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Abstract

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.