Abstract
Let \({M \subset H(\mathbb{B})}\) be a homogeneous submodule of the n-shift Hilbert module on the unit ball in \({\mathbb{C}^{n}}\). We show that a modification of an operator inequality used by Guo and Wang in the case of principal submodules is equivalent to the existence of factorizations of the form \({[M_{z_j}^*,P_M] = (N+1)^{-1}A_j}\), where N is the number operator on \({H(\mathbb{B})}\). Thus a proof of the inequality would yield positive answers to conjectures of Arveson and Douglas concerning the essential normality of homogeneous submodules of \({H(\mathbb{B})}\). We show that in all cases in which the conjectures have been established the inequality holds and leads to a unified proof of stronger results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arveson W.: Subalgebras of C*-algebras. III. Multivariable operator theory. Acta Math. 181, 159–228 (1998)
Arveson W.: The curvature invariant of a Hilbert module over \({\mathbb{C}[z_1, \ldots, z_n]}\). J. Reine Angew. Math. 522, 173–236 (2000)
Arveson W.: The Dirac operator of a commuting tuple. J. Funct. Anal. 189, 53–79 (2002)
Arveson, W.: Several problems in operator theory (2003). http://www.math.berkely.edu/arveson
Arveson W.: p-summable commutators in dimension d. J. Oper. Theory 54, 101–117 (2005)
Arveson W.: Quotients of standard Hilbert modules. Trans. Am. Math. Soc. 359, 6027–6055 (2007)
Douglas, R.G.: A new kind of index theorem. In: Analysis, Geometry and Topology of Elliptic Operators (Roskilde, Denmark, 2005). World Scientific Publishing, Singapore (2006)
Douglas R.G.: Essentially reductive Hilbert modules. J. Oper. Theory 55, 117–133 (2006)
Douglas, R.G., Paulsen, V.: Hilbert modules over function algebras. In: Pitman Research Notes in Mathematics, vol. 217 (1989)
Douglas, R.G., Sarkar, J.: Essentially reductive weighted shift Hilbert modules. J. Oper. Theory (2010) (to appear)
Eschmeier J.: Grothendieck’s comparison theorem and multivariable Fredholm theory. Arch. Math. 92, 461–475 (2009)
Eschmeier, J., Putinar, M.: Spectral decompositions and analytic sheaves. London Mathematical Society, New Series, vol. 10. Clarendon Press, Oxford (1996)
Guo K., Wang K.: Essentially normal Hilbert modules and K-homology. Math. Ann. 340, 907–934 (2008)
Shalit, O.: Stable polynomial division and essential normality of graded Hilbert modules, Preprint, U. Waterloo (2009)
Yang R.W.: The Berger–Shaw theorem in the Hardy module over the bidisc. J. Oper. Theory 42, 379–404 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Eschmeier, J. Essential Normality of Homogeneous Submodules. Integr. Equ. Oper. Theory 69, 171–182 (2011). https://doi.org/10.1007/s00020-010-1809-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-010-1809-4