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

Central State University, Wilberforce OH

Investigators

Abstract

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.