On the Lipschitz numerical index of Banach spaces
Geunsu Choi, Mingu Jung, Hyung-Joon Tag

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.
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Taxonomy
TopicsAdvanced Banach Space Theory · Optimization and Variational Analysis · Approximation Theory and Sequence Spaces
