Research

My interest primarily lies at the intersection of complex analysis and operator theory. Specifically, I am concerned with operators on Hilbert spaces in which the vectors are complex-valued analytic functions on some domain \Omega   (often \Omega   is the unit disk \mathbb{D}  in the complex plane, or the unit ball in \mathbb{C}^N   ). We will call such a space a Hilbert function space \mathcal{H}   if evaluation of the functions in the space at a point of \Omega   are all continuous linear functionals. In particular, I work with composition operators C_\phi  that are defined by C_\phi f = f \circ \phi   where f   is in the Hilbert space and \phi   is an analytic map of \Omega   into itself. The most celebrated example is the Hardy space H^2(\mathbb{D})   . We are often concerned with obtaining properties of C_{\phi}   acting on \mathcal{H}   from properties of \phi   . Generally, one may explore spectral properties, boundedness, and so forth through studying dynamical and function theoretic properties of \phi   .

My current research involves exploring solutions to the functional eigenvalue equation C_{\phi}f=\lambda f    where our vectors are analytic functions of several complex variables and \lambda    is a complex number. Solutions to this equation where the vectors are functions of one complex variable, trace their roots back to Koenigs’ solution to Schroeder’s equation in 1884. The solution depends on properties of \phi    such as its fixed point behavior. I am currently thinking about composition operators induced by analytic maps with attractive boundary fixed point in the unit ball \mathbb{C}^N  acting on a Hilbert function space \mathcal{H}   such as the Drury-Arveson space and the corresponding solutions to the above eigenvalue equation.