Extending Hilbert Space Operators

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Agler. Abs Proposal: DMS-9801461 Principal Investigator: Jim Agler Abstract: 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 myriad 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, and the theories of functions in one and several complex variables. A particular group of problems that we propose to attack involve 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 group involves deriving analogs of the theorem of Adamyan, Arov, and Krein on spaces more general than the classical Hardy space. Research intrinsic to operator theory that we will undertake includes issues involving model theory in one variable on nonsimply con- nected domains in the plane and in several variables on domains other than the bidisc. 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. LMIs, which currently are all the rage in several areas of engineering, are an extension of linear programming, which has made large scale resource allocation and economic prediction possible. Finally, the particular brand of functi on theory we propose to study forms the mathematical core of the recently developed 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.