Modified mixed Tsirelson spaces

The Bourgain ell ^1-index of mixed Tsirelson space

Suppose that (F_n)_{n=0}^{\infty} is a sequence of regular families of finite subsets of N such that F_0 contains all singletons, and (θ_n)_{n=1}^{\infty} is a nonincreasing null sequence in (0,1). In this paper, we compute the Bourgain \ell^1 - index of the mixed Tsirelson space T(F_0,(θ_n, F_n)_{n=1}^{\infty}). As a consequence, it is shown that if ηis a countable ordinal not of the form ω^ξfor some limit ordinal ξ, then there is a Banach space whose \ell^1-index is ω^η. This answers a question of Judd and Odell.

高阶Tsirelson空间与其修正版本是同构的

Higher Order Tsirelson Spaces and their Modified Versions are Isomorphic

该研究证明可数序数阶的Tsirelson空间$T_\xi$与其修正版通过恒等映射3-同构，同时证实对应阶Schreier族与其修正版相同，解决了相关公开问题，并应用该结果得出$T_\xi$上有界线性算子代数有$2^{\mathfrak c}$个闭理想。

On total incomparability of mixed Tsirelson spaces

We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T[(M_k,θ_k)_{k=1}^{\ell}]$ with index $i(M_k)$ finite are either $c_0$ or $\ell_p$ saturated for some $p$ and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i(M_k)$ and the parameter $θ_k$. Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T[(A_k,θ_k)_{k=1}^\infty]$ in terms of the asymptotic behaviour of the sequence $\Vert\sum_{i=1}^n e_i\Vert$ where $(e_i)$ is the…

\ell ^1-spreading models in mixed Tsirelson space

Suppose that (F_n)_{n=1}^{\infty} is a sequence of regular families of finite subsets of N and (θ_n)_{n=1}^{\infty} is a nonincreasing null sequence in (0,1). The mixed Tsirelson space T[(θ_{n}, F_n)_{n=1}^{\infty}] is the completion of $c_{00}$ with respect to the implicitly defined norm ||x|| = max{||x||_{c_0}, sup_n sup θ_n \sum_{i=1}^{j}||E_{i}x||}, where the last supremum is taken over all finite subsets E_{1},...,E_{j} of N such that E_1 < >... <E_j and {min E_1,...,min E_j} \in F_n. Necessary and sufficient conditions are obtained for the existence of higher order \ell ^1-spreading mode…

Stabilization of Tsirelson-type norms on $\ell_p$ spaces

We consider classical Tsirelson-type norms of $T[A_r,θ]$ and their modified versions on $\ell_p$ spaces. We show that for any $1<p<\infty$ there is a constant $λ_p$ such that considered Tsirelson-type norms do not $λ_p$-distort any of subspaces of $\ell_p$.

A norm for Tsirelson's Banach space

We give an expression for the norm of the space constructed by Tsirelson. The implicit equation satisfied by this norm is dual to the implicit equation for the norm of the dual of Tsirelson space given by Figiel and Johnson. The expression can be modified to give the norm of the dual of any mixed Tsirelson space. In particular, our results can be adapted to give the norm for the dual of Schlumprecht space.

A classification of Tsirelson type spaces

We give a complete classification of mixed Tsirelson spaces T[(F\_i, theta\_i)\_{i=1}^r ] for finitely many pairs of given compact and hereditary families F\_i of finite sets of integers and 0<theta\_i<1 in terms of the Cantor-Bendixson indexes of the families F\_i, and theta\_i (0< i < r+1). We prove that there are unique countable ordinal alpha and 0<theta<1 such that every block sequence of T[(F\_i, theta\_i)\_{i=1}^r ] has a subsequence equivalent to a subsequence of the natural basis of the T(S\_{omega^alpha},theta). Finally, we give a complete criterion of comparison in between two of th…

Regular subspaces of a Bourgain-Delbaen space $\mathscr B_{mT}$

The space $\mathscr B_{mt}[(m_j)_j,(n_j)_j]$ is a Bourgain-Delbaen space modelled on a mixed Tsirelson space $T[(m_j)_j,(n_j)_j]$ and is a slight modification of $\mathfrak B_{\text{mt}}[(m_j)_j,(n_j)_j]$, a space defined by S. Argyros and R. Haydon. We prove that in every infinite dimensional subspace of $\mathscr B_{mt}[(m_j)_j,(n_j)_j]$ there exists a basic sequence equivalent to a sequence of weighted basis averages of increasing length from $T[(m_j)_j,(n_j)_j]$. We remark that the same is true for the original space $\mathfrak B_{\text{mt}}[(m_j)_j,(n_j)_j]$.