Symmetric operator extensions of composites of Higher order difference operator

Abstract

in this paper we have considered two higher order difference op- erators generated by two higher order difference functions on the Hilbert space of square summable sequences. By allowing the leading coefficients to be un- bounded and the other coefficients as constant functions, we have shown that the composites of two higher order 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 sym- metry cannot be weakened. Furthermore, we have shown that the deficiency index of the composite is equal to the sum of the deficiency indices of the in- dividual operators and that the spectra of the self-adjoint operator extensions is the whole of the real line