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

Almohammedali, Fadelah Abdulmohsen
http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1628507809885957

Abstract Details

2021, Doctor of Philosophy (Ph.D.), Bowling Green State University, Mathematics/Mathematics (Pure).
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.
Kit Chan, Ph.D. (Advisor)
Nicole Kalaf-Hughes, Ph.D (Other)
Xiangdong Xie, Ph.D (Committee Member)
Mihai Staic, Ph.D (Committee Member)
94 p.
Mathematics
composition operators; conformal maps; hypercyclic operator; universal operator; space of holomorphic functions; F\'rchet space

Recommended Citations

Citations

  • Almohammedali, F. A. (2021). Universality of Composition Operator with Conformal Map on the Upper Half Plane [Doctoral dissertation, Bowling Green State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1628507809885957

    APA Style (7th edition)

  • Almohammedali, Fadelah. Universality of Composition Operator with Conformal Map on the Upper Half Plane. 2021. Bowling Green State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1628507809885957.

    MLA Style (8th edition)

  • Almohammedali, Fadelah. " Universality of Composition Operator with Conformal Map on the Upper Half Plane." Doctoral dissertation, Bowling Green State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1628507809885957

    Chicago Manual of Style (17th edition)

bgsu1628507809885957
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