Positional graphs and conditional structure of weakly null sequences

TL;DR

This paper explores the structure of weakly null sequences in Banach spaces, establishing limitations on unconditional subsequences, and characterizing the structure of certain spaces with respect to classical sequence spaces.

Contribution

It introduces new results on the existence and structure of weakly null sequences of various lengths and densities, and characterizes the subspace structure of mixed Tsirelson spaces.

Findings

Existence of weakly null sequences of length _n without unconditional subsequences.

Minimal cardinal _ for sequences with unconditional subsequences.

Mixed Tsirelson spaces contain copies of c_0 or ll_p for uncountable densities.

Abstract

We prove that, unless assuming additional set theoretical axioms, there are no reflexive space without unconditional sequences of density the continuum. We give for every integer $n$ there are normalized weakly-null sequences of length $\om_n$ without unconditional subsequences. This together with a result of \cite{Do-Lo-To} shows that $\om_\omega$ is the minimal cardinal $\kappa$ that could possibly have the property that every weakly null $\kappa$-sequence has an infinite unconditional basic subsequence . We also prove that for every cardinal number $\ka$ which is smaller than the first $\om$-Erd\"os cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either $c_0$ or $\ell_p$, with $p\ge 1$.