Friedrichs Extensions Of A Class Of Singular Discrete Hamiltonian Systems

The self-adjoint extension problem is one of the most important problems in the spectral theory for linear differential and difference systems.Based on the classical GKN(Glazman-Krein-Naimark)theory,all self-adjoint extensions of symmetric differential operators with equal defect indices can be described in terms of boundary conditions which are given by square-integrable solutions of the corresponding equations.In the case that the associated differential operator is bounded from below,among all self-adjoint extensions there is a particular one which preserves the lower bound.This extension is known as the Friedrichs extension,which was initially constructed by Friedrichs for a densely defined operator[10]and plays an important role in many problems.In 2021,paper[49]characterize the Friedrichs extensions of a class of singular Hamiltonian systems,including symmetric and asymmetric cases,by imposing some constraints on each element of domains D(H)of the maximal operators.In this paper,the Friedrichs extension of a class of discrete Hamiltonian systems with a singular endpoint is studied based on the method of[49].First,Friedrichs extensions of symmetric Hamiltonian systems are characterized by imposing some constraints on each element of domains D(H).Furthermore,it is proved that the Friedrichs extension of each of a class of non-symmetric systems is also a restriction of the maximal relation H by using a closed sesquilinear form.Then,the corresponding Friedrichs extensions are characterized.In addition,T-self-adjoint Friedrichs extensions are studied,and a result is given for elements of D(H),which makes the expression of the Friedrich extension simpler.All results are finally applied to Sturm-Liouville equations with matrix-valued coefficients.In this paper,it is concerned with Friedrichs extensions for a class of discrete Hamiltonian systems with one singular endpoint.The content is organized as follows.In chapter 2,we introduce some basic concepts and results about linear relations and sesquilinear forms in a Hilbert space,and define the maximal,pre-minimal,and minimal relations associated with system(1.2).In chapter 3,we characterize the Friedrichs extension of discrete Hamiltonian systems.For symmetric systems,the Friedrichs extension can be described directly by imposing some constraints on each element of domains D(H)of the maximal relations H.For non-symmetric systems,it is proved that the Friedrichs extension is also a restriction of the maximal relation H by introducing a closed sectorial form.Based on it,Friedrichs extension of a class of non-symmetric Hamiltonian systems can also be given.In chapter 4,a sufficient condition for(x*u)(∞)=0 is given.In chapter 5,all results are finally applied to Sturm-Liouville equations with matrix-valued coefficients.