Skip to main content
Log in

Essentially normal Hilbert modules and K-homology IV: Quasi-homogenous quotient modules of Hardy module on the polydisks

Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we give a complete characterization for the essential normality of quasi-homogenous quotient modules of the Hardy modules \(H^2 \left( {\mathbb{D}^2 } \right)\). Also, we show that if d ≥ 3, then all the principle homogenous quotient modules of \(H^2 \left( {\mathbb{D}^d } \right)\) are not essentially normal.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atiyan M, MacDonland I. Introduction to Commutative Algebra. Menlo Park: Addison-Wesley, 1969

    Google Scholar 

  2. Arveson W. Quotients of standard Hilbert modules. Trans Amer Math Soc, 2007, 359: 6027–6055

    Article  MathSciNet  MATH  Google Scholar 

  3. Arveson W. p-Summable commutators in dimmension d. J Oper Theory, 2005, 54: 101–117

    MathSciNet  MATH  Google Scholar 

  4. Arveson W. The Dirac operator of a commuting d-tuple. J Funct Anal, 2002, 189: 53–79

    Article  MathSciNet  MATH  Google Scholar 

  5. Berezovskaja F, Medvedeva N. A complicated Singular point of “Centerfocus” type and the Newton diagram. Selecta Mathematica, 1994, 13: 1–15

    MathSciNet  Google Scholar 

  6. Chen X, Guo K. Analytic Hilbert Modules. New York: CRC Press, 2003

    Book  MATH  Google Scholar 

  7. Clark D N. Restrictions of H p functions in the polydisk. Amer J Math, 1988, 110: 1119–1152

    Article  MathSciNet  MATH  Google Scholar 

  8. Douglas R G, Misra G. Some Calculations for Hilbert modules. J Orissa Math Soc, 1993–96, 12–15: 75–85

    Google Scholar 

  9. Douglas R G. Essentially reductive Hilbert modules. J Oper Theory, 2006, 55: 117–133

    MATH  Google Scholar 

  10. Douglas R G. Invariants for Hilbert Modules. Proceedings of Symposia in Pure Mathematics Vol 51. Providence, RI: Amer Math Soc, 1990, 179-196

  11. Douglas R. A New Kind of Index Theorem Analysis, Geometry and Topology of Elliptic Operators. Singapore: World Scientific Publishing, 2005

    Google Scholar 

  12. Douglas R. Essentially reductive Hilbert modules (II). Hot topics in operator theory, 79–87, Theta Ser Adv Math, 9. Bucharest: Theta, 2008; arxiv:math.FA/0607722

    Google Scholar 

  13. Douglas R, Paulsen V. Hilbert Modules over Function algebras. Pitman Research Notes in Mathematics Series, 217. New York: Longman Scientific & Technical, 1989

    MATH  Google Scholar 

  14. Duan Y. Quasi-homogenous Hilbert modules. Int Equ Oper Theory, 2007, 58: 301–314

    Article  MATH  Google Scholar 

  15. Duan Y. Quotient modules for some Hilbert modules over the bidisk. J Math Anal Appl, 2010, 366: 486–493

    Article  MathSciNet  MATH  Google Scholar 

  16. Douglas R, Wang K. Essential normality of the cyclic submodule generated by any polynomial. J Funct Anal, 2011, 261: 3155–3180

    Article  MathSciNet  MATH  Google Scholar 

  17. Guo K. Defect operator for submodules of H 2d . J Reine Angew Math, 2004, 573: 181–209

    Article  MathSciNet  MATH  Google Scholar 

  18. Guo K. Equivalence of Hardy submodules generated by polynomials. J Funct Anal, 2000, 178: 343–371

    Article  MathSciNet  MATH  Google Scholar 

  19. Guo K, Wang P. Essentially normal Hilbert modules and K-homology, III: Homogenous quotient modules of Hardy module on the Bidisk. Sci China Ser A, 2007, 50: 387–411

    Article  MathSciNet  MATH  Google Scholar 

  20. Guo K, Wang K. Essentially normal Hilbert modules and K-homology. Math Ann, 2008, 340: 907–934

    Article  MathSciNet  MATH  Google Scholar 

  21. Guo K, Wang K. Essentially normal Hilbert modules and K-homology II: Quasi-homogeneous Hilbert modules over two dimensional unit ball. J Ramanujan Math Soc, 2007, 22: 259–281

    MathSciNet  MATH  Google Scholar 

  22. Guo K, Wang K, Zhang G. Trace formulas and p-essentially normal properties of quotient modules on the bidisk. J Operator Theory, in press

  23. Izuchi K, Yang R. N φ-type quotient modules on the torus. New York J Math, 2008, 14: 431–457

    MathSciNet  MATH  Google Scholar 

  24. Rudin W. Function Theory in the Unit Ball of C n. Grundlehren der Math, 241. New York: Springer, 1980

    Book  MATH  Google Scholar 

  25. Yang R. Operator theory in the Hardy space over the bidisk (II). Int Equ Oper Theory, 2002, 42: 99–124

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to PengHui Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guo, K., Wang, P. Essentially normal Hilbert modules and K-homology IV: Quasi-homogenous quotient modules of Hardy module on the polydisks. Sci. China Math. 55, 1613–1626 (2012). https://doi.org/10.1007/s11425-012-4395-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-012-4395-x

Keywords

MSC(2010)

Navigation