Some Research On Dynamics Of Linear Operators

In this paper, the object of study is dynamics of linear operators and we will give some results. The research of dynamics of linear operators involves five parts:Firstly, we give a sufficient condition for the existence of a common universal subspace for various countable families of universal sequences of linear operators. And we will show that under some conditions, the unitary orbit of the supercyclic operator contains a path of operators whose closure contains the entire unitary orbit with the strong operator topology, and common supercyclic vectors of this path of operators is a dense set.Secondly, we will provide a Subspace-Supercyclicity Criterion and offer two equiva-lent conditions of this criterion. At the same time, we also characterize other properties of subspace-supercyclic operators.Thirdly, we will show that the property of disjoint supercyclic operators satisfying d-Supercyclicity Criterion on the same Hilbert space is equivalent to disjointness in supercyclicity of the corresponding left multiplication operators induced in the strong operator topology.Fourthly, we will investigate conditions to ensure that finite many powers of dif-ferentiation operators are disjoint hypercyclic on generalized weighted Bergman spaces of entire functions. Besides, the disjoint hypercyclicity and disjoint supercyclicity of finitely many linear fractional composition operators acting on weighted Banach spaces will be discussed.Finally, we will characterize disjoint hypercyclic powers of weighted translation operators on the Lebesgue space in terms of the weights.