Mathematics Ph.D. Dissertations

Title

Chaotic Extensions for General Operators on a Hilbert Subspace

Date of Award

2014

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Kit Chan (Advisor)

Second Advisor

Juan Bes (Committee Member)

Third Advisor

So-Hsiang Chou (Committee Member)

Fourth Advisor

Peter Gorsevski (Committee Member)

Abstract

A continuous linear operator T : X→ X on a separable topological vector space X is said to be a hypercyclic operator if there exists a vector x in X whose orbit under T, given by orb(T,x) = {x,Tx,T2x,...}, is dense in X. We call such a vector x a hypercyclic vector. In contrast to such vectors, we may encounter vectors x whose orbit is finite. More precisely, a vector x in X is said to be a periodic point if there exists a natural number n for which Tnx = x. If an operator admits both a dense orbit and a dense set of periodic points we call such an operator a chaotic operator. In this dissertation we show that for any linear bounded operator A : M→ H, where H is an infinite-dimensional separable Hilbert space and M is an infinite codimensional, closed subspace of H, there exists a chaotic operator T : H →H whose restriction T|M = A.

If we replace the powers Tn in the a above definition of hypercyclicity by a sequence of operators Tn , we obtain the definition for a universality. That is, the sequence {Tn : X→X |n = 1} is said to be a universal sequence if there exists a vector x in X for which the orbit {x,T1x,T2x,...} is dense in X. We call such a vector a universal vector. In this area, we show that any sequence of bounded linear operators {An : M → H |n = 1}, with H and M as above, can be extended to a universal sequence {Tn : H → H |n = 1}. In other words, {Tn} is universal and Tn|M = An for every n.

We also investigate chaotic operators in the setting of the Frechet space h(G) of harmonic functions for a region G in the complex plane. Suppose G is a finitely connected region and L : h(G) →h(G) is a continuous linear operator that commutes with both differential operators ∂ and ∂bar, and its adjoint L* does not have an eigenvalue. Then we prove that the following are equivalent: (1) L is chaotic, (2) L is hypercyclic, and (3) G is simply connected.

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