Cowen will study the structure of bounded linear operators on the Hardy Hilbert space, the Bergman Hilbert space, and related spaces of analytic functions on the unit disk, approaching problems on concretely presented operators in order to gain insight into more general problems. He will work on unifying results concerning composition operators on Bergman and Hardy spaces into a theory of composition operators on spaces of analytic functions emphasizing similarities between spaces and minimizing reliance on special features of particular spaces. Operator theory is that part of mathematics that studies the infinite dimensional generalizations of matrices. In particular, when restricted to finite dimensional subspaces, an operator has the usual linear properties, and thus can be represented by a matrix. The central problem in operator theory is to classify operators satisfying additional conditions given in terms of associated operators (e.g. the adjoint) or in terms of the underlying space. Operator theory underlies much of mathematics, and many of the applications of mathematics to other sciences.
