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 (often
is the unit disk
in the complex plane, or the unit ball in
). We will call such a space a Hilbert function space
if evaluation of the functions in the space at a point of
are all continuous linear functionals. In particular, I work with composition operators
that are defined by
where
is in the Hilbert space and
is an analytic map of
into itself. The most celebrated example is the Hardy space
. We are often concerned with obtaining properties of
acting on
from properties of
. Generally, one may explore spectral properties, boundedness, and so forth through studying dynamical and function theoretic properties of
.
My current research involves exploring solutions to the functional eigenvalue equation where our vectors are analytic functions of several complex variables and
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
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
acting on a Hilbert function space
such as the Drury-Arveson space and the corresponding solutions to the above eigenvalue equation.