On matrix summation and the pointwise ergodic theorem

Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 532)


For any contraction T in a space L1 there exists a matrix summation method A stronger than the Cesàro method such that for all f∈L1 the A-limit of Tkf exists a.e.. Here we answer the question of existence of a universal A, which works for all T, negatively.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chacon R.V.: A class of linear transformations. Proc. Am. Math. Soc. 15, 560–564, (1964).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chacon R.V. and U. Krengel: Linear modulus of a linear operator. Proc. Am. Math. Soc. 16, 196–200, (1965).MathSciNetzbMATHGoogle Scholar
  3. 3.
    Friedman N.A.: Introduction to Ergodic Theory. Van Nostrand Reinhold Math. Studies Nr. 29, New York, (1970)Google Scholar
  4. 4.
    Krengel U.: Classification of states for operators. Proc. Fifth Berkeley Symp. Math. Stat. Probability. Vol. II, 2, 415–429, (1967).MathSciNetzbMATHGoogle Scholar
  5. 5.
    Krengel U.: On the global limit behaviour of Markov chains and of general nonsingular Markov processes. Zeitschr, Wahrscheinlichkeitstheorie verw. Geb. 6, 302–316, (1966).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Sucheston L.: Banach limits. Amer. Math. Monthly 74, 308–311, (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zeller K. und W. Beekmann: Theorie der Limitierungsverfahren. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 15, 2. Auflage; Berlin-Heidelberg-New York, Springer, (1970).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  1. 1.Institut für Mathematische Statistik34 GöttingenFederal Republic of Germany

Personalised recommendations