Friedrichs extensions of a class of discrete Hamiltonian systems with one singular endpoint

Abstract

This paper is concerned with Friedrichs extensions for a class of discrete Hamiltonian systems with one singular endpoint. First, Friedrichs extensions of symmetric Hamiltonian systems are characterized by imposing some constraints on each element of domains 𝐷(𝐻) of the maximal relations 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,  -self-adjoint Friedrichs extensions are studied, and two results are given for elements of 𝐷(𝐻), which make the expression of the Friedrichs extension simpler. All results are finally applied to Sturm-Liouville equations with matrix-valued coefficients.