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The positive and negative deficiency indices of formally self-adjoint difference equations. (English) Zbl 1493.39001

Summary: 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.

MSC:

39A05 General theory of difference equations
39A70 Difference operators
39A12 Discrete version of topics in analysis
47B39 Linear difference operators
34B20 Weyl theory and its generalizations for ordinary differential equations
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