Thomson decompositions of measures in the disk

TL;DR

This paper extends Thomson's decomposition theory for measures supported on the disk and circle, providing explicit decompositions for certain measures and exploring applications to Cauchy operators and de Branges-Rovnyak spaces.

Contribution

It introduces a new decomposition of measures on the disk and circle, explicitly calculating the Thomson decomposition for these measures.

Findings

Explicit Thomson decomposition formulas for measures supported on the disk and circle.

Identification of measure decompositions related to subsets of the unit circle.

Applications demonstrated in Cauchy integral operator theory and de Branges-Rovnyak spaces.

Abstract

We study the classical problem of identifying the structure of $P^2(\mu)$, the closure of analytic polynomials in the Lebesgue space $L^2(\mu)$ of a compactly supported Borel measure $\mu$ living in the complex plane. In his influential work, Thomson showed that the space decomposes into a full $L^2$-space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures $\mu$ supported on the closed unit disk $\overline{\mathbb{D}}$ which have a part on the open disk $\mathbb{D}$ which is similar to the Lebesgue area measure, and a part on the unit circle $\mathbb{T}$ which is the restriction of the Lebesgue linear measure to a general measurable subset $E$ of $\mathbb{T}$, we extend the ideas of Khrushchev and calculate the exact form of the Thomson decomposition of the space $P^2(\mu)$. It turns out that the space splits according to a certain decomposition of measurable subsets of $\mathbb{T}$ which we introduce. We highlight applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces.