#### Date of Graduation

8-2016

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics (PhD)

#### Degree Level

Graduate

#### Department

Mathematical Sciences

#### Advisor

John R. Akeroyd

#### Committee Member

Daniel H. Luecking

#### Second Committee Member

Maria Tjani

#### Abstract

Let $\phi$ be an analytic self-map of the unit disk $\mathbb{D}:=\{z:\lvert z\rvert<1\}$. The composition operator $C_\phi$ defined by $C_\phi(f)=f\circ\phi$ is a bounded linear operator on the Hardy space $H^2(\mathbb{D})$. It is well-known that if $C_\phi$ is compact on $H^2(\mathbb{D})$ then $\lVert\phi^{n}\rVert_{H^{2}(\mathbb{D})}\to 0$ as $n\to\infty$. But the converse doesn't necessarily hold. We discuss the decay rate of $\lVert\phi^{n}\rVert_{H^{2}(\mathbb{D})}$ in the case when $\phi$ maps the unit disk to a domain whose boundary touches the unit circle exactly at one point. We also investigate inheritance of closed-rangeness property of $C_\phi$ from a Banach space of analytic functions on $\mathbb{D}$ to a weighted subspace.

#### Recommended Citation

Dutta, Arnab, "On Compactness and Closed-Rangeness of Composition Operators" (2016). *Theses and Dissertations*. 1683.

http://scholarworks.uark.edu/etd/1683