In this dissertation we investigated the two topics hypercyclicity and the Aluthge transform. Each of these is related to the Invariant Subspace Problem.
On a topological vector space X, a linear operator T:X → X is said to be hypercyclic if there is a vector x for which the sequence x,Tx, T2x,T3x,... is dense in X.
We explored whether the dynamical properties of an operator are preserved by the Aluthge transform. We showed for bilateral weighted shifts, an operator T is mixing, chaotic, or hypercyclic if and only if the Aluthge transform of the operator has the same dynamical property. We also supplied conditions for when the Aluthge transform of an arbitrary operator T has the same dynamical properties as T.
In chapter three we provided a strong counterexample to a conjecture by Jung, Ko, and Percy. They conjectured that for every bounded linear operator T on a Hilbert space, the sequence of operators formed by iteratively applying the Aluthge transform to T would converge to a normal operator. We used a probabilistic argument to show that if T is any bilateral forward shift, then either the sequence of iterations of the Aluthge transform converges to a normal shift in the strong operator topology, or it fails to converge in a dramatic sense in that its set of strong operator topology subsequential limits is an “interval” of normal shifts. We then showed for any positive reals a < b, there is a bilateral weighted forward shift T for which the set of strong operator topology subsequential limits of the sequence of iterates of the Aluthge transform is the set of shifts of the form tS where S is the pure forward shift, and t is any number in the interval [a,b]. These results were extended to address complexly weighted shifts, and bilateral backward shifts.
In the last chapter, we address where “most” hypercyclic vectors are located relative to the range of a hypercyclic operator. If x is hypercyclic for T, then so is Tnx for every natural number n, and Tn x is in the range of T. Since moreover, the range of T is dense in X, one might expect that most if not all of an operators hypercyclic vectors lie in its range. To the contrary, we showed for every non-surjective hypercyclic operator T on a Banach space,the set of hypercyclic vectors for T that are not in its range is large in that it is a set of category II. We also provided a sense by which the range of an arbitrary hypercyclic operator is large in its set of hypercyclic vectors for T.