\ell ^1-spreading models in mixed Tsirelson space

Abstract

Suppose that (F_n)_{n=1}^{\infty} is a sequence of regular families of finite subsets of N and (\theta_n)_{n=1}^{\infty} is a nonincreasing null sequence in (0,1). The mixed Tsirelson space T[(\theta_{n}, F_n)_{n=1}^{\infty}] is the completion of $c_{00}$ with respect to the implicitly defined norm ||x|| = max{||x||_{c_0}, sup_n sup \theta_n \sum_{i=1}^{j}||E_{i}x||}, where the last supremum is taken over all finite subsets E_{1},...,E_{j} of N such that E_1 < >... <E_j and {min E_1,...,min E_j} \in F_n. Necessary and sufficient conditions are obtained for the existence of higher order \ell ^1-spreading models in every subspace generated by a subsequence of the unit vector basis of T[(\theta_{n}, F_n)_{n=1}^{\infty}.