On a generalization of Bourgain's tree index

Abstract

For a Banach space $X$, a sequence of Banach spaces $(Y_n)$, and a Banach space $Z$ with an unconditional basis, D. Alspach and B. Sari introduced a generalization of a Bourgain tree called a $({\oplus }_n{Y}_n{)}_Z$-tree in $X$. These authors also prove that any separable Banach space admitting a $({\oplus }_n{Y}_n{)}_Z$-tree with order ${\omega }_1$ admits a subspace isomorphic to $({\oplus }_n{Y}_n{)}_Z$. In this paper we give two new proofs of this result.