Abstract

For $1 \leq p < \infty$, consider the Hardy space $H^p(\mathbb{D}^n)$ on the unit polydisk. Beurling's theorem characterizes all shift cyclic functions in the Hardy spaces when $n = 1$. Such a theorem is not known to exist in most other analytic function spaces, even in the one variable case. Therefore, it becomes natural to ask what properties these functions satisfy in order to understand them better. The goal of this thesis is to showcase some important properties of cyclic functions in two different settings. 1. Fix $1 \leq p,q < \infty$ and $m, n \in \mathbb{N}$. Let $T : H^p(\mathbb{D}^n) \xrightarrow{} H^q(\mathbb{D}^m)$ be a bounded linear operator. Then $T$ preserves cyclic functions, i.e. $Tf$ is cyclic whenever $f$ is, if and only if $T$ is a weighted composition operator. 2. Let $\mathcal{H}$ be a normalized complete Nevanlinna-Pick space, and let $f, g \in \mathcal{H}$ be such that $fg \in \mathcal{H}$. Then $f$ and $g$ are multiplier cyclic if and only if their product $fg$ is. We also extend $(1)$ to a large class of analytic function spaces that includes the Dirichlet space, and the Drury-Arveson space on the unit ball $\mathbb{B}_n$ among others. Both of these properties generalize all previously known results of this type.

Committee Chair

Gregory Knese

Committee Members

John McCarthy

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

Spring 5-15-2022

Language

English (en)

Author's ORCID

http://orcid.org/0000-0001-7299-1307

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Mathematics Commons

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