Measure algebras associated with a second order differential operator

Abstract

Let L = d 2 dx 2 − q(x) be a Sturm-Liouville operator acting on functions defined on R. The authors have recently shown how to construct commutative associative algebras of distributions of compact support for which L is a centralizer (in the sense that L(f ∗ g) = (Lf) ∗ g for distributions f, g of compact support) when q is locally bounded. Here, it is assumed either that q is bounded and x → (1 + ¦ x ¦) q(x) is integrable, or that q is of bounded variation. A function ψ is then found such that M ψ={μ : μ is a measure on R and | μ |(ψ) < & infin;} becomes a Banach algebra containing the algebra of measures of compact support. The representation theory of M ψ is discussed and conditions for its semisimplicity are obtained.