Mathematical Sciences: Composition Operators, Slant Toeplitz Operators, and Matrix Analysis

9500870 Cowen研究者提出研究Hardy和Bergman Hilbert空间上有界线性算子的结构和矩阵理论的相关问题，探讨具体呈现算子的问题及其应用，以期深入了解更普遍的问题。特别是，他将继续研究组合算子、倾斜Toeplitz算子和矩阵分析中的相关问题。该项目还将涉及几名本科生的研究方向，即矩阵分析及其在一些工程问题上的应用，目的是吸引年轻人研究生学习和研究数学。在这个项目中，研究者和他的研究生将研究Bergman和Hardy空间上的复合算子的几个基本问题，包括研究大多数复合算子中意外的圆对称的解释，以及当符号是具有特殊结构的几个变量的函数时，寻求将算子分解成更简单的组成部分的方法。此外，研究者和他的研究生将研究斜Toeplitz算子的谱理论，这是一类加权组合算子，应用于小波的平滑准则。这些项目将使用希尔伯特空间算子理论和单变量和多变量解析函数理论中的工具，并将建立在研究者在研究Toeplitz算子和组合算子方面的经验基础上。＊＊＊

9500870 Cowen The investigator proposes to study the structure of bounded linear operators on the Hardy and Bergman Hilbert spaces and related problems of matrix theory, approaching problems on concretely presented operators and their applications in order to gain insight into more general problems. In particular, he will continue his investigations of composition operators, slant Toeplitz operators, and related problems from matrix analysis. The project will also involve the direction of research of several undergraduate students in matrix analysis and its applications to some engineering problems with the goal of attracting young people to graduate study and research in mathematics. In this project, the investigator and his graduate students will work on several fundamental questions about composition operators on Bergman and Hardy spaces including investigation into the explanations for unexpected circular symmetry in most composition operators and seeking ways to break operators into simpler components when the symbol is a function of several variables with special structure. Further, the investigator and his graduate students will investigate the spectral theory of slant Toeplitz operators, a class of weighted composition operators with applications to smoothness criteria for wavelets. These projects will use tools from the theory of operators on Hilbert space and the theory of analytic functions in one and several variables and will build on the investigator's experience in the study of Toeplitz operators and composition operators. ***