Modified mixed Tsirelson spaces

Abstract

We study the modified and boundedly modified mixed Tsirelson spaces $T_M[({\mathcal{F}}_{k_n},{\theta }_n{)}_{n=1}^{\infty }]$ and $T_{M(s)}[({\mathcal{F}}_{k_n},{\theta }_n{)}_{n=1}^{\infty }]$ respectively, defined by a subsequence $({\mathcal{F}}_{k_n})$ of the sequence of Schreier families $({\mathcal{F}}_n)$. These are reflexive asymptotic ${\ell }_1$ spaces with an unconditio- nal basis $(e_i{)}_i$ having the property that every sequence $\{x_i{\}}_{i=1}^n$ of normalized disjointly supported vectors contained in $⟨e_i{⟩}_{i=n}^{\infty }$ is equivalent to the basis of ${\ell }_1^n$. We show that if $\lim {\theta }_n^{1/n}=1$ then the space $T[({\mathcal{F}}_n,{\theta }_n{)}_{n=1}^{\infty }]$ and its modified variations are totally incomparable by proving that $c_0$ is finitely disjointly representable in every block subspace of $T[({\mathcal{F}}_n,{\theta }_n{)}_{n=1}^{\infty }]$. Next, we present an example of a boundedly modified mixed Tsirelson space $X_{M(1),u}$ which is arbitrarily distortable. Finally, we construct a variation of the space $X_{M(1),u}$ which is hereditarily indecomposable.

References in corpus (2)

• Complexity of weakly null sequences
• Distorting Mixed Tsirelson Spaces