| The main result of this dissertation is a simple proof of the noted theorem of Alfonso Montes-Rodríguez, who proved a sufficient condition for the existence of a closed infinite dimensional subspace consisting entirely, except for the zero vector, of hypercyclic vectors of a bounded linear operator T on a separable, infinite dimensional Banach space X. We accomplish this through developing the theory of hypercyclicity of a bounded linear mapping L on the operator algebra B( X). To this end, we need to establish the result that B( X) is separable in the strong operator topology, by exhibiting a specific countable dense subset of B(X) consisting of finite rank operators, which play an important role in our subsequent arguments.;Results in this dissertation also include a parallel theory of supercyclicity of the mapping L, in the case that the underlying space X of the operator algebra B(X) is indeed a separable infinite dimensional Hilbert space. In addition, we discuss the notions of multi-hypercyclicity and multi-supercyclcity for L. Finally we prove the result that whenever L is hypercylic, there exists a dense linear manifold consisting entirely, except for zero, of hypercyclic vectors of L. |
