There is no largest proper operator ideal
Abstract
An operator ideal is proper if the only operators of the form IdX 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-Gonz\'alez (2000), of an improjective but essential operator on Gowers-Maurey's shift space XS (1997), through a new analysis of the algebra of operators on powers of XS. We also prove that certain properties hold for general 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 Gonz\'alez-Herrera (2007). This gives us a frame to extend the negative answer to the question of Pietsch to the real setting.