Denseness of holomorphic functions attaining their numerical radii

Abstract

For two complex Banach spaces X and Y, $\mathcal{A}_\infty$ (B _X; Y) will denote the space of bounded and continuous functions from B _X to Y that are holomorphic on the open unit ball. The numerical radius of an element h in $\mathcal{A}_\infty$ (B _X; X) is the supremum of the set

$$\{ |x*(h(x))| : x \in X, x* \in X*, \parallel x*\parallel = \parallel x\parallel = x*(x) = 1\} $$

. We prove that every complex Banach space X with the Radon-Nikodým property satisfies that the subset of numerical radius attaining functions in $\mathcal{A}_\infty$ (B _X; X) is dense in $\mathcal{A}_\infty$ (B _X; X). We also show the denseness of the numerical radius attaining elements of $\mathcal{A}_u (B_{c_0 } ; c_0 )$ in the whole space, where $\mathcal{A}_u (B_{c_0 } ; c_0 )$ is the subset of functions in $\mathcal{A}_\infty (B_{c_0 } ; c_0 )$ which are uniformly continuous on the unit ball. For C(K) we prove a denseness result for the subset of the functions in $\mathcal{A}_\infty$ (B _C(K); C(K)) which are weakly uniformly continuous on the closed unit ball. For a certain sequence space X, there is a 2-homogenous polynomial P from X to X such that for every R > e, P cannot be approximated by bounded and numerical radius attaining holomorphic functions defined on RB _X . If Y satisfies some isometric conditions and X is such that the subset of norm attaining functions of $\mathcal{A}_\infty$ (B _X; ℂ) is dense in $\mathcal{A}_\infty$ (B _X; ℂ), then the subset of norm attaining functions in $\mathcal{A}_\infty$ (B _X; Y) is dense in the whole space.