A NOTE ON NUMERICAL RADIUS ATTAINING MAPPINGS

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