Multipliers on Spaces of Holomorphic Functions on Runge Domains in $\mathbb{C}^n

Abstract

We investigate multipliers on the space of holomorphic functions $H(\Omega)$, where $\Omega \subset \mathbb{C}^n$ is an open set. For Runge domains, we characterize these multipliers as convolutions with analytic functionals. Additionally, we explore Cartesian product domains, providing a representation of multipliers through germs of holomorphic functions. Finally, we identify the appropriate topology for analytic functionals, establishing a topological isomorphism with multipliers by utilizing the topology of uniform convergence on bounded sets inherited from the space of endomorphisms on $H(\Omega)$.