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Weighted spaces of holomorphic functions on balanced domains. (English) Zbl 0803.46023

This paper deals with radial weights on balanced domains $G\subset {ℂ}^{n}$, for which it is possible to apply methods involving the Taylor series of holomorphic functions about zero. The contractive properties of the Cesàro means of the Taylor series of functions in the disk algebra are used to derive remarkable consequences for spaces of holomorphic functions on arbitrary balanced open sets in ${ℂ}^{n}$. This leads to simple proofs that the spaces $H{V}_{0}\left(G\right)$ and ${𝒱}_{0}H\left(G\right)$ have the bounded approximation property whenever they contain the polynomials, and that then the polynomials are dense.

The second part of the paper is devoted to a related problem on $\epsilon$-tensor products with an arbitrary Banach space with applications to certain spaces of vector-valued holomorphic functions. In the last part some remarkable vector-valued generalizations of the (bi- )dualities $\left({\left(H{V}_{0}{\left(G\right)}_{b}^{\text{'}}\right)}_{b}^{\text{'}}=HV\left(G\right)$ and ${\left({\left({𝒱}_{0}H\left(G\right)\right)}_{b}^{\text{'}}\right)}_{i}^{\text{'}}=𝒱H\left(G\right)$ are established.

##### MSC:
 46E10 Topological linear spaces of continuous, differentiable or analytic functions 32A07 Special domains in ${ℂ}^{n}$ (Reinhardt, Hartogs, circular, tube) 46B28 Spaces of operators; tensor products; approximation properties 46M05 Tensor products of topological linear spaces