Distorting Mixed Tsirelson Spaces

Abstract

Any regular mixed Tsirelson space $T({\theta }_n,S_n{)}_{\mathbb{N}}$ for which $\frac{{\theta }_n}{{\theta }^n}\to 0$, where $\theta ={\lim }_n{\theta }_n^{1/n}$, is shown to be arbitrarily distortable. Certain asymptotic ${\ell }_1$ constants for those and other mixed Tsirelson spaces are calculated. Also a combinatorial result on the Schreier families $(S_{\alpha }{)}_{\alpha <{\omega }_1}$ is proved and an application is given to show that for every Banach space $X$ with a basis $(e_i)$, the two $\Delta$-spectrums $\Delta (X)$ and $\Delta (X,(e_i))$ coincide.

References in corpus (1)

• Complexity of weakly null sequences

Cited by in corpus (3)

• Proximity to ${\ell }_1$ and Distortion in Asymptotic ${\ell }_1$ Spaces
• Modified mixed Tsirelson spaces
• A problem on spreading models