A note on numerical radius attaining mappings

We prove that if every bounded linear operator (or N N -homogeneous polynomials) on a Banach space X X with the compact approximation property attains its numerical radius, then X 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 Galán [Q. J. Math. 54 (2003), pp. 1–10]. Finally, the denseness of weakly (uniformly) continuous 2 2 -homogeneous polynomials on a Banach space whose Aron-Berner extensions attain their numerical radii is obtained.