\ell^1-spreading models in subspaces of mixed Tsirelson spaces

Abstract

We investigate the existence of higher order \ell^1-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space X=T[(θ_n,S_n)_{n=1}^{\infty}] (1)Every block subspace of $X$ contains an \ell^1-S_ω-spreading model, (2)The Bourgain \ell^1-index I_b(Y) = I(Y) > ω^ω for any block subspace Y of X, (3)\lim_m\limsup_nθ_{m+n}/θ_n > 0 and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these conditions holds, then X is arbitrarily distortable.