Self-adjoint extensions for discrete linear Hamiltonian systems

This paper is concerned with self-adjoint extensions for a class of discrete linear Hamiltonian systems. By applying the generalized von Neumann theory and the GKN theory for Hermitian subspaces, a complete characterization of all the self-adjoint subspace extensions for the systems are obtained in terms of boundary conditions via linear independent square summable solutions. As a consequence, characterizations of all the self-adjoint subspace extensions are given in the limit point and limit circle cases. In addition, some sufficient conditions for the corresponding minimal subspace to be an operator or a densely defined operator are given, and consequently, characterizations of all the self-adjoint operator extensions for the system are obtained. In particular, even-order formally self-adjoint vector difference equations are discussed as a special case.