Mathematics Ph.D. Dissertations

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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