A harmonic analysis approach to essential normality of principal submodules

Abstract

Guo and the second author have shown that the closure $[I]$ in the Drury-Arveson space of a homogeneous principal ideal $I$ in $\mathbb{C}[z_1,...,z_n]$ is essentially normal. In this note, the authors extend this result to the closure of any principal polynomial ideal in the Bergman space. In particular, the commutators and cross-commutators of the restrictions of the multiplication operators are shown to be in the Schatten $p$-class for $p>n$. The same is true for modules generated by polynomials with vector-valued coefficients. Further, the maximal ideal space $X_I$ of the resulting $C^{\ast }$-algebra for the quotient module is shown to be contained in $Z(I)\cap \partial {\mathbb{B}}_n$, where $Z(I)$ is the zero variety for $I$, and to contain all points in $\partial {\mathbb{B}}_n$ that are limit points of $Z(I)\cap {\mathbb{B}}_n$. Finally, the techniques introduced enable one to study a certain class of weight Bergman spaces on the ball.

accepted by JFA

References in corpus (6)

• p-Summable Commutators in Dimension d
• Essentially Reductive Hilbert Modules II
• Stable polynomial division and essential normality of graded Hilbert modules
• Essentially Reductive Weighted Shift Hilbert Modules
• Rings with Topologies Induced by Spaces of Functions
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Cited by in corpus (7)

• Essential normality and the decomposability of algebraic varieties
• Essential Normality - a Unified Approach in Terms of Local Decompositions
• Geometric Arveson-Douglas Conjecture and Holomorphic Extension
• A brief survey on operator theory in $H^2({\mathbb{D}}^2)$
• Geometirc Arveson-Douglas Conjecture - Decomposition of Varieties
• An index theorem for quotients of Bergman spaces on egg domains
• Stable division and essential normality: the non-homogeneous and quasi homogeneous cases