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 parallel to Id + T parallel to = 1 + parallel to T parallel to. 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 study pairs of spaces X subset of Y and operators T : X --> Y satisfying parallel to J + T parallel to = 1 + parallel to T parallel to, where J : X --> Y is the natural embedding. This leads to the result 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 parallel to Id + T parallel to = 1 + parallel to T parallel to is as small as possible and give characterisations in terms of a smoothness condition.