Mathematical Sciences: Problems in Linear Analysis

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PROPOSAL NO.: DMS - 9500924 PROPOSAL TITLE: Problems in Linear Analysis PI/PD: Leiba Rodman CO-PI/PD: Hugo J. Woerdeman ABSTRACT The principal investigators plan to study a variety of interconnected problems in a highly active area of research in Mathematical Sciences that involves operator theory, matrix analysis, and their applications in science and technology, using also closely related areas of pure and applied mathematics: function theory, Fourier analysis. In addition to continuing work on other topics, specific problems are outlined in the following subjects which are the main focus of the research: 1. Interpolation problems for matrix and operator valued functions; 2. Completions of finite and infinite matrices and operators; 3. Riccati equations; 4. Krein space operators; 5.Applications. The principal investigators strive to develop new results that can be proved in an abstract mathematical context (e. g., in terms of *-algebras), and that at the same time yield new and useful information in various more specific settings. The principal investigators plan to relate the newly obtained abstract results to the original areas of application. Applications that are emphasized in the proposal include Advanced Manufacturing (engineering control systems) and High Performance Computing and Communications (tomography), which have been identified as two of the strategic areas with impact on national need. Other areas of relevant application are: engineering control systems, optics of polarized light, kinetic energy minimization in quantum chemistry, diffraction of waves, signal processing, and pattern recognition. Interactions with many researchers in different fields (mathematics, computer science, engineering), various countries besides the U. S. (Canada, The Netherlands, Italy, Israel, Norway, France, United Kingdom), and scientists in industry and government (the National Institute of Health, NEC Research Institute) are planned and form an integral part of this project.