Distortion and spreading models in modified mixed Tsirelson spaces

S. A. Argyros¹ ; I. Deliyanni² ; A. Manoussakis³

¹Department of Mathematics National Technical University of Athens Athens, Greece
²Department of Mathematics University of Crete Heraklion, Crete, Greece
³Department of Sciences Technical University of Crete Chania, Crete, Greece

Studia Mathematica, Tome 157 (2003) no. 3, pp. 199-236

Résumé

The results of the first part concern the existence of higher order $\ell_{1}$ spreading models in asymptotic $\ell_{1}$ Banach spaces. We sketch the proof of the fact that the mixed Tsirelson space $T[({\cal S}_{n},\theta_{n})_{n}]$, $\theta_{n+m}\geq\theta_{n}\theta_{m}$ and $\lim_{n}\theta_{n}^{1/n}=1$, admits an $\ell_{1}^{\omega}$ spreading model in every block subspace. We also prove that if $X$ is a Banach space with a basis, with the property that there exists a sequence $(\theta_{n})_{n}\subset (0,1)$ with $\lim_{n}\theta_{n}^{1/n}=1$, such that, for every $n\in{\Bbb N}$, $\Vert \sum_{k=1}^{m} x_{k}\Vert \geq \theta_{n}\sum_{k=1}^{m}\Vert x_{k}\Vert$ for every ${\cal S}_{n}$-admissible block sequence $(x_{k})_{k=1}^{m}$ of vectors in $X$, then there exists $c>0$ such that every block subspace of $X$ admits, for every $n$, an $\ell_{1}^{n}$ spreading model with constant $c$. Finally, we give an example of a Banach space which has the above property but fails to admit an $\ell_{1}^{\omega}$ spreading model.In the second part we prove that under certain conditions on the double sequence $(k_{n},\theta_{n})_{n}$ the modified mixed Tsirelson space $T_{M}[({\cal S}_{k_{n}},\theta_{n})_{n}]$ is arbitrarily distortable. Moreover, for an appropriate choice of $(k_{n},\theta_{n})_{n}$, every block subspace admits an $\ell_{1}^{\omega}$ spreading model.

Détail

DOI : 10.4064/sm157-3-1

Keywords: results first part concern existence higher order ell spreading models asymptotic ell banach spaces sketch proof the mixed tsirelson space cal theta theta geq theta theta lim theta admits ell omega spreading model every block subspace prove banach space basis property there exists sequence theta subset lim theta every bbb vert sum vert geq theta sum vert vert every cal admissible block sequence vectors there exists every block subspace admits every ell spreading model constant nbsp finally example banach space which has above property fails admit ell omega spreading model second part prove under certain conditions double sequence theta modified mixed tsirelson space cal theta arbitrarily distortable moreover appropriate choice theta every block subspace admits ell omega spreading model