Almost Daugavet centers

An operator G:X→Y is an almost Daugavet center if there exists a norming subspace Z⊂Y⁎ such that ‖G+T‖=‖G‖+‖T‖ for every rank-1 operator T:X→Y of the form T=x⁎⊗y where y∈Y and . This notion is both a generalization of the almost Daugavet property when G=I and X=Y, and a generalization of the notion of Daugavet centers when W=X⁎. We give a characterization of the almost Daugavet centers in terms of the thickness of an operator and in terms of canonical ℓ1-type sequences. We show that, for a separable space Y, an operator G:X→Y is similar to an almost Daugavet center if and only if G fixes an isomorphic copy of ℓ1. We also give some geometric characterizations of this property.