Extending Hilbert Space Operators

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ABSTRACTAglerAfter the spectral theorem it is difficult to think of a theorem that has had a more profound effect on the development of operator theory and its myriads of applications to mathematics and science than the Sz.-Nagy Dilation Theorem. The idea of representing a general operator in a specialized class of operators as a part of a nice operator in the class has had many successes and we seek to develop this point of view with a primary focus on applications to problems in the theory of functions in one and several complex 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. Another focus of our project will be the use of operator-theoretic methods to study function theory on analytic varieties. 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. This will include the continuation of our work on the symmetrized bidisc with a particular focus on applications to complex geometry and the mu-synthesis problem.Operator Theory, the particular type of mathematics that we are proposing to develop, 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 enrich the theory of Linear Matrix Inequalities. LMI 's, which currently are all the rage in several areas of engineering, are an extension of linear programming, a mathematics which has made possible not only the optimization of large scale resource allocation but the accurate prediction of economic markets as well. Finally, the particular branch of function theory we propose to study, forms the mathematical core of the 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.