Banach spaces with the Daugavet property

Abstract

A Banach space $X$ is said to have the Daugavet property if every operator $T:X\to X$ of rank~$1$ satisfies $\Vert Id+T\Vert =1+\Vert T\Vert$. 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 $T:X\to Y$ satisfying $\Vert J+T\Vert =1+\Vert T\Vert$, where $J:X\to Y$ 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 $\Vert Id+T\Vert =1+\Vert T\Vert$ is as small as possible and give characterisations in terms of a smoothness condition.