Thomson decompositions of measures in the disk

Abstract

We study the classical problem of identifying the structure of P 2 ( μ ) \mathcal {P}^2(\mu ) , the closure of analytic polynomials in the Lebesgue space L 2 ( μ ) L^2(\mu ) of a compactly supported Borel measure μ \mu living in the complex plane. In his influential work, Thomson [Ann. of Math. (2) 133 (1991), pp. 477–507] showed that the space decomposes into a full L 2 L^2 -space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures μ \mu supported on the closed unit disk D ¯ \overline {\mathbb {D}} which have a part on the open disk D \mathbb {D} which is similar to the Lebesgue area measure, and a part on the unit circle T \mathbb {T} which is the restriction of the Lebesgue linear measure to a general measurable subset E E of T \mathbb {T} , we extend the ideas of Khrushchev and calculate the exact form of the Thomson decomposition of the space P 2 ( μ ) \mathcal {P}^2(\mu ) . It turns out that the space splits according to a certain natural decomposition of measurable subsets of T \mathbb {T} which we introduce. We highlight applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces.