Hypercyclic operators on Banach spaces

A bounded linear operator T on a Banach space X is said to be hypercyclic if there exists some vector x in X so that its orbit {x,Tx,T 2x,T3x,...} is dense in X.;In this thesis, algebraic and spectral properties, in addition to others, of hypercyclic operators on complex Banach spaces are presented. In particular, the spectrum of a hypercyclic operator is analyzed in detail.;A classic hypercyclic result states that every component of the spectrum must intersect the unit circle. A modest generalization of this fact is used to establish new results about the cardinality of the spectrum of a hypercyclic operator. Specifically, it is proved that the set consisting of the absolute values of every point in the spectrum of a hypercyclic operator on a Banach space is at most countable, then it is the singleton {1}, meaning that the spectrum must lie on the unit circle. This extends a result done by Matache. An analysis of the states of the spectrum for a hypercyclic operator is also done.