Norm and Numerical Peak Holomorphic Functions on Banach Spaces

We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let $A_b(B_X:X)$ be the Banach space of all bounded continuous functions $f$ on the unit ball $B_X$ of a Banach space $X$ and their restrictions $f|_{B_X^\circ}$ to the open unit ball are holomorphic. In finite dimensional spaces, we show that the intersection of the set of all norm peak functions and the set of all numerical peak functions is a dense $G_\delta$ subset of $A_b(B_X:X)$. We also prove that if $X$ is a smooth Banach space with the Radon-Nikod\'ym property, then the set of all numerical strong peak functions is dense in $A_b(B_X:X)$. In particular, when $X=L_p(\mu)$ $(1<p<\infty)$ or $X=\ell_1$, it is shown that the intersection of the set of all norm strong peak functions and the set of all numerical strong peak functions is a dense $G_\delta$ subset of $A_b(B_X:X)$. In the meanwhile, we study the properties of the numerical radius of an holomorphic function and the numerical index of subspaces of $A_b(B_X:X)$. As an application, the existence and properties of numerical boundary of $A_b(B_X:X)$ are studied. Finally, the numerical peak function in $A_b(B_X:X)$ is characterized when $X=\ell_\infty^n$ and some negative results on the denseness of numerical (strong) peak holomorphic functions are given.