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

Keywords

Pure sciences; Closed-range; Compact; Composition operators; Weighted banach spaces

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.

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