On the Hadamard multiplication theorem in ℂ n

The Hadamard product of two holomorphic functions given by Taylor series ∑αaαzα and ∑αbαzα, defined on neighborhoods Ω1 and Ω2 of the origin, is the function whose Taylor series is ∑αaαbαzα. For n=1, it is known that the Hadamard product extends to a holomorphic function on the so-called *-product of Ω1 and Ω2. In this paper, we generalize this result to the case of several complex variables.