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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


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.


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

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