The Arveson-Douglas Conjecture
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Date
2018-05-31
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Abstract
The Arveson-Douglas Conjecture states that, closures of polynomial ideals in some analytic Hilbert modules, such as the Drury-Arveson module, Bergman module or Hardy module, are essentially normal. The conjecture has connections to multivariable operator theory, index theory and function theory. In this dissertation, we discuss an approach using tools from harmonic analysis and several complex variables. Two methods are introduced, approaching this problem from different aspects. Each method has given some interesting new results. Most of the theories developed in this dissertation concerns the Bergman space.
First, we introduce the Arveson-Douglas Conjecture, its background and applications.
Then we describe the first method. The main subject is a geometric version of the Arveson-Douglas Conjecture. For this method to work we need the variety to have nice properties, such as smoothness or transversality, around the boundary. Two types of varieties are considered.
Next, we develop the second method, which mainly treats principal submodules. This method gives weaker results but is more flexible. As a consequence, we are able to extend our discussion to strongly pseudoconvex domains and to the Hardy space. As an application, we apply this theory to the unit ball and obtain some nontrivial results.
Finally, we end this dissertation with some concluding remarks as well as future plans.
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Arveson-Douglas Conjecure, Essential Normality