 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  8. V. V. Ryzhikov, “Ergodic homoclinic groups, Sidon constructions and Poisson suspensions,” Trans. Moscow Math. Soc. 75, 77–85 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. A. Prikhod’ko, “On ergodic flows with simple Lebesgue spectrum,” Sb. Math. 211 (4), 594–615 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  10. A. G. Kachurovskii and V. V. Sedalishchev, “On the constants in the estimates of the rate of convergence in von Neumann’s ergodic theorem,” Math. Notes 87 (5), 720–727 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  11. A. G. Kachurovskii and V. V. Sedalishchev, “Constants in estimates for the rates of convergence in von Neumann’s and Birkhoff’s ergodic theorems,” Sb. Math. 202 (8), 1105–1125 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. G. Kachurovskii, “The rate of convergence in ergodic theorem,” Russian Math. Surveys 51 (4), 653–703 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  13. V. F. Gaposhkin, “Decrease rate of the probabilities \(\varepsilon\)-deviations for the means of stationary processes,” Math. Notes 64 (3), 316–321 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  14. W. Rudin, Principles of Mathematical Analysis (McGraw-Hill, New York, 1964).

    MATH  Google Scholar 

  15. A. N. Shiryaev, Probability (Nauka, Moscow, 1989) [in Russian].

    MATH  Google Scholar 

  16. F. A. Robinson, “Sums of stationary random variables,” Proc. Amer. Math. Soc. 11 (1), 77–79 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  17. P. L. Butzer and U. Westphal, “The mean ergodic theorem and saturation,” Indiana Univ. Math. J. 20, 1163–1174 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. F. Gaposhkin, “Convergence of series connected with stationary sequences,” Izv. Math. 39 (6), 1297–1321 (1975).

    Article  MathSciNet  Google Scholar 

  19. V. V. Sedalishchev, “Interrelation between the convergence rates in von Neumann’s and Birkhoff’s ergodic theorems,” Sib. Math. J. 55 (2), 336–348 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  20. F. E. Browder, “On the iteration of transformations in noncompact minimal dynamical systems,” Proc. Amer. Math. Soc. 9, 773–780 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Lin, “On the uniform ergodic theorem,” Proc. Amer. Math. Soc. 43 (2), 337–340 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  22. E. Yu. Emel’yanov, Non-Spectral Asymptotic Analysis of One-Parameter Operator Semigroups, Operator Theory: Adv. and Appl., Vol. 173 (Birkhäuser, Basel, 2007).

    MATH  Google Scholar 

  23. A. Gomilko, M. Haase, and Yu. Tomilov, “On rates in mean ergodic theorems,” Math. Res. Lett. 18 (2), 201–213 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  24. T. Eisner, “Embedding operators into strongly continuous semigroups,” Arch. Math. 92 (5), 451–460 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  25. A. M. Stepin and A. M. Eremenko, “Non-unique inclusion in a flow and vast centralizer of a generic measure-preserving transformation,” Sb. Math. 195 (12), 1795–1808 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  26. I. P. Kornfeld, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory (Nauka, Moscow, 1980) [in Russian].

    Google Scholar 

Download references

Acknowledgments

The author thanks the referee for remarks and suggestions for improving the paper.

Funding

This work was carried out in the framework of the state assignment at Institute of Mathematics, Siberian Branch of Russian Academy of Sciences (grant no. FWNF-2022-0004).

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences, Novosibirsk, 630090, Russia

    A. G. Kachurovskii & I. V. Podvigin

  2. Novosibirsk State University, Novosibirsk, 630090, Russia

    A. Zh. Khakimbaev

Authors
  1. A. G. Kachurovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. V. Podvigin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Zh. Khakimbaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. G. Kachurovskii.

Additional information

Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 713–730 https://doi.org/10.4213/mzm13739.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kachurovskii, A.G., Podvigin, I.V. & Khakimbaev, A.Z. Uniform Convergence on Subspaces in the von Neumann Ergodic Theorem with Discrete Time. Math Notes 113, 680–693 (2023). https://doi.org/10.1134/S0001434623050073

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434623050073

Keywords

Search

Navigation