Skip to main content
SpringerLink
Log in

Bernstein functions and rates in mean ergodic theorems for operator semigroups

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We present a functional calculus approach to the study of rates of decay in mean ergodic theorems for bounded strongly continuous operator semigroups. A central role is played by operators of the form g(A, where −A is the generator of the semigroup and g is a Bernstein function. In addition, we obtain some new results on Bernstein functions which are of independent interest.

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.

Similar content being viewed by others

References

  1. I. Assani and M. Lin, On the one-sided ergodic Hilbert transform, Ergodic Theory and Related Fields, Contemp. Math. 430, Amer. Math. Soc., Providence, RI, 2007, pp. 21–39.

    Chapter  Google Scholar 

  2. A. V. Balakrishnan, An operational calculus for infinitesimal generators of semigroups, Trans. Amer. Math. Soc. 91 (1959), 330–353.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Butzer and A. Gessinger, Ergodic theorems for semigroups and cosine operator functions at zero and infinity with rates; applications to partial differential equations. A survey, Mathematical Analysis, Wavelets, and Signal Processing, Contemp. Math. 190, Amer. Math. Soc., Providence, RI, 1995, pp. 67–94.

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. P. Clément and J. Prüss, Completely positive measures and Feller semigroups, Math. Ann. 287 (1990), 73–105.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Cohen, C. Cuny, and M. Lin, The one-sided ergodic Hilbert transform in Banach spaces, Studia Math. 196 (2010), 251–263.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Cohen, C. Cuny, and M. Lin, On convergence of power series of Lp contractions, Banach Center Publications, 2013, to appear.

  9. G. Cohen and M. Lin, Laws of large numbers with rates and the one-sided ergodic Hilbert transform, Illinois J. Math. 47 (2003), 997–1031.

    MathSciNet  MATH  Google Scholar 

  10. G. Cohen and M. Lin, Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory, Israel J. Math. 148 (2005), 41–86.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Cohen and M. Lin, The one-sided ergodic Hilbert transform of normal contractions, Characteristic Functions, Scattering Functions and Transfer Functions, Birkhäuser Verlag, Basel, 2010, pp. 77–98.

    Google Scholar 

  12. C. Cuny, Pointwise ergodic theorems with rate and application to limit theorems for stationary processes, Stoch. Dyn. 11 (2011), 135–155.

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Cuny, On the a.s. convergence of the one-sided ergodic Hilbert transform, Ergodic Theory Dynam. Systems 29 (2009), 1781–1788.

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Cuny and M. Lin, Pointwise ergodic theorems with rate and application to the CLT for Markov chains, Ann. Inst. Henri Poincaré, Probab. Statist. 45 (2009), 710–733.

    Article  MathSciNet  MATH  Google Scholar 

  15. R. deLaubenfels, Automatic extensions of functional calculi, Studia Math. 114 (1995), 237–259.

    MathSciNet  MATH  Google Scholar 

  16. Y. Derriennic, Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the “central limit theorem”, Discrete Contin. Dyn. Syst. 15 (2006), 143–158.

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Derriennic and M. Lin, Fractional Poisson equations and ergodic theorems for fractional coboundaries, Israel J. Math. 123 (2001), 93–130.

    Article  MathSciNet  MATH  Google Scholar 

  18. N. Dunford, Spectral theory. I. Convergence to projections, Trans. Amer. Math. Soc. 54 (1943), 185–217.

    MathSciNet  MATH  Google Scholar 

  19. K. J. Engel, and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, Berlin, 2000.

    MATH  Google Scholar 

  20. V. F. Gaposhkin, On the rate of decrease of the probabilities of ∈-deviations for means of stationary processes, Mat. Zametki 64 (1998), 366–372 (in Russian); translation in Math. Notes 64 (1998), 316–321.

    Article  MathSciNet  Google Scholar 

  21. J. Goldstein, C. Radin, and R. Showalter, Convergence rates of ergodic limits for semigroups and cosine functions, Semigroup Forum 16 (1978), 89–95.

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser, Basel, 2006.

    Book  MATH  Google Scholar 

  24. M. Haase and Yu. Tomilov, Domain characterizations of certain functions of power-bounded operators, Studia Math. 196 (2010), 265–288.

    Article  MathSciNet  MATH  Google Scholar 

  25. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, 3rd rev. ed., Amer. Math. Soc., Providence, RI, 1957.

    Google Scholar 

  26. F. Hirsch, Transformation de Stieltjes et fonctions opérant sur les potentiels abstraits, Théorie du potentiel et analyse harmonique, Lecture Notes in Math. 404, Springer, Berlin, 1974, pp. 149–163.

    Chapter  Google Scholar 

  27. F. Hirsch, Familles d’opérateurs potentiels, Ann. Inst. Fourier 254 (1975), 263–288.

    Article  Google Scholar 

  28. F. Hirsch, Domaines d’opérateurs représentés comme intégrales de résolvantes, J. Functional Analysis 23 (1976), 199–217.

    Article  MathSciNet  MATH  Google Scholar 

  29. A. G. Kachurovskiĭ, Rates of convergence in ergodic theorems, Uspekhi Mat. Nauk 51 (1996), 73–124 (in Russian); translation in Russian Math. Surveys 51 (1996), 653–703.

    Article  Google Scholar 

  30. A. G. Kachurovskiĭ and A. V. Reshetenko, On the rate of convergence in von Neumann’s ergodic theorem with continuous time, Mat. Sb. 201 (2010), 25–32 (in Russian); translation in Sb. Math. 201 (2010), 493–500.

    Article  Google Scholar 

  31. Th. Kaluza, Über die Koeffizienten reziproker Potenzreihen, Math. Z. 28 (1928), 161–170.

    Article  MathSciNet  MATH  Google Scholar 

  32. U. Krengel, Ergodic Theorems. With a supplement by Antoine Brunel, Walter de Gruyter, Berlin, 1985.

    Book  MATH  Google Scholar 

  33. U. Krengel and M. Lin, On the range of the generator of a Markovian semigroup, Math. Z. 185 (1984), 553–565.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Lin, On the uniform ergodic theorem. II, Proc. Amer. Math. Soc. 46 (1974), 217–225.

    Article  MathSciNet  MATH  Google Scholar 

  36. V. Nollau, Über den Logarithmus abgeschlossener Operatoren in Banachschen Räumen, Acta Sci. Math. (Szeged) 30 (1969), 161–174.

    MathSciNet  MATH  Google Scholar 

  37. R. S. Phillips, On the generation of semigroups of linear operators, Pacific J. Math. 2 (1952), 343–369.

    Article  MathSciNet  MATH  Google Scholar 

  38. W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, New York, 1991.

    MATH  Google Scholar 

  39. S. Y. Shaw, Convergence rates of ergodic limits and associated solutions, J. Approx. Theory 75 (1993), 157–166.

    Article  MathSciNet  MATH  Google Scholar 

  40. R. L. Schilling, Subordination in the sense of Bochner and a related functional calculus, J. Austral. Math. Soc. Ser. A 64 (1998), 368–396.

    Article  MathSciNet  MATH  Google Scholar 

  41. R. Schilling, R. Song, and Z. Vondraček, Bernstein Functions, Walter de Gruyter, Berlin, 2010.

    MATH  Google Scholar 

  42. U. Westphal, A generalized version of the Abelian mean ergodic theorem with rates for semigroup operators and fractional powers of infinitesimal generators, Results Math. 34 (1998), 381–394.

    Article  MathSciNet  MATH  Google Scholar 

  43. D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1941.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Nicolas Copernicus University, Chopin Str. 12/18, 87-100, Toruń, Poland

    Alexander Gomilko & Yuri Tomilov

  2. Delft Institute of Appliedmathematics, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Markus Haase

  3. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich Str. 8, 00-956, Warszawa, Poland

    Yuri Tomilov

Authors
  1. Alexander Gomilko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Markus Haase
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuri Tomilov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Gomilko.

Additional information

Dedicated to Michael Lin on the occasion of his retirement

The first and third authors were partially supported by the NCN grant DEC-2011/03/B/ST1/00407.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gomilko, A., Haase, M. & Tomilov, Y. Bernstein functions and rates in mean ergodic theorems for operator semigroups. JAMA 118, 545–576 (2012). https://doi.org/10.1007/s11854-012-0044-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-012-0044-0

Keywords

Search

Navigation