Research On The Essential Numerical Range Of Singular Discrete Hamiltonian Systems

Since the operators generated by the discrete Hamilton system may be multivalued and their minimum operators may not be dense,the situation where such linear operators are multi-valued can be studied under the framework of linear relations.The famous American mathematician von Neumann J.introduced the concept of linear relation for the first time when considering adjoints of nondensely defined linear differential operators,and with the continuous in-depth study of linear relations,it plays an important role in many major mathematical theoretical problems,such as nonlinear analysis,continuation theory of linear operators,differential equations,semigroups of degenerate operators,and optimization and control problems.This paper mainly studies the maximum and minimum relations generated by a singular discrete Hamilton system,and studies the essential numerical range of the maximum and minimum relations.For semi-bounded systems,a new singular sequence is used to characterize the essential numerical range of the minimum relation generated by discrete Hamilton systems.Secondly,the concept of perturbation at infinity is given,and the invariance of the intrinsic numerical domain of the system under such perturbations is proved on this basis.In this paper,the essential numerical range of singular discrete Hamiltonian systems is studied.The main research work is as follows:The first chapter mainly gives the research background and current status of the study of the essential numerical range and the problems considered in this article and the main work.The second chapter is the preliminary,which gives the essential spectrum and essential numerical range of linear relations and the basic concepts and properties,as well as their relationships.In the third chapter,we give the maximum,pre-minimum,and minimum relations of the discrete Hamiltonian system and derive its essential numerical range,and then we give the concept of perturbations small at infinity and prove the stability of the essential numerical range under this perturbation.In the fourth chapter,some sufficient conditions for the invariance of essential numerical range of some systems are obtained in terms of coefficients of systems and perturbations terms.In the fifth chapter,all the findings are applied to the Sturm-Liouville difference equation.