ORCID

0000-0003-1921-4394

Date of Award

5-9-2024

Author's School

Graduate School of Arts and Sciences

Author's Department

Mathematics

Degree Name

Doctor of Philosophy (PhD)

Degree Type

Dissertation

Abstract

This thesis is concerned with the treatment of three different research topics, all lying at the interface of complex analysis and operator theory. The first chapter is based on two distinct research projects, both revolving around complete Pick spaces. These are reproducing kernel Hilbert spaces that host an analogue of the Pick interpolation theorem for multipliers. First, we study a generalized inner-outer factorization in the setting of a particular complete Pick space over the annulus. The second project deals with the characterization of interpolating sequences for multipliers between certain pairs of function spaces that enjoy an analogue of the complete Pick property. In particular, we show that a sequence is interpolating for a pair of such spaces if and only if it generates a Carleson measure with respect to the first space and is n-weakly separated by the kernel of the second space, for any n greater than 1. We also construct counterexamples to show that n-weak separation cannot, in general, be replaced by weak separation, thus answering a question of Aleman, Hartz, McCarthy and Richter. The second chapter deals with operator inequalities over the annulus. In particular, we consider three different classes of operators associated with the annulus and offer estimates for the norm of functions of such operators. Each class requires separate treatment: for the first one, we construct a certain extremal weighted shift operator, while for the second one, we convert the operator norm into the multiplier norm of a certain Hilbert function space. Further, to handle the third class, we employ a technique due to Crouzeix and Greenbaum that involves the double-layer potential integral operator. Finally, we note that this chapter also contains material that has not been submitted for publication; in Section 2.4, we construct a counterexample to a question of Bello and Yakubovich concerning the class of operators that have the annulus as a spectral set. Finally, the third chapter focuses on the behavior of the iterates of holomorphic self-maps F of the bidisk that do not have any interior fixed points. It is well-known that, unlike the single-variable case, the sequence of iterates will, in general, diverge. However, it turns out that the limiting behavior of the sequence is heavily influenced by the differentiability properties of F at certain boundary fixed points, which we term Denjoy-Wolff points following the classical setting. In fact, we show that if F possesses Denjoy-Wolff points with particular properties, then its iterates will have to converge. To obtain these results, we employ a certain operator-theoretic representation of holomorphic functions on the bidisk due to Agler.

Language

English (en)

Chair and Committee

John McCarthy

Download

Share

COinS