Distorting Mixed Tsirelson Spaces

TL;DR

This paper demonstrates that certain regular mixed Tsirelson spaces are arbitrarily distortable, computes asymptotic constants, proves a combinatorial result on Schreier families, and relates $ ext{Delta}$-spectrums of Banach spaces.

Contribution

It establishes the arbitrary distortability of specific mixed Tsirelson spaces and provides new combinatorial and spectral results in Banach space theory.

Findings

Regular mixed Tsirelson spaces with $rac{ heta_n}{ heta^n} o 0$ are arbitrarily distortable.

Calculated asymptotic $ ext{ell}_1$ constants for these spaces.

Proved a combinatorial result on Schreier families and related $ ext{Delta}$-spectrums.

Abstract

Any regular mixed Tsirelson space $T(\theta_n,S_n)_{\N}$ 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.