Problems in Matrix and Operator Theory

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0600859 LiAbstractMatrix and operator theory have connections and applications to many pure and applied subjects. In this project, the investigator will work with collaborators and students to study problems in matrix and operator theory arising in different branches of sciences. The emphasis will be on establishing connections and stimulating interactions among researchers in different areas. In the study, the theory of generalized numerical ranges and numerical radii will be developed to solve problems in engineering, quantum computing, and numerical analysis; techniques in operator theory, character theory, group theory, and combinatorial theory will be used to develop new results in the study of induced operators on symmetry classes of tensors associated with non-linear characters; matrix theory and algebraic combinatorics methods will be used to solve matrix inequality problems arising in perturbation analysis and approximation problems; algebra, analytic and geometric techniques will be established to study preservers problems, which concern the characterizations of transformations on matrices or operators with some special properties. The research will lead to the advance of difference branches of sciences. The study will yield solutions of open problems and new results. New techniques and insights will be developed for future research. Moreover, the study will establish connections among different operator theory topics and to other research areas so that operator theory methods can be used to solve problems in other topics, and vice versa. Since the proposed research will involve researchers in different disciplines, the results and techniques will be disseminated to different circles so that they can be readily applied. The involvement of student researchers from other areas in the project will enable more people to use operator theory results and methods to other subjects. This will facilitate collaborations and interdisciplinary research, which are important ingredients for the advance of science and technology.