$\ell^{1}$-Spreading models in subspaces of mixed Tsirelson spaces
Volume 172 / 2006
Studia Mathematica 172 (2006), 47-68
MSC: Primary 46B20; Secondary 46B07, 46B15.
DOI: 10.4064/sm172-1-3
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 $X = T[(\theta _{n},{\mathcal{S}}_{n})^{\infty}_{n=1}]$:
(1) Every block subspace of $X$ contains an $\ell^{1}$-${\mathcal{S}}_{\omega}$-spreading model,
(2) The Bourgain $\ell^{1}$-index $I_{b}(Y) = I(Y) > \omega^{\omega}$ for any block subspace $Y$ of $X$,(3) $\lim_{m}\limsup_{n}\theta_{m+n}/\theta_{n} > 0$ 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.
