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George T. Dissertation.pdf (374.37 KB)

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Hypercyclic Extensions Of Bounded Linear Operators

Turcu, George R
http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1386189984

Abstract Details

2013, Doctor of Philosophy (Ph.D.), Bowling Green State University, 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 Tp z = 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.
Kit Chan (Advisor)
Ron Lancaster (Committee Member)
Juan Bes (Committee Member)
Craig Zirbel (Committee Member)
56 p.
Mathematics
hypercyclic operators; extensions of hypercyclic operators; chaotic operators; hypercyclic vectors; periodic points; Banach space; Hilbert space; linear operator; hypercyclicity; operator theory; orthogonal projection

Recommended Citations

Citations

  • Turcu, G. R. (2013). Hypercyclic Extensions Of Bounded Linear Operators [Doctoral dissertation, Bowling Green State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1386189984

    APA Style (7th edition)

  • Turcu, George . Hypercyclic Extensions Of Bounded Linear Operators. 2013. Bowling Green State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1386189984.

    MLA Style (8th edition)

  • Turcu, George . "Hypercyclic Extensions Of Bounded Linear Operators." Doctoral dissertation, Bowling Green State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1386189984

    Chicago Manual of Style (17th edition)

bgsu1386189984
872
© 2013, all rights reserved.
This open access ETD is published by Bowling Green State University and OhioLINK.
Release 3.2.12