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

TL;DR

This paper constructs hereditarily indecomposable, separable 0 spaces with  duals that have a specific operator structure, including a nilpotent operator with unique properties, addressing a question in operator theory.

Contribution

It introduces new hereditarily indecomposable 0 spaces with  duals, featuring a nilpotent operator with special properties, expanding understanding of operator structures on such spaces.

Findings

Existence of hereditarily indecomposable 0 spaces with  duals.

Construction of a non-compact, strictly singular operator with specific nilpotency.

Every bounded operator has a specific form involving the nilpotent operator and compact operators.

Abstract

Given a natural number $k \geq 2$, we construct a hereditarily indecomposable, $\mathscr{L}_{\infty}$ 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 $S^j (0 \leq j \leq k-1)$ is not a compact perturbation of any linear combination of $S^l, l \neq j$. 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").