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

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Abstract

Suppose that inline image 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 inline image is weakly null, we prove that there exists a subtree T of S such that for any infinite chain β of T the sequence inline image is nearly (resp., convexly) unconditional. In the case where inline image 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 inline image is unconditional. Finally, in the more general setting where for any chain β, inline image is a Schauder basic sequence, we obtain a dichotomy result concerning the semi-boundedly completeness of the sequences inline image.

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