Extending Hilbert Space Operators

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After 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 many applications to mathematics and science than the Sz.-Nagy Dilation Theorem. The idea of representing a general operator in a specified 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 unsolved problems in matrix theory, operator theory, the theories of functions in one and several complex variables, and the development of holomorphic function theory on analytic varieties. One group of problems that we propose to investigate involve the generalizations to several complex variables of 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 group involves extending the Caratheodory-Julia Theory to several variables with a long term goal of building a geometric approach to the boundary regularity of holomorphic mappings. Research intrinsic to operator theory that we will undertake includes issues involving model theory in one variable on non-simply connected domains in the plane and in several variables on domains other than the bidisc as well as the generalization to several variables of the work of Loewner on operator monotone functions.Operator 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 and Quadratic Programming. Linear Matrix Inequalities and Quadratic Programming, are a far reaching extension of Linear Programming, which has made large scale resource allocation and economic prediction possible. In addition, to being a powerful tool in engineering, they have been used to develop state of the art algorithms for global positioning with incomplete sensor location data . Finally, the particular brand 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.