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

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 $\left | {||x|| - ||Vx||} \right | < \epsilon$ for each $x \in F’$ with $||x|| \leq 1$. $X$ 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 $(x - a\bar x) \in \overline {(I - T)X}$.