  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.
