Interpolating hereditarily indecomposable Banach spaces

The following dichotomy is proved.

Every Banach space either contains a subspace isomorphic to $\ell^1$, or it has an infinite-dimensional closed subspace which is a quotient of a Hereditarily Indecomposable (H.I.) separable Banach space.

In the particular case of $L^p(\lambda), 1<p<\infty$, it is shown that the space itself is a quotient of a H.I. space. The factorization of certain classes of operators, acting between Banach spaces, through H.I. spaces is also investigated. Among others it is shown that the identity operator $I: L^{\infty}(\lambda)\to L^1(\lambda)$ admits a factorization through a H.I. space. The same result holds for every strictly singular operator $T: \ell^p\to \ell^q, 1<p,q<\infty$.

Interpolation methods and the geometric concept of thin convex sets together with the techniques concerning the construction of Hereditarily Indecomposable spaces are used to obtain the above mentioned results.