Some combinatorial principles for trees and applications to tree families in Banach spaces

Abstract

Suppose that (xs)s∈S is a normalized family in a Banach space indexed by the dyadic tree S. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree‐families. More precisely, assuming that for any infinite chain β of S the sequence (xs)s∈β is weakly null, we prove that there exists a subtree T of S such that for any infinite chain β of T the sequence (xs)s∈β is nearly (resp., convexly) unconditional. In the case where (fs)s∈S is a family of continuous functions, under some additional assumptions, we prove the existence of a subtree T of S such that for any infinite chain β of T, the sequence (fs)s∈β is unconditional. Finally, in the more general setting where for any chain β, (xs)s∈β is a Schauder basic sequence, we obtain a dichotomy result concerning the semi‐boundedly completeness of the sequences (xs)s∈β.