My research interests center on the study of bounded operators on a Hilbert space, particularly those parts that connect with the theory of analytic functions. I like to look at problems in operator theory that are susceptible to an application of complex function theory, and I have specialized in those operators where this naturally occurs. A prime example of such operator is the class of subnormal operators. These are operators that are the restriction of a normal operator to an invariant subspace. The theory of normal operators, which is very well understood and essentially complete, is based on measure theory.
Subnormal operators are asymmetric.
One could say that normal operators are to subnormal operators as continuous functions are to analytic functions. Typical examples of subnormal operators arise from analytic functions. One such example is the unilateral shift. Another is the Bergman shift, defined as follows. I also have an interest in non-abelian approximation of operators on Hilbert space.