Trace ideal criteria for embeddings and composition operators on model spaces

Let Kϑ be a model space generated by an inner function ϑ. We study the Schatten class membership of composition operators Cφ:Kϑ→H2(D) with a holomorphic function φ:D→D, and, more generally, of embeddings Iμ:Kθ→L2(μ) with a positive measure μ in D¯. In the case of one-component inner functions ϑ we show that the problem can be reduced to the study of natural extensions of I and Cφ to the Hardy-Smirnov space E2(D) in some domain D⊃D. In particular, we obtain a characterization of Schatten membership of Cφ in terms of Nevanlinna counting function. By example this characterization does not hold true for general ϑ.