Spreading Models in Banach Space Theory

TL;DR

This paper generalizes the concept of spreading models in Banach space theory by introducing a transfinite hierarchy based on regular thin families, expanding the theoretical framework and providing illustrative examples.

Contribution

It introduces a new hierarchy of spreading models using $$-sequences and plegma families, extending classical Banach space theory.

Findings

Established a transfinite hierarchy of spreading sequences.

Provided examples illustrating the hierarchy and its limitations.

Extended classical definitions to a broader, more complex framework.

Abstract

We extend the classical Brunel-Sucheston definition of the spreading model by introducing the $\mathcal{F}$-sequences $(x_s)_{s\in\mathcal{F}}$ in a Banach space and the plegma families in $\mathcal{F}$ where $\mathcal{F}$ is a regular thin family. The new concept yields a transfinite increasing hierarchy of classes of spreading sequences. We explore the corresponding theory and we present examples establishing this hierarchy and illustrating the limitation of the theory.