Shift operators, Cauchy integrals and approximations

This article consists of two connected parts. In the first part, we study the shift invariant subspaces in certain $\mathcal{P}^2(\mu)$-spaces, which are the closures of analytic polynomials in the Lebesgue spaces $\mathcal{L}^2(\mu)$ defined by a class of measures $\mu$ living on the closed unit disk $\overline{\mathbb{D}}$. The measures $\mu$ which occur in our study have a part on the open disk $\mathbb{D}$ which is radial and decreases at least exponentially fast near the boundary. Our focus is on those shift invariant subspaces which are generated by a bounded function in $H^\infty$. In this context, our results are definitive. We give a characterization of the cyclic singular inner functions by an explicit and readily verifiable condition, and we establish certain permanence properties of non-cyclic ones which are important in the applications. The applications take up the second part of the article. We prove that if a function $g \in \mathcal{L}^1(\mathbb{T})$ on the unit circle $\mathbb{T}$ has a Cauchy transform with Taylor coefficients of order $\mathcal{O}\big(\exp(-c \sqrt{n})\big)$ for some $c > 0$, then the set $U = \{x \in \mathbb{T} : |g(x)| > 0 \}$ is essentially open and $\log |g|$ is locally integrable on $U$. We establish also a simple characterization of analytic functions $b: \mathbb{D} \to \mathbb{D}$ with the property that the de Branges-Rovnyak space $\mathcal{H}(b)$ contains a dense subset of functions which, in a sense, just barely fail to have an analytic continuation to a disk of radius larger than 1. We indicate how close our results are to being optimal and pose a few questions.