A harmonic analysis approach to essential normality of principal submodules |
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Authors: | Ronald G Douglas |
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Institution: | a Department of Mathematics, Texas A&M University, College Station, TX, USA b School of Mathematical Sciences, Fudan University, Shanghai, PR China |
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Abstract: | Guo and the second author have shown that the closure I] in the Drury-Arveson space of a homogeneous principal ideal I in Cz1,…,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 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 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. |
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Keywords: | Essentially normal Hilbert module Arveson?s conjecture Covering lemma |
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