On the numerical radius of Lipschitz operators in Banach spaces

TL;DR

This paper investigates the Lipschitz numerical index in Banach spaces, providing characterizations, conditions for index 1, and demonstrating stability and specific space examples with index 1.

Contribution

It introduces the Lipschitz numerical index, characterizes it, and establishes conditions and stability properties for various Banach spaces.

Findings

Banach spaces with Lipschitz numerical index 1 characterized

Lush and C-rich spaces have Lipschitz numerical index 1

Stability of the index under sums and function space constructions

Abstract

We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index 1. As an application, we show that real lush spaces and $C$-rich subspaces have Lipschitz numerical index 1. Moreover, using the G$\hat{a}$teaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Finally, we prove that the Lipschitz numerical index has the stability properties for the $c_0$-, $l_1$-, and $l_\infty$-sums of spaces and vector-valued function spaces. From this, we show that the $C(K)$ spaces, $L_1(\mu)$-spaces and $L_\infty(\nu)$ spaces have Lipschitz numerical index 1.