On discrete symplectic systems: Associated maximal and minimal linear relations and nonhomogeneous problems

TL;DR

This paper investigates the properties of discrete symplectic systems, focusing on their definiteness, associated linear relations, and solutions to nonhomogeneous problems, providing new insights into their mathematical structure and solution spaces.

Contribution

It introduces the concepts of minimal and maximal linear relations for discrete symplectic systems and explores their properties, including deficiency indices and conditions for operator existence.

Findings

Characterization of definiteness in discrete symplectic systems

Relationship between square summable solutions and defect subspace dimension

Sufficient condition for the existence of a densely defined operator

Abstract

In this paper we characterize the definiteness of the discrete symplectic system, study a nonhomogeneous discrete symplectic system, and introduce the minimal and maximal linear relations associated with these systems. Fundamental properties of the corresponding deficiency indices, including a relationship between the number of square summable solutions and the dimension of the defect subspace, are also derived. Moreover, a sufficient condition for the existence of a densely defined operator associated with the symplectic system is provided.