Limit Theorems for Numerical Index

TL;DR

This paper advances the understanding of numerical index limit theorems for Banach spaces, including vector-valued and sum spaces, by establishing new limit relations and conditions that ensure convergence.

Contribution

It introduces a modified numerical index and proves its limit behavior, also identifying conditions under which the numerical index converges for Banach spaces.

Findings

Proved $n_1(X) = \\lim_m n_1(X_m)$ for a modified numerical index.

Established that the numerical index converges under the local characterization condition.

Provided an example of a Banach space satisfying the local characterization condition.

Abstract

We improve upon on a limit theorem for numerical index for large classes of Banach spaces including vector valued $\ell_p$-spaces and $\ell_p$-sums of Banach spaces where $1\leq p \leq \infty$. We first prove $ n_1(X) = \displaystyle \lim_m n_1(X_m)$ for a modified numerical index $n_1(\, .\,)$. Later, we establish if a norm on $X$ satisfies the local characterization condition, then $n(X) = \displaystyle\lim_m n(X_m).$ We also present an example of a Banach space where the local characterization condition is satisfied.