Mathematics Ph.D. Dissertations

Title

Cyclic behavior of holomorphic functions on a Runge region

Date of Award

2021

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Kit Chan (Committee Chair)

Second Advisor

Monica Longmore (Other)

Third Advisor

Mihai Staic (Committee Member)

Fourth Advisor

Xiangdong Xie (Committee Member)

Abstract

Let ℂ ⊆ ℂN be a Runge region and let H(ℂ) denote the Fréhet space of holomorphic functions on ℂ. In this dissertation, we explore the cyclic behavior of various operators defined on H(ℂ). First, we provide extensions of some earlier results regarding nonscalar continuous linear operators on H(ℂ) commuting with each partial differentiation operator ∂/∂zk, where 1 ≤ k ≤ N. Specifically, we demonstrate that all such operators are hypercyclic and share a dense set of common cyclic vectors. Motivated by our results, we introduce a class of finite sets of Fréchet space operators patterned after the partial differentiation operators, called backward multi-shifts, and show that any nonscalar operator in the commutant of such a finite set is supercyclic. Lastly, we demonstrate the existence of a dense set of common cyclic vectors for all nonscalar operators in the commutant of a weighted differentiation operator Bλ : H(ℂ) → H(ℂ) defined by Bλ(z) = d/dz[f(λz)], where λ ≠ 0. To do so, we need to make use of the method of spectral synthesis, as well as more classical techniques for different values of λ.

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