Investigation of somewhere dense and dense orbits on complex Hilbert space

This dissertation form part of the study of operates theory results for somewhere dense orbits and hypercyclic operators. Somewhere dense orbits are one of the generalizations of the notion of dense orbits. In this study, first we present the existence of n-tuple of operators on complex Hilbert space that has a somewhere dense orbit that is not dense. We give the solution to the question which was posed by Feldman 2008: ‘’ Is there n-tuple of operators on a complex Hilbert space that has a somewhere dense orbit and is not dense?’’ we do so by extending the results due to Feldman 2008 from real Hilbert space to complex Hilbert space. The second topic major that we investigate in this study is hypercylicity. Hypercylicity is the study of linear and continuous operators that possess dense orbits. We extended the results due to Costakis et al. 2009 by showing the existence of hypercyclic (non diagonalizable) n- tuple of matrices in Jordan form on C0. In doing so, we modified some lemmas from 2 x2 matrices in Jordan form to 2 x 2 matrices in lower triangular form and to 3 x 3 matrices in Jordan form and some propositions and theorems from R to C. Next, let T1, T2 be continuous linear operators acting on a Banach space X and (T1, T2) be a pair of operators. We answered the question raised by Feldman 2003 which states that: ‘’If (T1,T2 ) is hypercyclic pair, is (T1⨁T1T2 ⨁ T2) Also a hypercyclic?’’ we showed that indeed that is (T1⨁T1T2 ⨁ T2) hypercyclic pair and it satisfies the hypercyclic criterion. Illustrative example of somewhere dense orbits and hypercyclicy operators are given to support the results.