A note on weighted Banach spaces of holomorphic functions.

*(English)*Zbl 1047.46018The authors deal with weighted spaces of holomorphic functions. Let $G$ be an open subset of ${\u2102}^{N}$ and $v:G\to \mathbb{R}$ a continuous, strictly positive function, called a weight. Then the main result of the paper states that

$${\left(Hv\right)}_{0}\left(G\right)=\{f:G\to \u2102:\phantom{\rule{4pt}{0ex}}f\phantom{\rule{4.pt}{0ex}}\text{holomorphic,}\phantom{\rule{4.pt}{0ex}}v|f\left|\phantom{\rule{4.pt}{0ex}}\text{vanishes}\phantom{\rule{4.pt}{0ex}}\text{at}\phantom{\rule{4.pt}{0ex}}\infty \right\}$$

endowed with the weighted sup-norm

$${\parallel f\parallel}_{v}=\underset{z\in G}{sup}v\left(z\right)\left|f\left(z\right)\right|$$

is always isomorphic to a subspace of ${c}_{0}$. This was previously known only for very restricted classes of weights and open sets $G$.

Reviewer: Wolfgang Lusky (Paderborn)