Daugavet centers

TL;DR

This paper generalizes the concept of Daugavet centers, showing that under certain conditions, the property can be preserved through renorming in a broader class of Banach spaces, with new geometric insights and examples.

Contribution

It extends the known results about Daugavet centers from specific cases to a more general setting involving subspaces and renorming, using a novel proof approach.

Findings

The main theorem shows preservation of Daugavet centers under renorming in Banach spaces.

Provides geometric characterizations of Daugavet centers.

Includes examples and extends known results to a broader context.

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.