Non-isomorphism of some algebras of holomorphic functions

Abstract

Suppose that $\mathcal X$ is a family of spaces of holomorphic functions such that each $X=X(D) \in \mathcal X$ can be defined on a domain $D$ belonging to some class $\mathcal D$ of domains. Then for any two concrete domains $D_1$ and $D_2 \in \mathcal D$ and $X \in \mathcal X$ one can ask the following natural question if corresponding spaces $X(D_1)$ and $X(D_2)$ are isomorphic as topological vector spaces. Similarly, for a fixed $D \in \mathcal D$ and two different spaces $X_1, X_2 \in \mathcal X$ one can consider the existence of an isomorphism between $X_1(D)$ and $X_2(D)$. We answer these questions when $\mathcal X$ consists of Hardy $N^p_*(D)$, maximal Hardy $MN^p_*(D)$, Bergman ${\mathbb N}^p(D)$, and Lumer's Hardy $LN^p_*(D)$ algebras, $p \geq 1$, and $\mathcal D = \{\mathbb B_n, \mathbb U^n, n \in \mathbb N\}$ is the family of the unit balls and the unit polydiscs in ${C}^n$.