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 Ab(BX : X) be the Banach space of all bounded continuous functions f on the unit ball BX of a Banach space X and their restrictions f |B◦ X 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 Gδ subset of Ab(BX : X). We also prove that if X is a smooth Banach space with the Radon-Nikodým property, then the set of all numerical strong peak functions is dense in Ab(BX : X). In particular, when X = Lp(μ) (1 < p < ∞) or X = l1, 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 Gδ subset of Ab(BX : X). In the meanwhile, we study the properties of the numerical radius of an holomorphic function and the numerical index of subspaces of Ab(BX : X). As an application, the existence and properties of numerical boundary of Ab(BX : X) are studied. Finally, the numerical peak function in Ab(BX : X) is characterized when X = l n ∞ and some negative results on the denseness of numerical (strong) peak holomorphic functions are given.