Hilbert spaces of analytic functions with a contractive backward shift

We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift f(z)↦[Formula presented] is a contraction on the space. We present a model for this operator and use it to prove the surprising result that functions which extend continuously to the closure of the disk are dense in the space. This has several applications, for example we can answer a question regarding reverse Carleson embeddings for these spaces. We also identify a large class of spaces which are similar to the de Branges–Rovnyak spaces and prove some results which are new even in the classical case.