The positive and negative deficiency indices of formally self-adjoint difference equations

TL;DR

This paper investigates the properties of formally self-adjoint difference equations, revealing that their order is always even and that their positive and negative deficiency indices are equal, which impacts their self-adjoint extensions.

Contribution

It establishes the even order of such difference equations and demonstrates the equality of their deficiency indices, highlighting a key difference from differential equations.

Findings

Order of formally self-adjoint difference equations is always even

Positive and negative deficiency indices are equal

Existence of self-adjoint extensions for the associated linear relations

Abstract

This paper is concerned with formally self-adjoint difference equations and their positive and negative deficiency indices. It is shown that the order of any formally self-adjoint difference equation is even, and some characterizations of formally self-adjoint difference equations are established. Further, we show that the positive and negative deficiency indices are always equal, which implies the existence of the self-adjoint extensions of the minimal linear relations generated by the difference equations. This is an important and essential difference between formally self-adjoint difference equations and their corresponding differential equations in the spectral theory.