Banach spaces with the Daugavet property

Abstract

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