Cyclicity of bounded linear operators on separable banach spaces and closed subspaces - Muhemed Said, Nareen Sabih (2016)
The main focus of this thesis is to study some properties of diskcyclic operators. The similarities and differences between diskcyclic operators and the concepts of cyclicity are investigated. New classes of operators on the direct sum of Banach spaces, namely k-bitransitive, k-diskcyclic and k-compound operators, are defined to study the direct sum of diskcyclic operators. A weaker property than chaotic operators, namely semi chaotic operators, is defined and studied to show that a chaos for linear operators exists on finite dimensions. Diskcyclicity concept is extended to closed subspaces of Banach spaces, and such a concept is called a subspace-diskcyclic operator. The similarities and differences between subspace-diskcyclicity and the subspace-cyclicity are investigated. Finally, some properties of hypercyclic operators are extended to subspace-hypercyclic operators to solve some open problems in the literature. This study shows that diskcyclic, semi chaotic and subspace-diskcyclic operators have some properties that are not shared with the existing concepts of cyclicity of operators.