Invariance properties of homomorphisms on algebras of holomorphic functions with the Hadamard product

Let ${\displaystyle H\left(G_1\right)}$ be the set of all holomorphic functions on the domain ${\displaystyle G_1.}$ Two domains ${\displaystyle G_1}$ and ${\displaystyle G_2}$ are called Hadamard-isomorphic if ${\displaystyle H\left(G_1\right)}$ and ${\displaystyle H\left(G_2\right)}$ are isomorphic algebras with respect to the Hadamard product. Our main result states that two admissible domains are Hadamard-isomorphic if and only if they are equal.