This research concerns several problems which lie at the interface of operator theory and function theory. The first part of the project involves polynomial approximation directed at understanding certain measures on the closed unit disc. The other part of the project involves a study of the composition operators on various spaces of analytic functions. This includes work on identifying the spectrum of a composition operator, continuing work on the study of composition operators on large weighted Bergman spaces, and the question of identifying the components in the space of composition operators in the uniform operator topology. This research is in the general area of modern analysis and, as noted above, lies at the interface of operator theory and function theory. The research involves the study of composition operators on spaces of analytic functions in one or several complex variables and is a central topic in the modern theory of complex analysis.
