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