On G-Cyclicity of Operators

Abstract

In this thesis, we focus our study on a part of cyclic phenomena, namely Gcyclic operators on an infinite dimensional separable complex Hilbert space. We study some properties of cyclic, supercyclic, and hyupersyclic operators, then we give some examples that explain the relationship between them, where we find that, supercyclicity stands in the midway between hypercyclicity and cyclicity. In the first step we give a necessary and sufficient conditions for an operator to be G-cyclic, we show that every G-cyclic operator is supercyclic but the converse need not be true in general. Then we discuss some of the properties of the spectrum of G-cyclic operators. In the second step, as examples of G-cyclic operators we define disk-cyclic and codisk-cyclic operators, and state and prove the Disk-Codisk cyclicity criterion. Finally we give applications of this result.