The Disjoint Cyclic Operator And Numerically Hypercyclic Operator

Finitely many cyclic operators T1, T2, ..., TN acting on a topological vector space X are called disjoint if their direct sum has a cyclic vector formed as (x,x,......,x) in the diagnal space XN. In this disserta-tion, we firstly provide the definition of Kitai’s criterion for disjoint cyclic operators, and we prove that operators satisfying disjoint cyclic criterion are disjoint cyclic by two different methods. Which general-izes some well-known connections between the disjoint cyclic criterion, disjoint topologically transitive and disjoint densely universal. We al-so prove that operators satisfying the disjoint cyclic criterion have a densely disjoint cyclic linear manifold.Numerical hypercyclicity is another active direction of generaliza-tion of hypercyclicity. A continuous linear operator T on a normed space E is called numerically hypercyclic if the numerical orbit {x*(Tn(x)):n≥0} is dense in C for some (x,x*) ∈ Ⅱ(E). Here we generalize the definition of topological transitivity, topological mixing, hereditary hypercyclicity, hereditary dense hypercyclicity to numerical hypercyclicity. We also prove the equivalence of numerical transitivity and numerrical hypercyclicity, similarly the equivalence of numerical mixing and hereditary dense numerical hypercyclicity. Then we char-acterize the hereditarily numerically hypercyclic weighted shifts on two classical sequence spaces and the product of numerically mixing operator on l2 space.