A hereditarily indecomposable Banach space with rich spreading model structure

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 $T:{\mathfrak{X}}_{{}_{{}^{\text{usm}}}}\to {\mathfrak{X}}_{{}_{{}^{\text{usm}}}}$ 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.

arXiv comment39 pages, one figure