On the Lipschitz numerical index of Banach spaces

TL;DR

This paper explores the Lipschitz numerical index and radius in Banach spaces, providing new renorming results, introducing a concept of radius attaining functions, and analyzing the index in various Banach space constructions.

Contribution

It introduces new results on the Lipschitz numerical index, defines Lipschitz numerical radius attaining functions, and studies the index in vector-valued, sum, and function spaces.

Findings

Renorming results for Lipschitz numerical index

Failure of denseness of radius attaining functions in general Banach spaces

Analysis of Lipschitz numerical index in various Banach space classes

Abstract

We provide some new consequences on the Lipschitz numerical radius and index which were introduced recently. More precisely, we give some renorming results on the Lipschitz numerical index, introduce a concept of Lipschitz numerical radius attaining functions in order to show that the denseness fails for an arbitrary Banach space, and study a Lipschitz version of Daugavet centers. Furthermore, we discuss the Lipschitz numerical index of vector-valued function spaces, absolute sums of Banach spaces, the K\"othe-Bochner spaces, and Banach spaces which contain a dense union of increasing family of one-complemented subspaces.