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 C[z1, . . . , zn] 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 crosscommutators 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 XI of the resulting C∗-algebra for the quotient module is shown to be contained in Z(I ) ∩ ∂Bn, where Z(I ) is the zero variety for I , and to contain all points in ∂Bn that are limit points of Z(I )∩Bn. Finally, the techniques introduced enable one to study a certain class of weight Bergman spaces on the ball. © 2011 Elsevier Inc. All rights reserved.