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Weak-type normal families of holomorphic mappings in Banach spaces and characterization of the Hilbert ball by its automorphism group. (English) Zbl 1039.32003

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 K.-T. Kim and 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.

MSC:

32A19 Normal families of holomorphic functions, mappings of several complex variables, and related topics (taut manifolds etc.)
46C15 Characterizations of Hilbert spaces
46G20 Infinite-dimensional holomorphy

Citations:

Zbl 1007.32002
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References:

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