A hereditarily indecomposable Banach space with rich spreading model structure

We present a reflexive Banach space \(\mathfrak{X}_{usm}\) which is Hereditarily Indecomposable and satisfies the following properties. In every subspace Y of \(\mathfrak{X}_{usm}\) 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 \(\mathfrak{X}_{usm}\) there exists a seminormalized block sequence {z _n } and \(T:\mathfrak{X}_{usm} \to \mathfrak{X}_{usm}\) an isomorphism such that for every 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.