There is no largest proper operator ideal

An operator ideal is proper if the only invertible operators it contains have finite rank. We answer a problem posed by Pietsch (Operator ideals, North-Holland, Amsterdam, 1980) by proving (i) that the ideal of inessential operators is not maximal among proper operator ideals, and (ii) that there is no largest proper operator ideal. Our proof is based on an extension of the construction by Aiena and Gonzalez (Math Z 233:471-479, 2000), of an improjective but essential operator on Gowers-Maurey's shift space X-S (Math Ann 307:543-568, 1997), through a new analysis of the algebra of operators on powers of X-S. 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 operators belonging to the ideals of strictly singular, strictly cosingular, or inessential operators, answering a question of Gonzalez and Herrera (Stud Math 183(1):1-14, 2007). This gives us a frame to extend the negative answer to the problem of Pietsch to the real setting.

Automated summary

This article studies the properties of proper operator ideals. By extending the construction of Aiena and Gonzalez and analyzing the powers of X-S, it is proven that the ideal of inessential operators is not maximal among proper operator ideals, and that there is no largest proper operator ideal. It is also demonstrated that certain properties hold for general C-linear operators if and only if they hold for these operators seen as real.