English

Equivalence of quotient Hilbert modules -- II

Functional Analysis 2007-05-23 v2 Complex Variables Spectral Theory

Abstract

For any open, connected and bounded set ΩCm\Omega \subseteq \mathbb C^m, let A\mathcal A be a natural function algebra consisting of functions holomorphic on Ω\Omega. Let M\mathcal M be a Hilbert module over the algebra A\mathcal A and M0M\mathcal M_0\subseteq \mathcal M be the submodule of functions vanishing to order kk on a hypersurface ZΩ\mathcal Z \subseteq \Omega. Recently the authors have obtained an explicit complete set of unitary invariants for the quotient module Q=MM0\mathcal Q = \mathcal M \ominus \mathcal M_0 in the case of k=2k=2. In this paper, we relate these invariants to familiar notions from complex geometry. We also find a complete set of unitary invariants for the general case. We discuss many concrete examples in this setting. As an application of our equivalence results, we characterise certain homogeneous Hilbert modules over the bi-disc algebra.

Keywords

Cite

@article{arxiv.math/0507553,
  title  = {Equivalence of quotient Hilbert modules -- II},
  author = {Ronald G. Douglas and Gadadhar Misra},
  journal= {arXiv preprint arXiv:math/0507553},
  year   = {2007}
}

Comments

The introduction has been revised and streamlined. The material has been reorganized slightly

R2 v1 2026-07-22T17:22:34.079Z