Research Initiation Award: Spectra of composition operators on analytic function spaces

研究启动奖为历史上黑人学院和大学的初级和中级职业教师提供支持，他们正在建立新的研究项目或重新定向和重建现有的研究项目。预计该奖项将有助于进一步提高教师的研究能力和效率，改善家庭机构的研究和教学，并使本科生参与研究经验。中央州立大学的奖项调查了复杂函数理论中的一系列问题，旨在了解复合算子的谱结构。该项目是与托莱多大学合作开展的，并以有意义的方式让本科生参与研究。本计画研究复单位圆盘上定义的全纯函数空间上的复合算子的谱与数值域。这些算子的谱和数值范围取决于定义算子的基本符号和空间的不动点的位置和性质。最初，重点将放在非自守符号上，这些符号固定了复杂单位盘中的一个点。该项目的主要目标是：表征这些运营商的频谱定义在某些Banach和希尔伯特空间的全纯函数;调查的一般结构的频谱类的组成运营商的作用于哈代空间;和描述的数值范围和数值半径的组成运营商的作用于哈代空间。该研究将采用的技术，使用的工具，包括基本光谱，基本角导数，德约-沃尔夫点，Navalinna计数函数，Carleson措施，谐波措施，和复杂的插值。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Research Initiation Awards provide support for junior and mid-career faculty at Historically Black Colleges and Universities who are building new research programs or redirecting and rebuilding existing research programs. It is expected that the award helps to further the faculty member's research capability and effectiveness, improves research and teaching at the home institution, and involves undergraduate students in research experiences. The award to Central State University investigates a cluster of problems in complex function theory aimed at understanding the structure of spectra for composition operators. The project is carried out in collaboration with the University of Toledo and involves undergraduate students in research in a meaningful way. This project examines the spectra and numerical ranges of composition operators acting on the spaces of holomorphic functions defined on the complex unit disc. Spectra and numerical ranges of these operators depend on locations and nature of the fixed points of the underlying symbol and space on which the operators are defined. Initially, the focus will be on non-automorphic symbols that fix a point in the complex unit disc. The main goals of the project are to: characterize the spectra of these operators defined on certain Banach and Hilbert spaces of holomorphic functions; investigate the general structure of the spectra of classes of composition operators acting on the Hardy space; and describe the numerical ranges and numerical radius of the composition operators acting on the Hardy space. The study will employ techniques that use the tools including essential spectrum, essential angular derivatives, Dejoy-Wolff points, Navalinna Counting functions, Carleson measure, Harmonic measure, and complex interpolation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.