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 $C_{\varphi }:K_{\theta }\to H^2(\mathbb{D})$ with a holomprphic function $\varphi :\mathbb{D}\to \mathbb{D}$. 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 $C_{\varphi }$ 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 $C_{\varphi }$ in terms of Nevanlinna counting function. By example this characterization does not hold true for general $\varphi$.

20 pages

References in corpus (2)

• Compact embeddings of model subspaces of the Hardy space
• Some examples of compact composition operators on $H^2$

Cited by in corpus (3)

• Duality theorems for coinvariant subspaces of $H^1$
• A Characterization of One-component Inner Functions
• Volterra type integral and composition operators on model spaces