Skip to main content
SpringerLink
Log in

A van der Corput-type lemma for power bounded operators

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We prove a van der Corput-type lemma for power bounded Hilbert space operators. As a corollary we show that \(N^{-1}\sum _{n=1}^N T^{p(n)}\) converges in the strong operator topology for all power bounded Hilbert space operators T and all polynomials p satisfying \(p(\mathbb {N}_0)\subset \mathbb {N}_0\). This generalizes known results for Hilbert space contractions. Similar results are true also for bounded strongly continuous semigroups of operators.

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. Berend, D., Lin, M., Rosenblatt, J., Tempelman, A.: Modulated and subsequential ergodic theorems in Hilbert and Banach spaces. Ergod. Theory Dyn. Syst. 22, 1653–1665 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bergelson, V., Leibman, A., Moreira, C.G.: From discrete-to continuous-time ergodic theorems. Ergod. Theory Dyn. Syst. 32, 386–426 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blum, J., Eisenberg, B.: Generalized summing sequences and the mean ergodic theorem. Proc. Am. Math. Soc. 42, 423–429 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  4. Einsiedler, M., Ward, T.: Ergodic Theory with a View Towards Number Theory. Graduate Texts in Mathematics, vol. 259. Springer, London (2011)

    MATH  Google Scholar 

  5. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York (2000)

    MATH  Google Scholar 

  6. Jones, L.K., Lin, M.: Unimodular eigenvalues and weak mixing. J. Funct. Anal. 35, 42–48 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kunszenti-Kovács, D., Nittka, R., Sauter, M.: One the limits of Cesàro means of polynomial powers. Math. Z. 268, 771–776 (2011)

    Article  MATH  Google Scholar 

  8. Krengel, U.: Ergodic Theorems. De Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter & Co., Berlin (1985)

    Book  Google Scholar 

Download references

Acknowledgments

The paper was written during the second named author’s stay at the University of Auckland. He wishes to thank for the warm hospitality and perfect working conditions there. The stay was supported by Grants IRSES, 14-07880S of GA ČR and RVO: 67985840.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Auckland, Auckland, New Zealand

    A. F. M. ter Elst

  2. Mathematical Institute, Czech Academy of Sciences, Zitna 25, 115 67, Prague 1, Czech Republic

    V. Müller

Authors
  1. A. F. M. ter Elst
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Müller.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

ter Elst, A.F.M., Müller, V. A van der Corput-type lemma for power bounded operators. Math. Z. 285, 143–158 (2017). https://doi.org/10.1007/s00209-016-1701-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-016-1701-2

Keywords

Mathematics Subject Classification

Search

Navigation