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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 N , let Hv(G) and Hv 0 (G) denote the Banach spaces, under the norm f=sup{v(z)|f(z)|:zG}, 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 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 and v is a radial weight function vanishing at the boundary; that is, v(z)=v(|z|) for all z and lim |z|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:
46E15Banach spaces of continuous, differentiable or analytic functions
46A70Saks spaces and their duals
46B10Duality and reflexivity in normed spaces
46E10Topological linear spaces of continuous, differentiable or analytic functions