Contribution to the Hadamard multiplication theorem

Abstract

In this article we define a binary linear operator T for holomorphic functions in given open sets \(A\) and \(B\) in the complex plane under certain additional assumptions. It coincides with the classical Hadamard product of holomorphic functions in the case where \(A\) and \(B\) are the unit disk. We show that the operator T exists provided \(A\) and \(B\) are simply connected domains containing the origin. Moreover, T is determined explicitly by means of an integral form. To this aim we prove an alternative representation of the star product \(A*B\) of any sets \(A,B\subset\mathbb{C}\) containing the origin. We also touch the problem of holomorphic extensibility of Hadamard product.