$\ell^{1}$-Spreading models in subspaces of mixed Tsirelson spaces

Denny H. Leung¹ ; Wee-Kee Tang²

¹ Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543
² Mathematics and Mathematics Education National Institute of Education Nanyang Technological University 1 Nanyang Walk Singapore 637616

Studia Mathematica, Tome 172 (2006) no. 1, pp. 47-68

Résumé

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.

Détail

DOI : 10.4064/sm172-1-3

Keywords: investigate existence higher order ell spreading models subspaces mixed tsirelson spaces instance following conditions equivalent mixed tsirelson space theta mathcal infty every block subspace contains ell mathcal omega spreading model bourgain ell index omega omega block subspace lim limsup theta theta every block subspace contains block sequence equivalent subsequence unit vector basis moreover hence these conditions holds arbitrarily distortable