## Mathematics Ph.D. Dissertations

#### Title

Hypercyclic Extensions of an Operator on a Hilbert Subspace with Prescribed Behaviors

#### Date of Award

2013

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Mathematics

#### First Advisor

Kit Chan

#### Second Advisor

Juan Bes (Committee Member)

#### Third Advisor

So-Hsiang Chou (Committee Member)

#### Fourth Advisor

Rachel Reinhart (Committee Member)

#### Abstract

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) = {T ^{n}x : n ≥ 0 } = { x, Tx, T^{2}x, ..... }* 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

*T*for some positive integer

^{n}x = x*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 |*. In this dissertation, we provide a criterion for the existence of an invertible chaotic extension. Indeed, we show that a bounded linear operator

_{M}= A*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.

#### Recommended Citation

Kadel, Gokul, "Hypercyclic Extensions of an Operator on a Hilbert Subspace with Prescribed Behaviors" (2013). *Mathematics Ph.D. Dissertations*. 9.

https://scholarworks.bgsu.edu/math_diss/9