A harmonic analysis approach to essential normality of principal submodules

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.