## Mathematics Ph.D. Dissertations

#### Title

Chaotic Extensions for General Operators on a Hilbert Subspace

2014

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Mathematics

Juan Bes (Committee Member)

So-Hsiang Chou (Committee Member)

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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