Hilbert spaces of analytic functions with a contractive backward shift

TL;DR

This paper studies Hilbert spaces of analytic functions with a contractive backward shift, providing a model for the operator, showing density of continuous functions, and exploring applications like reverse Carleson embeddings and connections to de Branges-Rovnyak spaces.

Contribution

It introduces a new model for the backward shift operator in these spaces and demonstrates the density of functions extending continuously to the boundary, with applications to embeddings and space classifications.

Findings

Functions continuous on the closure are dense in the space.

Answered a question on reverse Carleson embeddings.

Identified spaces similar to de Branges-Rovnyak spaces.

Abstract

We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift $f(z) \mapsto \frac{f(z)-f(0)}{z}$ 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.