Geometric models and compactness of composition operators

J. Functional Analysis 127 (1995) 21--62

Abstract: 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 H^2, 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 ``Koenigs 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.

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