Compact embeddings of model subspaces of the Hardy space

We study embeddings of model (star-invariant) subspaces $K^p_{\Theta}$ of the Hardy space $H^p$, associated with an inner function $\Theta$. We obtain a criterion for the compactness of the embedding of $K^p_{\Theta}$ into $L^p(\mu)$ analogous to the Volberg--Treil theorem on bounded embeddings and answer a question posed by Cima and Matheson. The proof is based on Bernstein inequalities for functions in $K^p_{\Theta}$. Also we study measures $\mu$ such that the embedding operator belongs to a Schatten--von Neumann ideal.