On the Convergence of Numerical Index via Operator Openings and Ultraproducts

Abstract

The numerical index of a Banach space is a geometric constant relating the numerical radius of bounded linear operators to their standard operator norm. In this paper, we study the continuity of the numerical index under two distinct notions of subspace convergence. First, we establish a full limit theorem in the operator opening topology: if $\{X_n\}_{n \in \mathbb{N}}$ and $X$ are closed subspaces of a Banach space $Y$ with $X_n \to X$ in the operator opening, then $\lim_{n \to \infty} n(X_n) = n(X)$. Second, we develop ultraproduct methods for the numerical index, proving that the numerical radius is exactly preserved by ultraproduct operators, i.e., $v((T_n)_{\mathcal{U}}) = \lim_{\mathcal{U}} v(T_n)$. As a consequence, we show that $n(X_{\mathcal{U}}) \le n(X)$ for every ultrapower $\mathcal{U}$.