Regular methods of summability on tree-sequences in Banach spaces

Abstract

Suppose that X is a Banach space, 〈a ij 〉 is a regular method of summability and (x s ) s∈S is a bounded sequence in X indexed by the dyadic tree S. We prove that there exists a subtree S' ⊆ S such that: either (a) for any chain β of S' the sequence (x s ) s∈β is summable with respect to 〈a ij 〉 or (b) for any chain β of S' the sequence (x s ) s∈β is not summable with respect to 〈a ij 〉. Moreover, in case (a) we prove the existence of a subtree T ⊆ S' such that if β is any chain of T, then all the subsequences of (x s ) s∈β are summable to the same limit. In case (b), provided that 〈a ij 〉 is the Cesaro method of summability and that for any chain β of S' the sequence (x s ) s∈β is weakly null, we prove the existence of a subtree T C S' such that for any chain β of T any spreading model for the sequence (x s ) s∈β 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).