Banach spaces with the Daugavet property

A Banach space $X$ is said to have the Daugavet property if every operator $T:\allowbreak X\to X$ of rank $1$ satisfies $\|\operatorname{Id}+T\| = 1+\|T\|$. We show that then every weakly compact operator satisfies this equation as well and that $X$ contains a copy of $\ell _{1}$. However, $X$ need not contain a copy of $L_{1}$. We also study pairs of spaces $X\subset Y$ and operators $T:\allowbreak X\to Y$ satisfying $\|J+T\|=1+\|T\|$, where $J:\allowbreak X\to Y$ 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 $\|\operatorname{Id}+T\|=1+\|T\|$ is as small as possible and give characterisations in terms of a smoothness condition.