The Lipschitz injective hull of Lipschitz operator ideals and applications

Abstract

We introduce and study the Lipschitz injective hull of Lipschitz operator ideals defined between metric spaces. We show some properties and apply the results to the ideal of Lipschitz p-nuclear operators, obtaining the ideal of Lipschitz quasi p-nuclear operators. Also, we introduce in a natural way the ideal of Lipschitz Pietsch p-integral operators and show that its Lipschitz injective hull coincide with the ideal of Lipschitz p-summing operators defined by Farmer and Johnson. Finally, we consider both ideals as Lipschitz operator ideals between a metric space and a Banach space, showing that these ideals are not of composition type. Their maximal hull and minimal kernel are also studied.