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Transactions of the American Mathematical Society

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Equivalence of quotient Hilbert modules-II


Authors: Ronald G. Douglas and Gadadhar Misra
Journal: Trans. Amer. Math. Soc. 360 (2008), 2229-2264
MSC (2000): Primary 46E22, 32Axx, 32Qxx, 47A20, 47A65, 47B32, 55R65
DOI: https://doi.org/10.1090/S0002-9947-07-04434-0
Published electronically: October 22, 2007
MathSciNet review: 2366981
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Abstract: For any open, connected and bounded set $ \Omega\subseteq \mathbb{C}^m$, let $ \mathcal A$ be a natural function algebra consisting of functions holomorphic on $ \Omega$. Let $ \mathcal M$ be a Hilbert module over the algebra $ \mathcal A$ and let $ \mathcal M_0\subseteq \mathcal M$ be the submodule of functions vanishing to order $ k$ on a hypersurface $ \mathcal Z \subseteq \Omega$. Recently the authors have obtained an explicit complete set of unitary invariants for the quotient module $ \mathcal Q=\mathcal M \ominus \mathcal M_0$ 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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Additional Information

Ronald G. Douglas
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: rdouglas@math.tamu.edu

Gadadhar Misra
Affiliation: Statistics and Mathematics Unit, Indian Statistical Institute, R. V. College Post, Bangalore 560 059, India
Address at time of publication: Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
Email: gm@isibang.ac.in

DOI: https://doi.org/10.1090/S0002-9947-07-04434-0
Keywords: Hilbert modules, complex geometry, jet bundles, curvature, homogeneous operators
Received by editor(s): August 30, 2005
Received by editor(s) in revised form: October 4, 2006
Published electronically: October 22, 2007
Additional Notes: The research of both authors was supported in part by a grant from the DST - NSF Science and Technology Cooperation Programme.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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