Schauder Bases Having Many Good Block Basic Sequences
Cory A. Krause
TL;DR
This work analyzes when a Schauder basis in a Banach space yields a wealth of good block-structured sequences by linking block-tree branches and NCCB sequences to the space's asymptotic geometry. It proves that having a good-branch property for all normalized block trees is equivalent to being $1$-asymptotic $\ell_p$ for some $1\le p\le \infty$, and it explores the stronger but non-characterizing condition that every block basic sequence is good. A main contribution is a Ramsey–theoretic stabilization framework (via Hindman–Milliken–Taylor) that produces a basic sequence whose normalized constant-coefficient block basic sequences are all good and share a common spreading model, with unconditionality leading to spreading models equivalent to $c_0$ or $\ell_p$. The results yield both structural characterizations of spaces with many good block sequences and constructive stabilization tools for spreading models in the asymptotic regime, enriching the understanding of Banach-space geometry and its Ramsey-theoretic underpinnings.
Abstract
In the study of asymptotic geometry in Banach spaces, a basic sequence which gives rise to a spreading model has been called a good sequence. It is well known that every normalized basic sequence in a Banach space has a subsequence which is good. We investigate the assumption that every normalized block tree relative to a basis has a branch which is good. This combinatorial property turns out to be very strong and is equivalent to the space being $1$-asymptotic $\ell_p$ for some $1\leq p\leq\infty$. We also investigate the even stronger assumption that every block basic sequence of a basis is good. Finally, using the Hindman-Milliken-Taylor theorem, we prove a stabilization theorem which produces a basic sequence all of whose normalized constant coefficient block basic sequences are good, and we present an application of this stabilization.