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Biduals of weighted Banach spaces of analytic functions. (English) Zbl 0801.46021
For \(v\) a positive continuous weight function on an open subset \(G\) of \(\mathbb C^ N\), let \(Hv(G)\) and \(Hv_ 0(G)\) denote the Banach spaces, under the norm \(\| f \| = \sup \{v(z) | f(z) | : z \in G\}\), of holomorphic functions \(f\) on \(G\) such that \(vf\) is bounded and \(vf\) vanishes at infinity on \(G\), respectively. The principal question addressed in the paper under review is whether \(Hv(G)\) is (isometrically isomorphic to) the bidual of \(Hv_ 0 (G)\). This problem may be viewed as an analogue to weighted function spaces because of the fact that \(\ell_ \infty\) is the bidual of \(c_ 0\).
An affirmative result (that \(Hv(G)\) is the bidual of \(Hv_ 0 (G))\) was obtained in 1970 by Rubel and Shields for the case where \(G\) is the open unit disc in \(\mathbb C\) and \(v\) is a radial weight function vanishing at the boundary; that is, \(v(z) = v( | z |)\) for all \(z\) and \(\lim_{| z | \to 1^ -} v(z) = 0\). This result, cited by the authors here, occurs as a special case of the main result of this paper. Specifically, Theorem 1.1 shows that \(Hv(G)\) is always (isometrically isomorphic to) the dual space of a Banach space and that \(Hv(G)\) is the bidual of \(Hv_ 0 (G)\) if and only if the unit ball of \(Hv_ 0 (G)\) is dense in the unit ball of \(Hv(G)\) in the compact-open topology. This density criterion is then used to obtain the Rubel-Shields result as well as other more general examples. Theorem 2.3 shows that, if the set \(G\) is balanced and if \(v\) is radial, then \(Hv(G)\) is the bidual of \(Hv_ 0 (G)\) whenever the latter space contains all polynomials.
The final section of the paper concerns the predual of \(Hv(G)\) and the structure of \(Hv(G)\) when equipped with various topologies, most notably the bounded weak star topology.

46E15 Banach spaces of continuous, differentiable or analytic functions
46A70 Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.)
46B10 Duality and reflexivity in normed linear and Banach spaces
46E10 Topological linear spaces of continuous, differentiable or analytic functions