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A weak criterion for vector-valued holomorphy. (English) Zbl 1055.46026

The author presents a new approach to a theorem, originally proved in K.–G. Große–Erdmann [“The Borel-Okada theorem revisited” (Univ. Hagen, FB Math. (Habil.)) (1992; Zbl 0953.30500)], which is a generalization of Dunford’s theorem (weak holomorphy implies holomorphy for Banach space valued functions). Here the image space is a locally complete space E. The main tool in the proof is J. Wolff’s theorem [C. R. Acad. Sci. Paris 173, 1327–1328 (1921; JFM 48.0320.01)].

It follows that a function f defined on a subdomain ΩK of Ω admits a unique holomorphic continuation to Ω if f is locally bounded and the functions x ' f admit a holomorphic continuation, where x ' runs through a separating subset of E ' , roughly speaking. This is also true under similar assumptions if E is a Fréchet space.

In what follows, the author provides generalizations of his theorem to meromorphic functions and to functions defined on a domain of a locally convex spaces. Two results concerning a problem of Wrobel and a remark concerning real analytic functions conclude the paper.


MSC:
46G20Infinite dimensional holomorphy
30B40Analytic continuation (one complex variable)