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Extension of bounded vector-valued functions. (English) Zbl 1177.46024
This article studies the following problem about the extension of bounded holomorphic or harmonic functions. Let $\Omega$ be an open set and let $E$ be a locally complete locally convex space. Find conditions on $A \subset \Omega$, and $H \subset E'$ to assure that every bounded function $f:A \rightarrow E$ such that $u \circ f:A \rightarrow \mathbb{C}$ which has an extension in $\mathcal{H}^{\infty}(\Omega)$ for each $u \in H$ is the restriction to $A$ of a vector valued function $F \in \mathcal{H}^{\infty}(\Omega,E)$. The authors show that the extension is possible if (1) $A$ is a set of uniqueness for $\mathcal{H}^{\infty}(\Omega)$ and $H$ determines boundedness in $E$, or if (2) $A$ is a sampling set for $\mathcal{H}^{\infty}(\Omega)$ and $H$ is weak*-dense in $E'$. As a corollary, they obtain a vector valued Blaschke theorem which complements a result due to {\it W. Arendt} and {\it N. Nikolski} [Math. Z. 234, 777--805 (2000; Zbl 0976.46030)]. The proofs of these results are based on deep functional analytic techniques which continue an approach started by the first two authors and the reviewer in [Stud. Math. 183, 225--248 (2007; Zbl 1141.46017)].

46E40Spaces of vector- and operator-valued functions
46E10Topological linear spaces of continuous, differentiable or analytic functions
46A32Spaces of linear operators; topological tensor products; approximation properties
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