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

研究启动奖为传统黑人学院和大学的初级和中期职业教师提供支持，他们正在建立新的研究项目或重新指导和重建现有的研究项目。期望该奖项有助于提高教师的研究能力和效率，改善所在机构的研究和教学，并使本科生参与研究经验。该奖项授予中央州立大学，研究了复函数理论中的一系列问题，旨在理解复合算子的光谱结构。该项目是与托莱多大学合作开展的，以一种有意义的方式让本科生参与研究。本课题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和数值范围。这些算子的谱和数值范围取决于底层符号不动点的位置和性质以及定义算子的空间。最初，重点将放在固定复单位圆盘上的一个点的非自同构符号上。本课题的主要目标是：刻画在全纯函数的Banach和Hilbert空间上定义的这些算子的谱；研究作用于Hardy空间上的复合算子类谱的一般结构；描述作用于Hardy空间的复合算子的数值范围和数值半径。本研究将使用包括基本谱、基本角导数、Dejoy-Wolff点、Navalinna计数函数、Carleson测度、谐波测度和复插值在内的工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.