The Friedrichs extension of a class of discrete symplectic systems

TL;DR

This paper characterizes the Friedrichs extension of a class of discrete symplectic systems with spectral parameter dependence, generalizing previous results for linear operators and differential operators using recessive solutions.

Contribution

It extends the characterization of Friedrichs extensions to discrete symplectic systems with spectral dependence, broadening the scope of prior operator and differential operator results.

Findings

Friedrichs extension characterized via recessive solutions.

Generalization of previous results for banded matrices and differential operators.

Applicable to discrete symplectic systems with spectral parameter dependence.

Abstract

The Friedrichs extension of minimal linear relation being bounded below and associated with the discrete symplectic system with a special linear dependence on the spectral parameter is characterized by using recessive solutions. This generalizes a similar result obtained by Do\v{s}l\'y and Hasil for linear operators defined by infinite banded matrices corresponding to even-order Sturm--Liouville difference equations and, in a certain sense, also results of Marletta and Zettl or \v{S}imon Hilscher and Zem\'anek for singular differential operators.