Abstract:
We have established the necessary and sufficient conditions for
any two even higher order symmetric difference maps to generate commuting
minimal difference operators. We have done this through construction of ap-
propriate comparison algebras of the self-adjoint operator extensions of the
minimal operators generated and application of asymptotic summation. The
results show that if the first difference on the coefficients tends to zero when-
ever the coefficients are allowed to be unbounded and that the difference maps
considered have the same order, then they generate minimal operators that
commute and the corresponding self-adjoint operators commute too. We have
further shown that the self-adjoint operator extensions of the respective mini-
mal operators can be expressed as the composite of the independent self-adjoint
operator extensions if the generated minimal difference operators have closed
ranges. Finally, we have shown that the spectra of these self-adjoint opera-
tor extensions are the whole of the real line if the coefficients are unbounded.
These results therefore, extend the existing results in the continuous case to
discrete settin
