Multipliers on vector spaces of holomorphic functions

Abstract

Let $G$ be a domain in the complex plane containing zero and $ H(G) $ be the set of all holomorphic functions on $G$. In this paper the algebra $M(H(G))$ of all coefficient multipliers with respect to the Hadamard product is studied. Central for the investigation is the domain $\widehat{G}$ introduced by Arakelyan which is by definition the union of all sets ${1\over w }G$ with $w \in G^c$. The main result is the description of all isomorphisms between these multipliers algebras. As a consequence one obtains: If two multiplier algebras $M(H(G_1 ))$ and $M(H(G_2 ))$ are isomorphic then $\widehat{G_1 }$ is equal to $\widehat{G_2}$. Two algebras $H(G_1 )$ and $H(G_2)$ are isomorphic with respect to the Hadamard product if and only if $G_1$ is equal to $G_2$. Further the following uniqueness theorem is proved: If $G_1$ is a domain containing $0$ and if $M(H(G))$ is isomorphic to $H(G_1 )$ then $G_1$ is equal to $\widehat{G}$.