Mathematics Ph.D. Dissertations

Title

Universality of Composition Operator with Conformal Map on the Upper Half Plane

Date of Award

2021

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics/Mathematics (Pure)

First Advisor

Kit Chan (Advisor)

Second Advisor

Nicole Kalaf-Hughes (Other)

Third Advisor

Xiangdong Xie (Committee Member)

Fourth Advisor

Mihai Staic (Committee Member)

Abstract

The main theme of this dissertation is the dynamical behavior of composition operators on the F´rechet space H(P) of holomorphic functions on the upper half-plane P. In this dissertation, we prove a new version of the Seidel and Walsh Theorem [21] for the F´rechet space H(P). Indeed, we obtain a necessary and sufficient condition for the sequence of linear fractional transformations σn such that the sequence of composition operators {Cσn } for the F´rechet space H(P) is universal. For that, we use the Riemann Mapping Theorem to transfer dynamical results on the space H(D) of holomorphic functions on D to the space of holomorphic functions H(P). Furthermore, we generalize our first result by proving equivalent conditions for a sequence of composition operators in the space H(D) to be universal. Consequently, taking the point of view that hypercyclicity is a special case of universality, we obtain a new criterion for a linear fractional transformation σ so that Cσ is hypercyclic on H(P). Indeed, we provide necessary and sufficient conditions in terms of the coefficients of a linear fractional transformation σ so that Cσ is hypercyclic on H(P). Moreover, we use this result to derive a necessary and sufficient condition so that Cϕ is hypercyclic on H(D) where ϕ is a linear fractional transformation defined on D. Motivated by the Denjoy-Wolff Theorem [23, p. 78], we further work on the conformal map σ of the upper half-plane P is to make a connection between the hypercyclicity and the limit of the iterations of σ. In particular, we give a complete characterization for the limit point of the iterations of σ in the extended boundary ∂∞P = ∂P ∪ {∞}. Similarly, we provide an analogous result for the unit disk D. Finally, we obtain a new universal criterion in the space H(Ω) of holomorphic functions on a bounded simply connected region Ω that is not the whole complex plane C. We show that a sequence of composition operators {Cσn } on H(Ω) is universal if and only if there are a boundary limit point w ∈ ∂Ω and a subsequence {σnk }k of {σn}n such that σnk → w uniformly on compact subsets of Ω. Our last result extends a result of Grosse-Erdmann, and Manguillot in a particular case when the complement C \ Ω of Ω has a nonempty interior.

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