Higher Order Spreading Models

TL;DR

This paper introduces higher order spreading models in Banach spaces, forming a transfinite hierarchy based on regular thin families and plegma families, expanding the understanding of spreading models.

Contribution

It defines higher order spreading models using $f$-sequences and establishes their hierarchical structure, which was not previously known.

Findings

Higher order spreading models form an increasing transfinite hierarchy.

Each level contains models generated by sequences of a specific order.

The hierarchy's fundamental properties are thoroughly studied.

Abstract

We introduce the higher order spreading models associated to a Banach space $X$. Their definition is based on $\ff$-sequences $(x_s)_{s\in\ff}$ with $\ff$ a regular thin family and the plegma families. We show that the higher order spreading models of a Banach space $X$ form an increasing transfinite hierarchy $(\mathcal{SM}_\xi(X))_{\xi<\omega_1}$. Each $\mathcal{SM}_\xi (X)$ contains all spreading models generated by $\ff$-sequences $(x_s)_{s\in\ff}$ with order of $\ff$ equal to $\xi$. We also provide a study of the fundamental properties of the hierarchy.