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Mean ergodicity vs weak almost periodicity

Volume 248 / 2019

Moritz Gerlach, Jochen Glück Studia Mathematica 248 (2019), 45-56 MSC: Primary 47B65; Secondary 47A35, 46B42, 46A45. DOI: 10.4064/sm170918-20-3 Published online: 22 February 2019

Abstract

We provide explicit examples of positive and power-bounded operators on $c_0$ and $\ell ^\infty $ which are mean ergodic but not weakly almost periodic. As a consequence we prove that a countably order complete Banach lattice on which every positive and power-bounded mean ergodic operator is weakly almost periodic is necessarily a KB-space. This answers several open questions from the literature. Finally, we prove that if $T$ is a positive mean ergodic operator with zero fixed space on an arbitrary Banach lattice, then so is every power of $T$.

Authors

  • Moritz GerlachInstitut für Mathematik
    Universität Potsdam
    Karl-Liebknecht-Straße 24–25
    14476 Potsdam, Germany
    e-mail
  • Jochen GlückInstitut für Angewandte Analysis
    Universität Ulm
    89069 Ulm, Germany
    e-mail

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