Advertisement

Mathematische Zeitschrift

, Volume 268, Issue 3–4, pp 771–776 | Cite as

On the limits of Cesàro means of polynomial powers

  • Dávid Kunszenti-Kovács
  • Robin Nittka
  • Manfred Sauter
Article

Abstract

It is known that Cesàro means of polynomial powers of contractive operators in Hilbert spaces converge strongly. We address the question of whether the limit is a projection. We show that the only polynomials leading to projections for any operator are of degree at most one. Moreover, we find a spectral characterisation of operators in Hilbert spaces that have a projection as the limit of their polynomial Cesàro means for every reasonable polynomial.

Keywords

Polynomial Cesàro averages Unconventional ergodic means Projections Roots of unity Permutation polynomials 

Mathematics Subject Classification (2000)

47A35 37A30 11T06 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mustafa A.: Akcoglu and Louis Sucheston, on operator convergence in Hilbert space and in Lebesgue space. Period. Math. Hung. 2, 235–244 (1972) (Collection of articles dedicated to the memory of Alfréd Rényi, I)CrossRefGoogle Scholar
  2. 2.
    Apostol T.M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics. Springer, New York (1976)Google Scholar
  3. 3.
    Berend D., Lin M., Rosenblatt J., Tempelman A.: Modulated and subsequential ergodic theorems in Hilbert and Banach spaces. Ergod. Theory Dyn. Syst. 22(6), 1653–1665 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bergelson, V.: Ergodic Ramsey theory—an update, ergodic theory of \({\mathbb{Z}^d}\) actions (Warwick, 1993–1994). In: Lecture Note Series, vol. 228, pp. 1–61. Cambridge University Press, Cambridge (1996)Google Scholar
  5. 5.
    Bourgain J.: On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61(1), 39–72 (1988)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dunford N., Schwartz J.T.: Linear operators. Part II: spectral theory. In: SelfadjointoperatorsinHilbertspace. William, G.B., Robert, G.B. (eds) Pure and Applied Mathematics, Wiley, New York (1963)Google Scholar
  7. 7.
    Foguel S.R.: Powers of a contraction in Hilbert space. Pacific J. Math. 13, 551–562 (1963)MATHMathSciNetGoogle Scholar
  8. 8.
    Furstenberg H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton (1981)MATHGoogle Scholar
  9. 9.
    Jones L., Kuftinec V.: A note on the Blum–Hanson theorem. Proc. Am. Math. Soc. 30, 202–203 (1971)MATHMathSciNetGoogle Scholar
  10. 10.
    Kunszenti-Kovács, D.: On the limit of square-Cesàro means of contractions in Hilbert spaces. Arch. Math. (2010, to appear)Google Scholar
  11. 11.
    Lidl R., Niederreiter H.: Finite fields. In: Encyclopedia of Mathematics and its Applications, 2nd edn., vol. 20. Cambridge University Press, Cambridge (1997)Google Scholar
  12. 12.
    Lin M.: Mixing for Markov operators Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 19, 231–242 (1971)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Dávid Kunszenti-Kovács
    • 1
  • Robin Nittka
    • 2
  • Manfred Sauter
    • 3
  1. 1.Institute of MathematicsUniversity of TübingenTübingenGermany
  2. 2.Institute of Applied AnalysisUniversity of UlmUlmGermany
  3. 3.Department of MathematicsThe University of AucklandAucklandNew Zealand

Personalised recommendations