Skip to main content
SpringerLink
Log in

Approximate representation of Bergman submodules

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

In the present paper, the author shows that if a homogeneous submodule M of the Bergman module L 2 a (B d ) satisfies

$${P_M} - \sum\limits_i {{M_{{z^i}}}} {P_M}M_{{z^i}}^* \leqslant \frac{c}{{N + 1}}{P_M}$$

for some number c > 0, then there is a sequence {f j } of multipliers and a positive number c′ such that \(c'{P_M} \leqslant \sum\limits_j {{M_{{f_j}}}} M_{{f_j}}^* \leqslant {P_M}\), i.e., M is approximately representable. The author also proves that approximately representable homogeneous submodules are p-essentially normal for p > d.

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

Access this article

Log in via an institution

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. Arveson, W., The curvature invariant of a Hilbert module over C[z 1, · · ·, z d], J. Reine Angew. Math., 522, 2000, 173–236.

    MathSciNet  MATH  Google Scholar 

  2. Arveson, W., p-summable commutators in dimension d, J. Operator Theory, 54(1), 2005, 101–117.

    MathSciNet  MATH  Google Scholar 

  3. Arveson, W., Quotients of standard Hilbert modules, Trans. Amer. Math. Soc., 359(12), 2007, 6027–6055.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  5. Douglas, R. and Sarkar, J., Essentially reductive weighted shift Hilbert modules, J. Oper. Theory, 65, 2011, 101–133.

    MathSciNet  Google Scholar 

  6. Douglas, R. and Wang, K., A harmonic analysis approach to essential normality of principal submodules, J. Funct. Anal., 261(11), 2011, 3155–3180.

    Article  MathSciNet  MATH  Google Scholar 

  7. Douglas, R. and Wang, K., Some remarks on essentially normal submodules, Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation, Springer-Verlag, Basel, 2014, 159–170.

    Google Scholar 

  8. Engliš, M., Density of algebras generated by Topelitz operators on Bergman spaces, Ark. Mat., 30(1–2), 1992, 227–240.

    Article  MathSciNet  MATH  Google Scholar 

  9. Engliš, M., Operator models and Arveson’s curvature invariant, Topological Algebras, Their Applications, and Related Topics, Banach Center Publ., 67, 2003, 171–183.

    Google Scholar 

  10. Engliš, M., Some problems in operator theory on bounded symmetric domains, Acta Appl. Math., 81(1), 2004, 51–71.

    Article  MathSciNet  MATH  Google Scholar 

  11. Engliš, M. and Eschmeier, J., Geometric Arveson-Douglas conjecture, arXiv:1312.6777 [math.FA].

  12. Eschmeier, J., Essential normality of homogeneous submodules, Int. Eq. Oper. Ther., 69(2), 2011, 171–182.

    Article  MathSciNet  MATH  Google Scholar 

  13. Fang, Q. and Xia, J., Commutators and localization on the Drury-Arveson space, J. Funct. Anal., 260(3), 2011, 639–673.

    Article  MathSciNet  MATH  Google Scholar 

  14. Fang, Q. and Xia, J., Essential normality of polynomial-generated submodules: Hardy space and beyond, J. Funct. Anal., 265(12), 2013, 2991–3008.

    Article  MathSciNet  MATH  Google Scholar 

  15. Gleason, J., Richter, S. and Sundberg, C., On the index of invariant subspaces in spaces of analytic functions in several complex variables, J. Reine. Angew. Math., 587, 2005, 49–76.

    Article  MathSciNet  MATH  Google Scholar 

  16. Guo, K., Defect oeprators for submodules of H2 d, J. Reine Angew. Math., 573, 2004, 181–209.

    MathSciNet  MATH  Google Scholar 

  17. Guo, K. and Wang, K., Essentially normal Hilbert modules and K-homology, Math. Ann., 340(4), 2008, 907–934.

    Article  MathSciNet  MATH  Google Scholar 

  18. Guo, K. andWang, K., Essentially normal Hilbert modules and K-homology II: Quasi-homogeneous Hilbert modules over the two dimensional unit ball, J. Ramanujan Math. Soc., 22(3), 2007, 259–281.

    MathSciNet  MATH  Google Scholar 

  19. Guo, K. and Zhao, C., p-essential normality of quasi-homogeneous Drury-Arveson submodules, J. London Math. Soc., 87(3), 2013, 899–916.

    Article  MathSciNet  MATH  Google Scholar 

  20. Kennedy, M., Essential normality and decomposability of homogeneous submodules, Trans. Amer. Math. Soc., 367(1), 2015, 293–311.

    Article  MathSciNet  MATH  Google Scholar 

  21. Kennedy, M. and Shalit, O., Essential normality and the decomposability of algebraic varieties, New York J. Math., 18, 2012, 877–890.

    MathSciNet  MATH  Google Scholar 

  22. McCullough, S., Invariant subspaces and Nevanlinna-Pick kernels, J. Funct. Anal., 178(1), 2000, 226–249.

    Article  MathSciNet  MATH  Google Scholar 

  23. Rudin, W., Function theory in the unit ball of Cn, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], Vol. 241, Springer-Verlag, New York, Berlin, 1980.

    Google Scholar 

  24. Shalit, O., Stable polynomial division and essential normality of graded Hilbert modules, J. London Math. Soc., 83, 2010, 273–289.

    Article  MathSciNet  Google Scholar 

  25. Zhao, C. and Yu, J., Trace estimation of commutators of multiplication operators on function spaces, Illinois J. Math., 56(2), 2012, 617–632.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Chong Zhao

Authors
  1. Chong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chong Zhao.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11271075, 11371096), Shandong Province Natural Science Foundation (No. ZR2014AQ009) and the Fundamental Research Funds of Shandong University (No. 2015GN017).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, C. Approximate representation of Bergman submodules. Chin. Ann. Math. Ser. B 37, 221–234 (2016). https://doi.org/10.1007/s11401-016-0964-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-016-0964-6

Keywords

2000 MR Subject Classification

Search

Navigation