In this proposal we study the structural properties of a particular, yet important, collection of examples of unbounded subnormal operators; namely, the unbounded Bergman operators. The main goal of this proposal is to develop new techniques and ideas from analytic function and operator theory to answer questions with respect to self--commutator, compactness, cyclicity, invariant structure, and functional calculus of unbounded Bergman operators. As a by product of our proposal, we will also obtain interesting results regarding the unbounded Toeplitz operators. Another focus of the proposed research lies in the area of approximation in the mean by polynomials and rational functions on unbounded regions which is closely related to the concept of cyclicity. Our goal here is to extend the Caratheodory--Farrell--Markusevic Theorem to Bergman spaces over unbounded regions.
The growing body of recent work on unbounded subnormal operators has provided a rich theory with great deal of challenging problems in both operator theory and function theory. Questions involving unbounded subnormal operators are deep, and, in many cases, the techniques needed to attack these problems are nontrivial and incorporate many interdisciplinary areas of analysis. One of the major inspirational source in investigating the unbounded Bergman operators comes from the existence of many interesting concrete examples of unbounded subnormal operators. In fact, it has been shown that one of the fundamental operators in physics and quantum mechanics, the "creation operator", is an example of an unbounded subnormal operator. Furthermore it is not hard to prove that the creation operator can be modeled as an unbounded multiplication operator on a certain space of square integrable entire functions with respect to the two dimensional Gaussian measure. Consequently, progress in the proposed project has important consequences in our better understanding of the structure of the creation operator.
