Dynamical Properties Of Tuples Of Linear Operators

From a purely mathematical point of view,the linear operator dynamics and function space operator theory,complex analysis,operator algebra and matrix theory,and other fields have close and profound link.From the perspective of applied mathematics,the research of linear operator dynamics is widely applied to the differential equation and dynamical system,matrix analysis.In addition,the kinetics of linear operator to promote the development of chaotic cryptography and other fields also played an important role.The dissertation will be divided into six chapters:In chaper 1,we mainly introduce the background,domestic and foreign research present situation,analysis of developing trend,and then statethe content of our dissertation.In chaper 2,in the first part,we introduce the definition of（tuple of）hypercyclic operators,（tuple of）weakly mixing operators,（tuple of）topological mixing operators,（tuple of）Devaney chaotic operators and their criterions,respectively.In the second part,we mainly introduce some basic notions on hypercyclic semigroup and their criterions.In chaper 3,based on the results of Feldman and Costakis,we mainly characterize the hypercyclicity of tuple operators on finite complex spacen Cⁿ,and obtain a sufficient and necessary condition for an m-tuple of commutative complex upper triangular Toeplitz matrices to be hypercyclic.In addition,we extend the result to more general case.In charpter 4,our main consideration is the dynamical properties of tuple of operators on Hardy spaceH²（D）,sequence space e^p（l≤p<∞） and c₀.Using joint point spectrum,we firstly obtain the eigenvalue criterion for tuple of operators.During the process,we find the new kind of operator that between weakly mixing and mixing,we call it S-mixing operator,and depict the relationship between the S-mixing operator and the composition operators of the several operator in the constitute the tuple of operators.In addition,we further study the relationship between the S-mixing properties and topological mixing,weakly mixing,hypercyclicity and chaos.In charpter 5,we mainly consider the dynamic properties of（weighted）backward shift operators on the double sequence spaces.The content is inspired by the research on single weighted shift operators and double sequence space,using the ideas of Feldman,replace the action of single operator by semigroup action,to promote the study of hypercyclicity of tuple of operators on the double sequence spaces.First,we rearrange the double sequence as the form of infinite dimensional matrix,based on this rearrangement,we define two shift operators B^u,B^e,one is the left shift and another up shift,and they are collectively referred to “backward” shift operators.Based on the definition of hypercyclic operator,it is clear that they are not hypercyclic,thus we further consider the nature of the weighted shift operators B_w^u,B_v^e,according to the commutativity,and the research is divided into two cases.One case they can exchange with each other.In the presence of exchangeable,we present the fully characterization on pair of operators(B_w^u,B_v^e)to be hypercyclic,mixing and chaotic.In the case of noncommutative,the study of the operator more troublesome,because most of the time and is noncommutative,w={w_i,j},v={v_i,j} are the right of bounded positive weight sequences.Research this kind of situation,we give the definition of a continuous path,on the premise of specified path,we obtain the characterization on the pair of operators (B_w^u,B_v^e) to be hypercylic and topological mixing.On this basis,using the properties of qusiconjugate mapping which keep the nature of operator,the results above can be generalized to weighted sequence space.In charpter 6,we summarize the full text and analyze the deficiency in this paper,and then give the further research questions.