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