Banach spaces with the Daugavet property

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.