A hereditarily indecomposable Banach space with rich spreading model structure

TL;DR

This paper constructs a reflexive hereditarily indecomposable Banach space with a rich structure of spreading models, demonstrating complex internal geometric properties and providing new examples in Banach space theory.

Contribution

It introduces a new hereditarily indecomposable Banach space with a universal spreading model structure and specific isomorphic properties, expanding understanding of HI space complexity.

Findings

Every subspace contains sequences generating all subsymmetric spreading models.

Existence of a block sequence with an isomorphism swapping pairs.

The space is hereditarily indecomposable and not tight by range.

Abstract

We present a reflexive Banach space $\mathfrak{X}_{_{^\text{usm}}}$ which is Hereditarily Indecomposable and satisfies the following properties. In every subspace $Y$ of $\mathfrak{X}_{_{^\text{usm}}}$ there exists a weakly null normalized sequence $\{y_n\}_n$, such that every subsymmetric sequence $\{z_n\}_n$ is isomorphically generated as a spreading model of a subsequence of $\{y_n\}_n$. Also, in every block subspace $Y$ of $\mathfrak{X}_{_{^\text{usm}}}$ there exists a seminormalized block sequence $\{z_n\}$ and $T:\mathfrak{X}_{_{^\text{usm}}}\rightarrow\mathfrak{X}_{_{^\text{usm}}}$ an isomorphism such that for every $n\in\mathbb{N}$ $T(z_{2n-1}) = z_{2n}$. Thus the space is an example of an HI space which is not tight by range in a strong sense.