Mean ergodicity of power-bounded operators in countably order complete Banach lattices
Our goal is to prove that given a countably order complete Banach lattice E, if every power-bounded positive operator \(T: E\to E\) is mean ergodic (i.e., the sequence (1/n\(\sum^{n-1}_{i=0}T^ ix)_{n\in {\mathbb{N}}}\) converges in E for every \(x\in E)\), then E is reflexive. The proof is in a certain sense a constructive one; for every non- reflexive countably order complete Banach lattice E we construct a positive power-bounded operator \(T: E\to E\) which is not mean ergodic. The result obtained in this article is related to the following still open problem suggested by Louis Sucheston in 1975: Let E be a Banach space and let us assume that every contraction of E is mean ergodic. Decide whether E is reflexive or not.
scientific article; zbMATH DE number 3919150