Lipschitz vs Linear Numerical Index in certain Banach spaces

TL;DR

This paper proves that in certain real Banach spaces, the Lipschitz and linear numerical indices are equal, advancing understanding of their relationship and addressing a question in functional analysis.

Contribution

It demonstrates the equality of Lipschitz and linear numerical indices in separable and dual Banach spaces, providing partial evidence for a broader conjecture.

Findings

Lipschitz and linear numerical indices coincide in separable spaces.

Lipschitz and linear numerical indices coincide in dual spaces.

The result supports the conjecture that these indices are equal in all real Banach spaces.

Abstract

We show that for real Banach spaces that are either separable or dual spaces, the Lipschitz numerical index coincides with the classical (linear) numerical index. This result provides partial evidence toward the question posed by Wang, Huang, and Tan (2014) of whether these two quantities coincide for every real Banach space. Our approach relies on two standard linearization techniques for Lipschitz maps: differentiation via convolution with Gaussian probability measures, and invariant means.