Equivalence of quotient Hilbert modules -- II

2005/07/27 by Ronald G. Douglas, Douglas, Ronald G., Gadadhar Misra +1
Mathematics · #32Axx #32Qxx #46E22 #47A20 #47A65 #47B32 and 55R65 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.math/0507553

openalex publication_date 2005/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any open, connected and bounded set Ω⊆ \mathbb C^m, let \mathcal A be a natural function algebra consisting of functions holomorphic on Ω. Let \mathcal M be a Hilbert module over the algebra \mathcal A and \mathcal M₀⊆ \mathcal M be the submodule of functions vanishing to order k on a hypersurface \mathcal Z ⊆ Ω. Recently the authors have obtained an explicit complete set of unitary invariants for the quotient module \mathcal Q = \mathcal M \ominus \mathcal M₀ in the case of k=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.

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