Mathematics Ph.D. Dissertations

Hypercyclic Extensions Of Bounded Linear Operators

Date of Award

2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Kit Chan (Advisor)

Second Advisor

Ron Lancaster (Committee Member)

Third Advisor

Juan Bes (Committee Member)

Fourth Advisor

Craig Zirbel (Committee Member)

Abstract

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 find 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 finnite 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.

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