BANACH-SPACES WITH THE DAUGAVET-PROPERTY

A Banach space X is said to have the Daugavet property if every operat or T : X --> X of rank 1 satisfies //Id + T// = 1 + //T//. We show tha t then every weakly compact operator satisfies this equation as well a nd 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 do es not embed into a space with an unconditional basis. In another dire ction, we investigate spaces where the set of operators with //Id + T/ / = 1 + //T// is as small as possible and give characterizations in te rms of a smoothness condition.