Geometric Models and Compactness of Composition Operators

This work explores some of the terrain between functional equations, geometric function theory, and operator theory. It is inspired by the fact that whenever a composition operator or one of its powers is compact on the Hardy space H2, then its eigenfunctions cannot grow too quickly on the unit disc. The goal is to show that under certain natural (and necessary) additional conditions there is a converse: slow growth of eigenfunctions implies compactness. We interpret the slow-growth condition in terms of the geometry of the principal eigenfunction of the composition operator (the "Königs function" of the inducing map). We emphasize throughout the importance of this eigenfunction in providing a simple geometric model for the operator′s inducing map.