Thesis Defense: Complete Pick Spaces and Operator Inequalities

Abstract:

We will survey three different research topics, all lying at the interface of complex analysis and operator theory.

Our first topic concerns operator inequalities over the annulus. In particular, we will consider three different classes of operators associated with the annulus and offer estimates for the norm of functions of such operators.

For our second topic, we will focus on the dynamics of holomorphic self-maps $F$ of the bidisk that do not have any interior fixed points.  It turns out that the limiting behavior of the iterates of $F$ can be heavily influenced by the existence of certain boundary fixed points, which we term Denjoy-Wolff points following the classical setting. 

We will conclude with two research projects that revolve around complete Pick spaces. These are reproducing kernel Hilbert spaces that host an analogue of the Pick interpolation theorem for multipliers. First, we will study a generalized inner-outer factorization in the setting of a particular complete Pick space over the annulus. Then, we will discuss a characterization of interpolating sequences for multipliers between certain pairs of function spaces that enjoy an analogue of the complete Pick property.

Host: John McCarthy, Spencer T. Olin Professor in Arts & Sciences