On the Convergence of Numerical Index via Operator Openings and Ultraproducts

TL;DR

This paper investigates the stability of the numerical index of Banach spaces under subspace convergence and ultraproducts, establishing limit theorems and preservation properties using operator topologies and ultraproduct techniques.

Contribution

It introduces new limit theorems for the numerical index under operator opening topology and ultraproduct methods, advancing understanding of its continuity and invariance properties.

Findings

Numerical index is continuous under operator opening convergence.

Ultraproducts preserve the numerical radius exactly.

Ultrapowers do not increase the numerical index.

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 $\mathcal{U}$.