PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

The Mean Ergodic Theorem in symmetric spaces

Volume 245 / 2019

Fedor Sukochev, Aleksandr Veksler Studia Mathematica 245 (2019), 229-253 MSC: Primary 47A35. DOI: 10.4064/sm170311-31-10 Published online: 7 September 2018

Abstract

We investigate the validity of the Mean Ergodic Theorem in a symmetric Banach function space $E$ associated to an atomless Lebesgue probability space $(\Omega , \nu )$. We show that the Mean Ergodic Theorem holds if and only if $E$ is separable. That is, if $T:\Omega \to \Omega $ is a measure preserving bijection then the Cesàro averages of $\{ f \circ T^k \}_{k \ge 0}$ converge in a symmetric Banach function space $E$ for every $f \in E$ if and only if $E$ is separable. When $E$ is non-separable the Cesàro averages may converge in $E$ for some $f \in E$, but not all. It is also possible that every $f \in E$ can have an equimeasurable copy whose Cesàro averages do converge in $E$. We demonstrate this using sufficient conditions intimately connected with the theory of singular traces.

Authors

  • Fedor SukochevSchool of Mathematics and Statistics
    University of New South Wales
    Kensington, 2052, NSW, Australia
    e-mail
  • Aleksandr VekslerV. I. Romanovskiy Institute of Mathematics
    Uzbekistan Academy of Sciences
    Tashkent, Uzbekistan
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image