Regular methods of summability on tree-sequences in Banach spaces

Abstract:

Suppose that $X$ is a Banach space, $\langle a_{ij}\rangle$ is a regular method of summability and $(x_{s})_{s\in S}$ is a bounded sequence in $X$ indexed by the dyadic tree $S$. We prove that there exists a subtree $S’\subseteq S$ such that: either (a) for any chain $\beta$ of $S’$ the sequence $(x_{s})_{s\in \beta }$ is summable with respect to $\langle a_{ij}\rangle$ or (b) for any chain $\beta$ of $S’$ the sequence $(x_{s})_{s\in \beta }$ is not summable with respect to $\langle a_{ij}\rangle$. Moreover, in case (a) we prove the existence of a subtree $T\subseteq S’$ such that if $\beta$ is any chain of $T$, then all the subsequences of $(x_{s})_{s\in \beta }$ are summable to the same limit. In case (b), provided that $\langle a_{ij}\rangle$ is the Cesàro method of summability and that for any chain $\beta$ of $S’$ the sequence $(x_{s})_{s\in \beta }$ is weakly null, we prove the existence of a subtree $T\subseteq S’$ such that for any chain $\beta$ of $T$ any spreading model for the sequence $(x_{s})_{s\in \beta }$ has a basis equivalent to the usual $l_{1}$-basis. Finally, we generalize the theory of spreading models to tree-sequences. This also allows us to improve the result of case (b).