A note on numerical radius attaining mappings

TL;DR

This paper investigates conditions under which Banach spaces have finite dimensions based on numerical radius attainment properties of operators and polynomials, and improves existing theorems related to numerical radius.

Contribution

It establishes that numerical radius attainment for operators implies finite dimensionality and enhances the polynomial James' theorem for numerical radius.

Findings

Spaces where all operators attain numerical radius are finite dimensional

Improved polynomial James' theorem for numerical radius

Denseness of certain polynomials with radius-attaining Aron-Berner extensions

Abstract

We prove that if every bounded linear operator (or $N$-homogeneous polynomials) with the compact approximation property attains its numerical radius, then $X$ is a finite dimensional space. Moreover, we present an improvement of the polynomial James' theorem for numerical radius proved by Acosta, Becerra Guerrero and Gal$\'a$n in 2003. Finally, the denseness of weakly (uniformly) continuous $2$-homogeneous polynomials on a Banach space whose Aron-Berner extensions attain their numerical radii is obtained.