Abstract
Given a Banach lattice E that fails to be countably order complete, we construct a positive compact operator A: E → E for which T = I - A is power-bounded and not mean ergodic. As a consequence, by using the theorem of R. Zaharopol, we obtain that if every power-bounded operator in a Banach lattice is mean ergodic then the Banach lattice is reflexive.
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Emel'yanov, E.Y. Banach Lattices on Which Every Power-Bounded Operator is Mean Ergodic. Positivity 1, 291–296 (1997). https://doi.org/10.1023/A:1009764031312
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DOI: https://doi.org/10.1023/A:1009764031312