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Duals of Banach spaces of entire functions. (English) Zbl 0769.46011
Let \(w\) be a strictly positive function on \(\mathbb{C}\) and let \(H_ \infty^ w\), respectively \(H_ 0^ w\), denote the Banach spaces of those entire functions \(\varphi(z)\) with \(|\varphi(z)|=O(w(z))\) and \(|\varphi(z)|=o(w(z))\). In this generality, these spaces may contain only constants, but for many functions \(w(z)\) these will be interesting Banach spaces with norm \[ \| \varphi\|_ w=\sup\{|\varphi(z)|/w(z):\;z\in\mathbb{C}\}. \] We study two specific problems.
(A) For which weight functions \(w\) is \(H_ \infty^ w\) isomorphic, possibly isometrically, to \((H_ 0^ w)^{**}\)?
(B) For which weight functions \(w\) can the first dual \((H_ 0^ w)^*\) be identified with a space of functions analytic on some subset of \(\mathbb{C}\)?

MSC:
46E15 Banach spaces of continuous, differentiable or analytic functions
46B10 Duality and reflexivity in normed linear and Banach spaces
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References:
[1] Rubel, J. Austral. Math. Soc 11 pp 276– (1970)
[2] Anderson, J. Reine Angew. Math. 270 pp 12– (1974)
[3] Clunie, Canad. J. Math 20 pp 7– (1968) · Zbl 0164.08602
[4] DOI: 10.1007/BF02392162 · Zbl 0229.46049
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