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Essential Normality of Homogenous Quotient Modules Over the Polydisc: Distinguished Variety Case

  • Penghui Wang
  • Chong Zhao
Article

Abstract

In the present paper, we investigate the essential normality of quotient modules over the polydisc. Let I be a homogeneous ideal in \(\mathbb {C}[z_1,\ldots ,z_d]\), we show that if the homogeneous quotient module \([I]^\bot \) of \(H^2(\mathbb D^d)\) is essentially normal, then \(\dim _{\mathbb {C}}Z(I)\le 1\). It is shown that if Z(I) is distinguished, then \([I]^\bot \) is \((1,\infty )\)-essentially normal, i.e. the \([S_{z_i}^*, S_{z_j}]\)’s are not necessarily trace class operators but indeed belong to the interpolation ideal \(\mathcal L^{(1,\infty )}\), see the monograph “Noncommutative Geometry” of Connes. This result leads to the answer to the polydisc version of Arveson–Douglas problem. Moreover, we study the boundary representation of \([I]^\bot \).

Keywords

Essential normality Hardy space over polydisc Quotient module Boundary representation 

Mathematics Subject Classification

Primary 47A13 Secondary 46H25 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanThe People’s Republic of China

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