Weak-type normal families of holomorphic mappings in Banach spaces and characterization of the Hilbert ball by its automorphism group (Q1407742)

The following theorem is proved: if a convex domain \(\Omega\) in a separable Hilbert space \({\mathcal H}\) admits a \(C^2\) strongly pseudoconvex boundary point at which a holomorphic automorphism orbit accumulates, then, \(\Omega\) is biholomorphic to the unit open ball in \({\mathcal H}\). This result generalizes the recent work of \textit{K.-T. Kim} and \textit{S. G. Krantz} [Trans. Am. Math. Soc. 354, No.7, 2797--2818 (2002; Zbl 1007.32002)] to the case when domain \(\Omega\) is not necessarily bounded. As one of the tools, the authors prove and use a version of the weak-type normal family theorem for holomorphic mappings in the infinite dimensional Banach spaces.