The study of composition and Toeplitz operators on spaces of analytic functions is an active research area that lies in the intersection of the fields of operator theory, complex analysis, operator algebras, and functional analysis. Both composition and Toeplitz operators are induced by underlying functions, and the properties of the operators are closely tied to the properties of the associated maps. One approach for studying these operators is to investigate the unital C*-algebras generated by collections of the operators. This strategy has been used to analyze Toeplitz operators for over forty years, but its use in the analysis of composition operators only began in the last decade. For this reason, many open questions remain about the structures of unital C*-algebras generated by composition operators.
My research seeks to answer some of these questions in the setting of composition operators acting on the Hardy space of the disk. In my dissertation work, I am investigating the unital C*-algebras generated by collections of composition operators induced by linear-fractional self-maps of the open unit disk ID that fix a given point on the unit circle. I am also studying the C*-algebras generated by these collections together with either the compact operators or the Toeplitz operators induced by continuous functions. The two main goals of my work are: