There is no largest proper operator ideal

Abstract

An operator ideal is proper if the only operators of the form $I{d}_X$ it contains have finite rank. We answer a question posed by Pietsch (1979) by proving that there is no largest proper operator ideal. Our proof is based on an extension of the construction by Aiena-González (2000), of an improjective but essential operator on Gowers-Maurey's shift space $X_S$ (1997), through a new analysis of the algebra of operators on powers of $X_S$. We also prove that certain properties hold for general $\mathbb{C}$-linear operators if and only if they hold for these operators seen as real: for example this holds for the ideals of strictly singular, strictly cosingular, or inessential operators, answering a question of González-Herrera (2007). This gives us a frame to extend the negative answer to the question of Pietsch to the real setting.

25 pages