Trace ideal criteria for embeddings and composition operators on model spaces

Abstract

Let \(K_\theta\) be a model space generated by an inner function \(\theta\). We study the Schatten class membership of embeddings \(I : K_\theta \to L^2(\mu)\), \(\mu\) a positive measure, and of composition operators \(C_\phi:K_\theta\to H^2(\mathbb D)\) with a holomprphic function \(\phi:\mathbb D\rightarrow \mathbb D\). In the case of one-component inner functions \(\theta\) we show that the problem can be reduced to the study of natural extensions of \(I\) and \(C_\phi\) to the Hardy-Smirnov space \(E^2(D)\) in some domain \(D\supset \mathbb D\). In particular, we obtain a characterization of Schatten membership of \(C_\phi\) in terms of Nevanlinna counting function. By example this characterization does not hold true for general \(\phi\).