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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An ergodic super-property of Banach spaces defined by a class of matrices


Authors: A. Brunel, H. Fong and L. Sucheston
Journal: Proc. Amer. Math. Soc. 49 (1975), 373-378
MSC: Primary 47A35; Secondary 28A65
DOI: https://doi.org/10.1090/S0002-9939-1975-0365180-0
MathSciNet review: 0365180
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Abstract: A matrix $ ({a_{ni}})$ is called an $ R$-matrix if (A) $ {\Sigma _i}{a_{ni}} \nrightarrow 0$, and (B) $ {\lim _n}{a_{ni}} = 0$ for each $ i$. A Banach space $ X$ is called $ R$-ergodic if for each isometry $ T$ and each $ x \in X$, there is an $ R$-matrix $ ({a_{ni}})$ such that $ {\Sigma _i}{a_{ni}}{T^i}x\xrightarrow{{\text{W}}}$ (converges weakly). Given two Banach spaces $ F$ and $ X$, write $ F{\text{ fr }}X$ if for each finite-dimensional subspace $ F'$ of $ F$ and $ \epsilon > 0$, there is an isomorphism $ V$ from $ F'$ onto a subspace of $ X$ such that $ \left\vert {\vert\vert x\vert\vert - \vert\vert Vx\vert\vert} \right\vert < \epsilon $ for each $ x \in F'$ with $ \vert\vert x\vert\vert \leq 1$. $ X$ is called super-$ R$-ergodic if $ F$ is $ R$-ergodic for each $ F{\text{ fr }}X$.

Theorem. $ X$ is super-$ R$-ergodic if and only if $ X$ is super-reflexive.

The proof is based on the following:

Theorem. Let $ T$ be a linear operator on $ X,({a_{ni}})$ a matrix satisfying (A), $ x \in X$ such that $ {\Sigma _i}{a_{ni}}{T^i}x\xrightarrow{{\text{W}}}\bar x$. Then there is a constant $ \alpha $ such that $ (x - a\bar x) \in \overline {(I - T)X} $.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0365180-0
Keywords: $ R$-matrices, super-properties, $ R$-ergodic, reflexive, stable
Article copyright: © Copyright 1975 American Mathematical Society