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.