Distortion and spreading models in modified mixed Tsirelson spaces
Tom 157 / 2003
Streszczenie
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.
