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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.

##### MSC:
 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