Limit Theorems for Numerical Index

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 $1\le p\le \infty$. We first prove $n_1(X)=\underset{m}{\lim }{n}_1(X_m)$ for a modified numerical index $n_1(.)$. Later, we establish if a norm on $X$ satisfies the local characterization condition, then $n(X)=\underset{m}{\lim }n(X_m).$ We also present an example of a Banach space where the local characterization condition is satisfied.