Essential Closures and AC Spectra for Reflectionless CMV, Jacobi, and Schr\"odinger Operators Revisited

We provide a concise, yet fairly complete discussion of the concept of essential closures of subsets of the real axis and their intimate connection with the topological support of absolutely continuous measures. As an elementary application of the notion of the essential closure of subsets of $\bbR$ we revisit the fact that CMV, Jacobi, and Schr\"odinger operators, reflectionless on a set E of positive Lebesgue measure, have absolutely continuous spectrum on the essential closure of the set E (with uniform multiplicity two on E). Though this result in the case of Schr\"odinger and Jacobi operators is known to experts, we feel it nicely illustrates the concept and usefulness of essential closures in the spectral theory of classes of reflectionless differential and difference operators.