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Mean ergodicity vs weak almost periodicity. (English) Zbl 07054000

Summary: 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 \(T\).

MSC:

47B65 Positive linear operators and order-bounded operators
47A35 Ergodic theory of linear operators
46B42 Banach lattices
46A45 Sequence spaces (including Köthe sequence spaces)
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