Compactness from Models

Abstract

Linear fractional models emerged in the last chapter as tools for determining the cyclic behavior of composition operators. In this one they figure in the study of compactness. Recall from §5.5 that a necessary (but by no means sufficient) condition for a composition operator to be compact is that its inducing map have a fixed point in U. Thus the model associated with such a map must arise from Königs’s solution to Schröder’s equation. We introduced this model in Chapter 6, and showed that compactness imposes severe restrictions on the geometry of the Königs domain of φ (the image of U under the Königs function σ). In particular:

If φ is univalent and C _φ is compact on H ², then the Königs domain contains no sector.

Our goal here is to turn this result around, and show that for a natural class of maps φ: “no sectors implies compactness.”