Embeddings of model subspaces of the Hardy space: compactness and Schatten–von Neumann ideals

We study properties of the embedding operators of model subspaces $K^p_{\Theta}$ (defined by inner functions) in the Hardy space $H^p$ (coinvariant subspaces of the shift operator). We find a criterion for the embedding of $K^p_{\Theta}$ in $L^p(\mu)$ to be compact similar to the Volberg–Treil theorem on bounded embeddings, and give a positive answer to a question of Cima and Matheson. The proof is based on Bernstein-type inequalities for functions in $K^p_{\Theta}$. We investigate measures $\mu$ such that the embedding operator belongs to some Schatten–von Neumann ideal.