Skip to main content
SpringerLink
Log in

Analogues of the “zero-two” law for positive linear contractions inL p andC(X)

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

LetT be a positive linear contraction inL p (1≦p<∞), then we show that lim ‖T pf −T n+1 f‖ p ≦(1 − ε)21/p (f∈L +p , ε>0 independent off) implies already limn n→∞ ‖T nf −T n+1 n+1f ‖p p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown.

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. Y. Derrienic,Lois «zero ou deux» pour les processus de Markov. Applications aux marches aléatoires, Ann. Inst. H. Ponicaré (B)12 (1976), 111–129.

    Google Scholar 

  2. S. R. Foguel,More on the “zero-two” law, Proc. Amer. Math. Soc.61 (1976), 262–264.

    Article  MathSciNet  Google Scholar 

  3. G. Greiner and R. Nagel,La loi “zéro ou deux” et ses conséquences pour le comportment asymptotique des opérateurs positifs, J. Math. Pures Appl.64 (1982), 261–273.

    MathSciNet  Google Scholar 

  4. Y. Katznelson and L. Tzafriri,On power bounded operators. J. Funct. Anal.68 (1986), 313–328.

    Article  MATH  MathSciNet  Google Scholar 

  5. U. Krengel,Ergodic Theorems, Walter de Gruyter, Berlin-New York, 1985.

    MATH  Google Scholar 

  6. M. Lin,On the “zero-two” law for conservative Markov processes, Z. Wahrscheinlichkeits-theor. Verw. Geb.61 (1982), 513–525.

    Article  MATH  Google Scholar 

  7. S. Orey,An ergodic theorem for Markov chains, Z. Wahrscheinlichkeitstheor. Verw. Geb.1 (1962), 174–176.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Ornstein and L. Sucheston,An operator theorem on L 1 convergence to zero with applications to Markov kernels, Ann. Math. Stat.41 (1970), 1631–1639.

    Google Scholar 

  9. M. Riesz,Sur les maxima des formes bilinéaires et sur les fonctionelles linéaires, Acta Math.49 (1962), 465–497.

    Article  MathSciNet  Google Scholar 

  10. H. Schaefer,Banach Lattices and Positive Operators, Springer-Verlag, Berlin-Heidelberg-New York, 1974.

    MATH  Google Scholar 

  11. R. Zaharopol,The modulus of a regular linear operator and the “Zero-Two” law in L p-spaces (1<p<∞,p ≠ 2), J. Funct. Anal.68 (1986), 300–312.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt, Ostenstraße 26-28, D-8078, Eichstätt, FRG

    Rainer Wittmann

Authors
  1. Rainer Wittmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wittmann, R. Analogues of the “zero-two” law for positive linear contractions inL p andC(X). Israel J. Math. 59, 8–28 (1987). https://doi.org/10.1007/BF02779664

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02779664

Keywords

Search

Navigation