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

- Publication Year:
- 2013

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- Bepress 25

- Repository URL:
- https://scholarworks.bgsu.edu/math_diss/9

- Author(s):

- Tags:
- Hypercyclicity; Periodic points; Chaotic extension; Adjoint operator; Dual hypercyclic operator; Hypercyclic extension; Fredholm extension; Hypercyclic subspace

##### thesis / dissertation description

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) = {Tnx : n ≥ 0 } = { x, Tx, T2x, ..... } 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 Tn x = x for some positive integer 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 |M = A. In this dissertation, we provide a criterion for the existence of an invertible chaotic extension. Indeed, we show that a bounded linear operator 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.