Problems in Matrix and Operator Theory

矩阵和算子理论与许多纯理论和应用学科都有联系和应用。在这个项目中，研究者将与合作者和学生一起研究矩阵和算子理论在不同科学分支中出现的问题。重点将是在不同领域的研究人员之间建立联系和促进互动。在这项研究中，广义数值范围和数值半径的理论将被发展，以解决工程，量子计算和数值分析中的问题;算子理论，特征理论，群论和组合理论中的技术将被用来发展新的结果，在研究与非线性特征相关的张量对称类上的诱导算子;矩阵理论和代数组合学方法将用于解决扰动分析和近似问题中出现的矩阵不等式问题;代数，分析和几何技术将被建立来研究几何问题，它涉及矩阵变换或具有某些特殊性质的算子的特征。 这项研究将导致不同学科的进步。该研究将产生开放问题的解决方案和新的结果。新的技术和见解将为未来的研究开发。此外，这项研究将建立不同的运营商理论的主题和其他研究领域之间的联系，使运营商理论的方法可以用来解决其他主题的问题，反之亦然。由于拟议的研究将涉及不同学科的研究人员，研究结果和技术将传播给不同的圈子，以便随时加以应用。来自其他领域的学生研究人员的参与将使更多的人将算子理论的结果和方法应用到其他学科中。这将促进合作和跨学科研究，这是推动科学和技术进步的重要因素。

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.