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

小行星9500870 研究者提出研究哈代和伯格曼希尔伯特空间上有界线性算子的结构以及矩阵论的相关问题， 接近具体提出的运营商及其应用的问题，以获得洞察到更一般的问题。 特别是，他会 继续他对合成算子的研究， 斜Toeplitz算子，以及矩阵分析中的相关问题。 该项目还将涉及几个研究方向， 本科学生在矩阵分析及其应用的一些 工程问题，目的是吸引年轻人 研究生学习和数学研究。 在这个项目中，研究人员和他的研究生将致力于 Bergman和哈代上复合算子的几个基本问题 空间，包括调查意外循环的解释 大多数复合算子中的对称性，并寻求将算子分解为 当符号是多个变量的函数时， 特殊结构。 此外，研究人员和他的研究生将 研究斜Toeplitz算子的谱理论， 加权复合算子及其在光滑性准则中的应用 小波 这些项目将使用的工具从算子理论上 希尔伯特空间与一元和多元解析函数理论 变量，并将建立在研究者的经验， Toeplitz算子和复合算子。 ***

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. ***