Sturm-Liouville operator functions

Many special functions are solutions of both a differential and a functional equation. We use this duality to solve a large class of abstract Sturm-Liouville equations on the non-negative real line, initiating a theory of Sturm-Liouville operator functions; cosine, Bessel, and Legendre operator functions are special cases. We investigate properties of the generator, uniformly continuous Sturm-Liouville operator functions, give a spectral inclusion theorem, and investigate existence of an exponential norm bound. Whenever such a bound exists, we present the resolvent formula and study the relation to C-0-semigroups and C-0-groups. This general theory part is supplemented by specific examples. We show connection formulas between different types of Sturm-Liouville operator functions, determine the generator of translation operator functions on homogeneous Banach spaces, and consider Sturm-Liouville operator functions generated by multiplication operators.