Modern Aspects of Multivariable Operator Theory and Matrix Analysis

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Operator theory and matrix analysis are both fundamental areas of mathematics. In large part, they were originally developed to provide a theoretical basis for quantum mechanics and other physical phenomena. An increasing number of application areas have emerged as a result of breakthroughs in operator theory and matrix analysis, which testify to their underlying importance to science and engineering. The areas of application closest to the research in this project are control systems engineering, electrical engineering, signal processing, image processing, and quantum computation. The principal investigator will build on his past work to further develop the interplay between the subdisciplines of free function theory, operator completions, statistical signal processing, matrix inequalities, and optimization. To accomplish this goal the principal investigator will continue existing collaborations as well as develop new ones, and engage actively with both graduate students and undergraduate students in the emerging research. The principal investigator will continue to maintain an intellectual environment fostering student involvement and development, providing the students with the skills, experience, and confidence to successfully pursue a career in the mathematical sciences. Thus, the project will yield both new, impactful mathematical results, as well as highly trained mathematicians prepared to join the scientific and educational workforce crucial to this nation. Many questions in system and control theory, filter design, signal and image processing come down to function theoretic questions. The case of several variables is a highly active research area where the techniques of multivariable operator theory are highly effective. The specific themes of the current project include (i) Matrix completions, (ii) Moment problems, (iii) Free function theory, (iv) Realizations, (v) Determinantal representations, (vi) Numerical range and radius, and their generalizations, (vii) Hypergeometric functions, and (viii) Inverse eigenvalue problems. This combination of areas will lead to new avenues of research that are of interest to different research groups. All projects also have a computational component, allowing for the implementation of the results and the potential to be used by researchers in all areas of science and engineering. The principal investigator will continue running an Analysis Seminar at Drexel University featuring local and international researchers, as well as Drexel students. The principal investigator and his students will disseminate the results via conference presentations and publications in a variety of leading mathematical journals and preprint servers.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.