Asymptotic $l\sb p$ hereditarily indecomposable Banach spaces

Abstract

For every $1 < p < \infty$ we construct an asymptotic $\ell_{p}$ Banach space which is hereditarily indecomposable and such that its dual is asymptotic $\ell_{q}$ hereditarily indecomposable, where $q$ is the conjugate of $p$. We prove that $c_{0}$ is finitely representable in these spaces and that every bounded linear operator on these spaces is a strictly singular perturbation of a multiple of the identity.