Extending Hilbert Space Operators

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Our research proposal concerns the interplay between Operator Theory and Function Theory in one and several variables. A particular group of problems that we propose to attack involves the generalizations to several complex variables of some of the classical moment and interpolation problems on the unit disc such as the interpolation theorem of Nevanlinna and Pick, and the moment theorems of Caratheodory and Herglotz. Other groups involve the generalization of geometric function theory to analytic varieties and the understanding of extremal mappings into the symmetrized bidisc. Research intrinsic to operator theory that we will undertake includes issues involving model theory in one variable on nonsimply connected domains in the plane and in several variables on domains other than the bidisc. Operator Theoretic Function Theory, the particular type of mathematics that we are proposing to investigate, has direct and concrete benefits for a number of areas of human endeavor. For example, the model theory aspects of our proposal all involve the generalization of the Commutant Lifting Structure which leads to an efficient algorithm for the discovery of oil from acoustical data taken on the surface of the earth. Other aspects would add to the theory of Linear Matrix Inequalities. The mathematical theory of LMI?s, which currently is of great importance in several areas of engineering, is an extension of linear programming, which has made large scale resource allocation and economic prediction possible. Finally, the particular brand of function theory we propose to study, forms the mathematical core of H-infinity control theory, which has been used to design control systems for fusion reactions inside Tokamaks and feedback stabilization systems for the space shuttle.