Mathematics Ph.D. Dissertations

Universal Composition Operators on the Hardy Space with Linear Fractional Symbols

Date of Award

2023

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Kit Chan (Committee Chair)

Second Advisor

Apollos Nwauwa (Other)

Third Advisor

Juan Bes (Committee Member)

Fourth Advisor

So-Hsiang Chou (Committee Member)

Abstract

In this dissertation, we study universality of linear fractional composition operators on the Fréchet spaces H(��) of holomorphic functions on the open unit disk �� and on the Hardy space H2. By definition, a sequence of operators Tn is universal if there is a vector x such that the sequence {x, T1x, T2x, …} is dense in the space. Specifically, we obtain necessary conditions and sufficient conditions for a sequence of composition operators Cϕn : H2 → H2 to be universal on the Hardy space H2 with linear fractional symbols ϕn(z) = (anz+bn)/(cnz+dn) taking the open unit disk �� into itself, where andn-bncn = 1. We show that if Cϕn is universal and the sequence dn/cn is bounded, then there is a subsequence ϕnk for which the sequences of coefficients, |ank|, |bnk|, |cnk|, |dnk| go to ∞, as k → ∞. In addition, the sequence |ank| - |cnk| goes to 0. Furthermore, we also provide sufficient conditions for Cϕn to be universal, in terms of the coefficients of ϕn. Lastly, in the special case when ϕn are automorphisms of the disk ��, we show that Cϕn : H2 → H2 is universal on the Hardy space H2 if and only if the corresponding sequence Cϕn : H(??) → H(��) is universal on the Fréchet spaces H(��)

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