Operators on subspaces of hereditarily indecomposable Banach spaces

Abstract

A space $X$ is said to be hereditarily indecomposable if no two (infinite dimensional) subspaces of $X$ are in a direct sum. In this paper, we show that if $X$ is a complex hereditarily indecomposable Banach space, then every operator from a subspace $Y$ of $X$ to $X$ is of the form $λI + S$, where $I$ is the inclusion map and $S$ is strictly singular.