Espaces de Banach ayant la propriété de Daugavet

Abstract

A Banach space X is said to have the Daugavet property if every operator T:X → X of rank 1 satisfies ∥Id + T∥=1 + ∥T∥. We show that then every weakly compact operator satisfies this equation as well and that X contains a copy of ℓ 1. However, X need not contain a copy of L 1. We also show that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with ∥Id + T∥ - 1 + ∥T∥ is as small as possible and give characterizations in terms of a smoothness condition.