Shift operators, Cauchy integrals and approximations

TL;DR

This paper explores shift invariant subspaces in certain measure-defined function spaces, characterizes cyclic singular inner functions, and links the decay rate of Cauchy transform Taylor coefficients to the properties of functions on the unit circle.

Contribution

It provides a definitive characterization of cyclic singular inner functions and establishes new links between Cauchy transform decay rates and function properties on the unit circle.

Findings

Characterization of cyclic singular inner functions.

Decay rate of Cauchy transform coefficients implies local integrability.

Description of functions with dense subsets in de Branges-Rovnyak spaces.

Abstract

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.