Dynamical Properties Of Several Classes Of Linear Operators

Dynamics of Linear Operators has a familiar connection with operator theory, ergodic theory, function theory, numerical theory, the geometry of Banach space, and several other fields. Our study on Dynamics of Linear Operators plays an important role in the development of these mathematical branches. In this thesis, we study the (frequent) hypercyclicity of operators acting on the entire function space and operators acting on general separable infinite dimensional space which commuts with one given operatorby applying the (Frequently) Hypercyclic Criterion. We divide the dissertation into six chapters:In Chaper 1, we mainly introduce the background, development of this thesis, and its connections with other fields, and then statethe content of our dissertation.In Chaper 2, some basic notions are introduced, including hypercyclic operators, weakly mixing operators, topological mixing operators, Devaney's chaotic operators, frequently hypercyclic operators, and their criterions.The third charpter was devoted to the studies of hypercyclicity, weakly mixing, topological mixing and Devaney chaos of the tensor products of two multiplication operators acting on the Hardy space by applying the Godefroy-Shapiro Criteion. Our research is motivated by the following Martinez-Gimenez and Peris'question:If T ? L(X) is hypercyclic, is T (?) T still hypercyclic? Using Godefroy-Shapiro's characterization on the hypercyclicity of mutiplication operator, we are able to gain an equivalent characterization on the hypercyclicity of tensor products of two mutiplication operators by Godefroy-Shapiro Criteion. For operatorsT satisfying the Hypercyclic Criterion and the hypercyclicity of T (?)I, Martinez-Gimenez and Peris'result is an equivalent characterization, now what we expect is thatthe same equivalent condition hold in the case offrequenthypercyclicity. By using the Frequently Hypercyclic Criterion itself, we present a sufficient condition.In charpter 4, our main consideration is the frequenthypercyclicity of an operator which commuts with one given operator. Our research is motivated by the following Costakis and Parissis'question:Is the permutation of identity by the weighted backward shiftI + Bw topologically multiply recurrent? Firstly, by using Shields'characterization on spectrum of the weighted backward shift, we are able to obtain a sufficient condition on f(Bw)to be frequently hypercyclic via perfectly spanning set of unimodular eigenvalue, in particular, a sufficient condition for the permutation of identity by the weighted backward shift to be frequently hypercyclic is obtained, further, by applying Costakis and Parissis'result, the same sufficient condition was obtained for the permutation of identity by weighted backward shift to be topological multiple recurrent. Secondly, we obtained a sufficient condition on f(B) acting on weighted sequence space lp(N,?)to be frequently hypercyclic via perfectly spanning set of unimodular eigenvalue, by applying the quasiconjugacy between f(B) and f(B?), then the same conclusion above was obtained. Finally, for operators T ?L(X) acting on a general separable infinite Banach space, a sufficient condition for f(T) to be frequently hypercyclic was also obtained. Moreover, for the frequent hypercyclicity of anoperatorwhich commuts with one given operator, we also present a sufficient condition.In charpter 5, we mainly consider the frequently hypercyclicity of non-convolution operator T?,b acting on the entire function space. Our study is motivated by the following Gupta and Mundayadan's question:Is the non-convolution operator frequently hypercyclic? Using Muro and Pinasco's equivalent characterization on the hypercyclicity of T?,b, we are able to obtain an equivalent condition on the frequent hypercyclicity of T?,b. Moreover, we modify the original Frequently Hypercyclic Criterion, and then give a sufficient condition for T?,b satisfying the modified Frequent Hypercyclic Criterion.In the final charpter, we summarize the main results of the thesis, analysison the facing difficults, and also introduce several problems that will be studied in the next.