Almost Daugavet centers

Abstract

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 x ⁎ ∈ W = G ⁎ ( Z ) ¯ . 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 ℓ 1 -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 ℓ 1 . We also give some geometric characterizations of this property.