Doctor of Philosophy (Ph.D.), Bowling Green State University, 2023, Mathematics

An interesting topic of study for a hypercyclic operator T on a topological vector space X has been whether X has an infinite-dimensional, closed subspaces consisting entirely, except for the zero vector, of hypercyclic vectors. These subspaces are called hypercyclic subspaces. The existence of a strictly weakly hypercyclic operator T, which is a weakly hypercyclic operator that is not norm hypercyclic on a Hilbert space H has been shown by Chan and Sanders. However, it is not known whether there exists a strictly weakly hypercyclic subspace of H. We first show that the left multiplication operator LT with the aforementioned strictly weakly hypercyclic operator T is a strictly WOT-hypercyclic operator on the operator algebra B(H). Then we obtain a sufficient condition for an operator T on a Hilbert space H to have a strictly weakly hypercyclic subspace. After that we construct an operator that satisfies these conditions and therefore prove the existence of a strictly weakly hypercyclic subspace.

Committee: Kit Chan Ph.D. (Committee Chair); Christopher Kluse Ph.D. (Other); Mihai Staic Ph.D. (Committee Member); Juan Bes Ph.D. (Committee Member) Subjects: Mathematics

Doctor of Philosophy (Ph.D.), Bowling Green State University, 2013, Mathematics

If X is a topological vector space and T : X → X is a continuous linear operator, then T is said to be hypercyclic when there is a vector x in X such that the set {Tnx : n = 0, 1, 2, … } is dense in X. If a hypercyclic operator has a dense set of periodic points it is said to be chaotic. This paper is divided into five chapters. In the first chapter we introduce the hypercyclicity phenomenon. In the second chapter we study the range of a hypercyclic operator and we find hypercyclic vectors outside the range. We also study arithmetic means of hypercyclic operators and their convergence. The main result of this chapter is that for a chaotic operator it is possible to approximate its periodic points by a sequence of arithmetic means of the first iterates of the orbit of a hypercyclic vector. More precisely, if z is a periodic point of multiplicity p, that is Tpz = z then there exists a hypercyclic vector of T such that An,px =(1/n)(z + Tpz +
...
+Tp(n-1)z) converges to the periodic point z. In the third chapter we show that for any given operator T : M → M on a closed subspace M of a Hilbert space H with finnite codimension it has an extension A : H → H that is chaotic. We conclude the chapter by observing that the traditional Rota model for operator theory can be put in the hypercyclicity setting. In the fourth chapter, we show that if T is an operator on a closed subspace M of a Hilbert space H, and P : H → M is the orthogonal projection onto M, then there is an operator A : H → H such that PAP = T, PA*P = T* and both A, A* are hypercyclic.

Committee: Kit Chan (Advisor); Ron Lancaster (Committee Member); Juan Bes (Committee Member); Craig Zirbel (Committee Member) Subjects: Mathematics

Doctor of Philosophy (Ph.D.), Bowling Green State University, 2013, Mathematics

A continuous linear operator T : X → X on an infinite dimensional separable topological vector space X is said to be hypercyclic if there is a vector x in X whose orbit under T, orb(T, x) = {Tnx : n ≥ 0 } = { x, Tx, T2x, ..... } is dense in X. Such a vector x is said to be a hypercyclic vector for T. While the orbit of a hypercyclic vector goes everywhere in the space, there may be other vectors whose orbits are indeed finite and not contain a zero vector. Such a vector is called a periodic point. More precisely, we say a vector x in X is a periodic point for T if Tn x = x for some positive integer n depending on x. The operator T is said to be chaotic if T is hypercyclic and has a dense set of periodic points. Let M be a closed subspace of a separable, infinite dimensional Hilbert space H with dim(H/M) = ∞ . We say that T : H → H is a chaotic extension of A : M → M if T is chaotic and T |M = A. In this dissertation, we provide a criterion for the existence of an invertible chaotic extension. Indeed, we show that a bounded linear operator A : M → M has an invertible chaotic extension T : H → H if and only if A is bounded below. Motivated by our result, we further show that A : M → M has a chaotic Fredholm extension T : H → H if and only if A is left semi-Fredholm. Our further investigation of hypercyclic extension results is on the existence of dual hypercyclic extension. The operator T : H → H is said to be a dual hypercyclic extension of A : M → M if T extends A, and both T : H → H and T* : H → H are hypercyclic. We actually give a complete characterization of the operator having dual hypercyclic extension on a separable, infinite dimensional Hilbert spaces. We show that a bounded linear operator A : M → M has a dual hypercyclic extension T : H → H if and only if its adjoint A* : M → M is hypercyclic.

Committee: Kit Chan (Advisor); Juan Bes (Committee Member); So-Hsiang Chou (Committee Member); Rachel Reinhart (Committee Member) Subjects: Mathematics

Doctor of Philosophy (Ph.D.), Bowling Green State University, 2021, Mathematics

Let ℂ ⊆ ℂN be a Runge region and let H(ℂ) denote the Frehet 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 Frechet 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.

Committee: Kit Chan (Committee Chair); Monica Longmore (Other); Mihai Staic (Committee Member); Xiangdong Xie (Committee Member) Subjects: Mathematics

Doctor of Philosophy (Ph.D.), Bowling Green State University, 2021, 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 an (open full item for complete abstract)

Committee: Kit Chan Ph.D. (Advisor); Nicole Kalaf-Hughes Ph.D (Other); Xiangdong Xie Ph.D (Committee Member); Mihai Staic Ph.D (Committee Member) Subjects: Mathematics

Doctor of Philosophy (Ph.D.), Bowling Green State University, 2020, Mathematics/Mathematics (Pure)

We prove that every pure quasinormal operator T : H→H on a separable, infinite-dimensional, complex Hilbert space H has a supercyclic adjoint (see Theorem 3.3.2 and Corollary 3.3.12). It follows that if an operator has a pure quasinormal extension then the operator has a supercyclic adjoint. Our result improves a result of Wogen [52] who proved in 1978 that every pure quasinormal operator has a cyclic adjoint. Feldman [26] proved in 1998 that every pure subnormal operator has a cyclic adjoint. Continuing with our result, it implies in particular that every pure subnormal operator having a pure quasinormal extension has a supercyclic adjoint (see Corollary 3.3.15). Hence improving Feldman's result in this special case. Indeed, we show that the adjoint T* of every pure quasinormal operator T is unitarily equivalent to an operator of the form Q : ⊕0∞L2(μ)→⊕0∞L2(μ) defined by Q(f0, f1, f2, . . .) = (A1f1 , A2f2 , A3f3 , . . .) for all vectors (f0, f1, f2, . . .)∈⊕0∞L2(μ), where each An : L2(μ)→L2(μ) is a left multiplication operator Mφn with symbol φn∈ L ∞(μ) satisfying φn=0 a.e. We constructively obtain a supercyclic vector for the operator Q and this then yields our result by the fact that unitary equivalence preserves supercyclicity. Furthermore, we prove that the adjoint T* of a pure quasinormal operator T : H→H is hypercyclic precisely when T is bounded below by a scalar α> 1 (see Theorem 2.6.4 and Corollary 2.6.8).

Committee: Kit Chan Ph.D. (Advisor); Jong Lee Ph.D. (Other); Jonathan Bostic Ph.D. (Committee Member); So-Hsiang Chou Ph.D. (Committee Member); Mihai Staic Ph.D. (Committee Member) Subjects: Mathematics

Doctor of Philosophy (Ph.D.), Bowling Green State University, 2024, Mathematics

We examine the dynamics of weighted composition operators on two function spaces. The first one is the space C∞(Ω, K) of K-valued smooth functions, where Ω ⊂ Rd is open and K is the real or complex scalar field. The second one is the space H(Ω) of holomorphic functions where Ω is a domain in the complex plane. The hypercyclicity of weighted composition operators on these two spaces has been completely characterized. The properties of weak mixing and mixing have also been characterized for this class of operators when acting on C∞(Ω, K). We study supercyclicity of these operators on these two spaces and compare it to other dynamical properties. We show that when acting on the space C∞(Ω, K), supercyclicity and weak mixing are equivalent properties, and when d = 1, such properties are also equivalent to mixing. When acting on the space H(Ω), we show that hypercyclicity coincides with mixing. Furthermore, when Ω is not conformally equivalent to the punctured unit disc, we show supercyclicity is equivalent to mixing. When Ω is conformally equivalent to an annulus or is n-connected for some n ≥ 3, then H(Ω) supports no supercyclic weighted composition operators, and when Ω is 2-connected, the properties of hypercyclicity, mixing and Devaney chaos are all equivalent.

Committee: Juan Bes PhD (Committee Chair); Robyn Miller PhD (Other); Benjamin Ward PhD (Committee Member); Kit Chan PhD (Committee Member) Subjects: Mathematics

Doctor of Philosophy (Ph.D.), Bowling Green State University, 2011, Mathematics and Statistics

In this dissertation we investigated the two topics hypercyclicity and the Aluthge transform. Each of these is related to the Invariant Subspace Problem. On a topological vector space X, a linear operator T:X → X is said to be hypercyclic if there is a vector x for which the sequence x,Tx, T2x,T3x,... is dense in X. We explored whether the dynamical properties of an operator are preserved by the Aluthge transform. We showed for bilateral weighted shifts, an operator T is mixing, chaotic, or hypercyclic if and only if the Aluthge transform of the operator has the same dynamical property. We also supplied conditions for when the Aluthge transform of an arbitrary operator T has the same dynamical properties as T. In chapter three we provided a strong counterexample to a conjecture by Jung, Ko, and Percy. They conjectured that for every bounded linear operator T on a Hilbert space, the sequence of operators formed by iteratively applying the Aluthge transform to T would converge to a normal operator. We used a probabilistic argument to show that if T is any bilateral forward shift, then either the sequence of iterations of the Aluthge transform converges to a normal shift in the strong operator topology, or it fails to converge in a dramatic sense in that its set of strong operator topology subsequential limits is an “interval” of normal shifts. We then showed for any positive reals a < b, there is a bilateral weighted forward shift T for which the set of strong operator topology subsequential limits of the sequence of iterates of the Aluthge transform is the set of shifts of the form tS where S is the pure forward shift, and t is any number in the interval [a,b]. These results were extended to address complexly weighted shifts, and bilateral backward shifts. In the last chapter, we address where “most” hypercyclic vectors are located relative to the range of a hypercyclic operator. If x is hypercyclic for T, then so is Tnx for every natural number n, and Tn x (open full item for complete abstract)

Committee: Juan Juan PhD (Advisor); Kit Chan PhD (Committee Member); John Hoag PhD (Committee Member); Craig Zirbel PhD (Committee Member) Subjects: Mathematics