Equivalence of quotient Hilbert modules

Abstract

Let $\cl{M}$ be a Hilbert module of holomorphic functions over a natural function algebra $\mathcal{A}(\Omega)$, where $\Omega \subseteq \bb{C}^m$ is a bounded domain. Let $\cl{M}_0\subseteq \cl{M}$ be the submodule of functions vanishing to order $k$ on a hypersurface $\cl{Z} \subseteq \Omega$. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modules $\cl{Q}=\cl{M} \ominus \cl{M}_0$. The invariants are given explicitly in the particular case of $k = 2$.