## Nagoya Mathematical Journal

### 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 $\widehat{G}$.

#### Article information

Source
Nagoya Math. J., Volume 159 (2000), 167-178.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631456

Mathematical Reviews number (MathSciNet)
MR1783568

Zentralblatt MATH identifier
0970.30028

#### Citation

Render, Hermann; Sauer, Andreas. Multipliers on vector spaces of holomorphic functions. Nagoya Math. J. 159 (2000), 167--178. https://projecteuclid.org/euclid.nmj/1114631456

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