Hadamard multipliers on spaces of holomorphic functions
Motivated by the work [\textit{M. Langenbruch} and \textit{P. Domański}, Analysis, München 32, No. 2, 137--162 (2012; Zbl 1288.46020)] where the problem of multipliers is considered on spaces of real analytic functions, the present paper is devoted to the study of Hadamard multipliers on spaces of holomorphic functions. The aim of the paper is to find a representation of all multipliers on \(H(\Omega)\) where \(\Omega\) is a Runge open set from \(\mathbb C\). The paper is organized in six sections, the first one presenting the main notations and results. Also here some examples of Hadamard multiplication operators and Euler differential operators are presented. In the second section the properties of the dilation set \(V(\Omega)\) of \(\Omega\) are analyzed, and in the third one the topology on \(H(V)\) and its dual is studied. The main representation theorems, presented in the first section, are proved in the fourth section, obtaining a description of all Hadamard multipliers on \(H(\Omega)\) via analytic functionals \(H(V(\Omega))'\) on the space of holomorphic germs on the dilation of \(\Omega\). The continuity problems are analyzed in the fifth section, and in the last section of the paper some remarks and comments on the obtained results are presented. Also, equivalent assertions for an Euler differential operator (Theorem 6.3) are proved.
scientific article; zbMATH DE number 6805235
scientific article