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

9500870考恩建议研究Hardy和Bergman Hilbert空间上的有界线性算子的结构和矩阵理论的相关问题，探讨具体提出的算子及其应用的问题，以便深入了解更一般的问题。特别是，他将继续研究复合算子、斜Toeplitz算子以及矩阵分析中的相关问题。该项目还将涉及几名本科生在矩阵分析方面的研究方向及其在一些工程问题上的应用，目的是吸引年轻人在数学方面进行研究生学习和研究。在这个项目中，研究者和他的研究生们将致力于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圆对称的解释，以及当符号是具有特殊结构的多个变量的函数时，如何将算子分解为更简单的分量。此外，研究人员和他的研究生将研究倾斜Toeplitz算子的谱理论，这是一类加权复合算子，应用于小波的光滑性准则。这些项目将使用希尔伯特空间上的算子理论和一元多变量解析函数理论中的工具，并将建立在研究Toeplitz算子和复合算子的研究经验的基础上。***

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. ***