On the -comparison algebra of a class of singular Sturm-Liouville expressions on the real line

Abstract

In this article we study a -comparison algebra in the sense of [C2] with generators related to the ordinary differential expression H on the full real line R where,with constants , (1.1) (Error rendering LaTeX formula) More precisely, the algebra, called , is generated by the multiplications , by functions and the (singular integral) operators (Error rendering LaTeX formula) ,and their adjoints. Here , the inverse positive square root of the unique self-adjoint realization H of the expression (1.1), in the Hilbert space .(We use the same notation for both, (1.1) and its realization.) The case of was discussed earlier in [Tg1], even for all n -dimensional problem. The commutators are compact and the Fredholm properties of operators in are determined by a complex-valued symbol on a symbol space homeomorphic to that of the usual Laplace comparison algebra on , although the symbol itself is calculated by different formulas.