Symmetric operator extensions of composites of higher order difference operators

Abstract

In this paper we have considered two higher order difference operators generated by two higher order difference functions  on the Hilbert space of square summable functions. By allowing the leading coefficients to be unbounded and the other coefficients as constant functions, we have shown that the composites of two symmetric difference operators are symmetric if the leading coefficients are scalar multiple of each other and the common divisor of their orders is 1. Using examples, we have shown that these conditions of symmetry cannot be weakened. Furthermore, We have shown that the deficiency indices of the composites is equal to the sum of the deficiency indices of the individual operators and that the spectra of the self-adjoint operator extensions is the whole of the real line.