Hereditarily indecomposable, separable L_\infty spaces with \ell_1 dual having few operators, but not very few operators

Abstract

Given a natural number $k\ge 2$, we construct a hereditarily indecomposable, ${\mathcal{L}}_{\infty }$ space, $X_k$ with dual isomorphic to ${\ell }_1$. We exhibit a non-compact, strictly singular operator $S$ on $X_k$, with the property that $S^k=0$ and $S^j(0\le j\le k-1)$ is not a compact perturbation of any linear combination of $S^l,l\ne j$. Moreover, every bounded linear operator on this space has the form ${\sum }_{i=0}^{k-1}{\lambda }_i{S}^i+K$ where the ${\lambda }_i$ are scalars and $K$ is compact. In particular, this construction answers a question of Argyros and Haydon ("A hereditarily indecomposable space that solves the scalar-plus-compact problem").