A Banach space dichotomy for quotients of subspaces

A Banach space $X$ with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable (QHI) property if $X\mathrm{/}Y$ is hereditarily indecomposable (HI) for any infinite codimensional subspace $Y$ with a successive finite-dimensional decomposition on the basis of $X$. A reflexive space with the restricted QHI property is in particular HI, has HI dual, and is saturated with subspaces which are HI and have HI dual. The following dichotomy theorem is proved: any infinite dimensional Banach space contains a quotient of subspace which either has an unconditional basis, or has the restricted QHI property.