Extension of vector-valued holomorphic and meromorphic functions
Let \(\emptyset \not= \Omega_1 \subseteq \Omega_2 \subseteq \mathbb{C}\) be any domains and let \(E\) be a locally convex space. Let \(f:\Omega_1 \to E\) be any holomorphic or meromorphic function such that all compositions \(u \circ f:\Omega_1 \to E \to \mathbb{C}\) by linear functionals \(u \in E'\) admit a holomorphic or meromorphic extension to \(\Omega_2\), respectively. Does there exist a holomorphic or meromorphic extension \(\tilde{f}:\Omega_2 \to E'\) of \(f\)? It is well known that the answer is yes for all Banach spaces \(E\) [see e.g., \textit{L. M. Hai, N. V. Khue} and \textit{N. T. Nga}, Colloq. Math. 64, No. 1, 65--70 (1993; Zbl 0824.46052)]. In the paper under review the author studies the problem for locally convex spaces. In the holomorphic case the answer is yes for all locally complete spaces \(E\), and in the meromorphic case the answer is yes under the additional assumption that \(E\) does not contain \(\mathbb{C}^{\mathbb{N}}\) and that \(E'_{\beta}\) is suprabarrelled or that \(E\) is a barrelled complete Schwartz space. As a main tool it is used that for locally complete spaces \(E\) not containing \(\mathbb{C}^{\mathbb{N}}\) the formula \(M(\Omega, E)=M(\Omega) \varepsilon E\) holds true.
scientific article; zbMATH DE number 2186499