Equivalence of quotient Hilbert modules–II

Douglas, Ronald; Misra, Gadadhar

doi:10.1090/S0002-9947-07-04434-0

Equivalence of quotient Hilbert modules–II
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by Ronald G. Douglas and Gadadhar Misra PDF

Trans. Amer. Math. Soc. 360 (2008), 2229-2264 Request permission

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.

References

N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437, DOI 10.1090/S0002-9947-1950-0051437-7
Bhaskar Bagchi and Gadadhar Misra, Homogeneous tuples of multiplication operators on twisted Bergman spaces, J. Funct. Anal. 136 (1996), no. 1, 171–213. MR 1375158, DOI 10.1006/jfan.1996.0026
Bhaskar Bagchi and Gadadhar Misra, Homogeneous operators and projective representations of the Möbius group: a survey, Proc. Indian Acad. Sci. Math. Sci. 111 (2001), no. 4, 415–437. MR 1923984, DOI 10.1007/BF02829616
E. Cartan, Leçons sur la géométrie des espaces de riemann, Les Grands Classiques Gauthier-Villars, 1988, Reprint of the second (1946) edition.
Xiao Man Chen and R. G. Douglas, Localization of Hilbert modules, Michigan Math. J. 39 (1992), no. 3, 443–454. MR 1182500, DOI 10.1307/mmj/1029004599
M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), no. 3-4, 187–261. MR 501368, DOI 10.1007/BF02545748
M. J. Cowen and R. G. Douglas, Operators possessing an open set of eigenvalues, Functions, series, operators, Vol. I, II (Budapest, 1980) Colloq. Math. Soc. János Bolyai, vol. 35, North-Holland, Amsterdam, 1983, pp. 323–341. MR 751007
M. J. Cowen and R. G. Douglas, Equivalence of connections, Adv. in Math. 56 (1985), no. 1, 39–91. MR 782542, DOI 10.1016/0001-8708(85)90084-2
Raúl E. Curto and Norberto Salinas, Generalized Bergman kernels and the Cowen-Douglas theory, Amer. J. Math. 106 (1984), no. 2, 447–488. MR 737780, DOI 10.2307/2374310
Ronald G. Douglas and Gadadhar Misra, Geometric invariants for resolutions of Hilbert modules, Nonselfadjoint operator algebras, operator theory, and related topics, Oper. Theory Adv. Appl., vol. 104, Birkhäuser, Basel, 1998, pp. 83–112. MR 1639650
Ronald G. Douglas and Gadadhar Misra, Equivalence of quotient Hilbert modules, Proc. Indian Acad. Sci. Math. Sci. 113 (2003), no. 3, 281–291. MR 1999257, DOI 10.1007/BF02829607
Ronald G. Douglas and Gadadhar Misra, Quasi-free resolutions of Hilbert modules, Integral Equations Operator Theory 47 (2003), no. 4, 435–456. MR 2021968, DOI 10.1007/s00020-003-1169-4
Ronald G. Douglas and Gadadhar Misra, On quasi-free Hilbert modules, New York J. Math. 11 (2005), 547–561. MR 2188255
Ronald G. Douglas, Gadadhar Misra, and Cherian Varughese, On quotient modules—the case of arbitrary multiplicity, J. Funct. Anal. 174 (2000), no. 2, 364–398. MR 1768979, DOI 10.1006/jfan.2000.3594
Ronald G. Douglas and Vern I. Paulsen, Hilbert modules over function algebras, Pitman Research Notes in Mathematics Series, vol. 217, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 1028546
R. G. Douglas, V. I. Paulsen, C.-H. Sah, and K. Yan, Algebraic reduction and rigidity for Hilbert modules, Amer. J. Math. 117 (1995), no. 1, 75–92. MR 1314458, DOI 10.2307/2375036
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
Sarah H. Ferguson and Richard Rochberg, Higher order Hilbert-Schmidt Hankel forms and tensors of analytic kernels, Math. Scand. 96 (2005), no. 1, 117–146. MR 2142876, DOI 10.7146/math.scand.a-14948
William Cohn, Sarah H. Ferguson, and Richard Rochberg, Boundedness of higher order Hankel forms, factorization in potential spaces and derivations, Proc. London Math. Soc. (3) 82 (2001), no. 1, 110–130. MR 1794259, DOI 10.1112/S0024611500012727
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
A. Koranyi and G. Misra, Homogeneous operators on Hilbert spaces of holomorphic functions- I, preprint.
Adam Korányi and Gadadhar Misra, New constructions of homogeneous operators, C. R. Math. Acad. Sci. Paris 342 (2006), no. 12, 933–936 (English, with English and French summaries). MR 2235613, DOI 10.1016/j.crma.2006.04.002
Gadadhar Misra and N. S. Narsimha Sastry, Homogeneous tuples of operators and representations of some classical groups, J. Operator Theory 24 (1990), no. 1, 23–32. MR 1086542
Lizhong Peng and Genkai Zhang, Tensor products of holomorphic representations and bilinear differential operators, J. Funct. Anal. 210 (2004), no. 1, 171–192. MR 2052118, DOI 10.1016/j.jfa.2003.09.006
Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
R. O. Wells, Differential analysis on complex manifolds, Springer, 1973.
David R. Wilkins, Homogeneous vector bundles and Cowen-Douglas operators, Internat. J. Math. 4 (1993), no. 3, 503–520. MR 1228586, DOI 10.1142/S0129167X93000261