Examples of k-iterated spreading models

TL;DR

The paper demonstrates that for any natural number k and a given spreading sequence generating a uniformly convex Banach space, there exists a uniformly convex space where this sequence appears as a (k+1)-iterated spreading model but not as a k-iterated one.

Contribution

It provides explicit examples showing the strict hierarchy of k-iterated spreading models in uniformly convex Banach spaces.

Findings

Existence of spaces with specific k-iterated spreading model properties

Separation between k-iterated and (k+1)-iterated spreading models

Advancement in understanding the structure of spreading models in Banach spaces

Abstract

It is shown that for every $k\in\mathbb{N}$ and every spreading sequence $\{e_n\}_{n\in\mathbb{N}}$ that generates a uniformly convex Banach space $E$, there exists a uniformly convex Banach space $X_{k+1}$ admitting $\{e_n\}_{n\in\mathbb{N}}$ as a $k+1$-iterated spreading model, but not as a $k$-iterated one.