The Positive and Negative Deficiency Indices of Formally Self-Adjoint Difference Equations

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.