Trace ideal criteria for embeddings and composition operators on model spaces

TL;DR

This paper investigates the conditions under which embeddings and composition operators on model spaces belong to Schatten classes, providing characterizations for one-component inner functions and highlighting limitations for general functions.

Contribution

It offers new criteria for Schatten class membership of embeddings and composition operators on model spaces, especially for one-component inner functions, using extensions to Hardy-Smirnov spaces.

Findings

Characterization of Schatten membership via Nevanlinna counting function.

Reduction of the problem to Hardy-Smirnov space extensions for one-component inner functions.

Counterexamples showing the characterization does not extend to all functions.

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 $\phi$.