Daugavet centers

Abstract

An operator $G : \allowbreak X \to Y$ is said to be a Daugavet center if $\|G + T\| = \|G\| + \|T\|$ for every rank-1 operator $T : \allowbreak X \to Y$. The main result of the paper is: if $G : \allowbreak X \to Y$ is a Daugavet center, $Y$ is a subspace of a Banach space $E$, and $J: Y \to E$ is the natural embedding operator, then $E$ can be equivalently renormed in such a way, that $J \circ G : X \to E$ is also a Daugavet center. This result was previously known for particular case $X=Y$, $G=\mathrm{Id}$ and only in separable spaces. The proof of our generalization is based on an idea completely different from the original one. We give also some geometric characterizations of Daugavet centers, present a number of examples, and generalize (mostly in straightforward manner) to Daugavet centers some results known previously for spaces with the Daugavet property.