Professor Cowen will study the structure of certain bounded linear operators on the Hardy space of the unit disk, looking at concretely presented operators in order to gain insight into more general problems. He will continue his investigations of composition operators and Toeplitz operators, the main issues being to identify commutants and decide sub- and hyponormality. The research envisioned here is about analytic functions, a central preoccupation of mathematics for well over a century. They can be defined variously as solutions of a certain simple system of partial differential equations, as maps which take planar regions to other planar regions in a way that preserves angles except at isolated singularities, or as limits of polynomials in a suitably precise sense. Corresponding to the diverse ways of describing them, analytic functions can be studied in a variety of ways. The approach favored in this project is operator-theoretic, building up Hilbert spaces of functions by taking polynomials and their limits in a particular fashion, then studying operators on the spaces that arise naturally from the functions, e.g. by multiplication or by composition. Conversely, these function-theoretic operators often turn out to be quite general in operator theory, so by studying the functions one can learn a great deal about operators in the abstract.
