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Cyclic behavior of holomorphic functions on a Runge region

Switlyk, Paul Matthew, Jr.
http://orcid.org/0000-0002-8784-4534
http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1633699612948776

Abstract Details

2021, Doctor of Philosophy (Ph.D.), Bowling Green State University, Mathematics.
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.
Kit Chan (Committee Chair)
Monica Longmore (Other)
Mihai Staic (Committee Member)
Xiangdong Xie (Committee Member)
66 p.
Mathematics
Runge region; hypercyclic operator; supercyclic operator; common cyclicity; generalized backward shift; backward multi-shift; weighted differentiation

Recommended Citations

Citations

  • Switlyk, Jr., P. M. (2021). Cyclic behavior of holomorphic functions on a Runge region [Doctoral dissertation, Bowling Green State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1633699612948776

    APA Style (7th edition)

  • Switlyk, Jr., Paul. Cyclic behavior of holomorphic functions on a Runge region. 2021. Bowling Green State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1633699612948776.

    MLA Style (8th edition)

  • Switlyk, Jr., Paul. "Cyclic behavior of holomorphic functions on a Runge region." Doctoral dissertation, Bowling Green State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1633699612948776

    Chicago Manual of Style (17th edition)

bgsu1633699612948776
179
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This open access ETD is published by Bowling Green State University and OhioLINK.
Release 3.2.12