Thomson decompositions of measures in the disk
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- by Bartosz Malman;
- Trans. Amer. Math. Soc. 376 (2023), 8529-8552
- DOI: https://doi.org/10.1090/tran/9018
- Published electronically: August 29, 2023
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Abstract:
We study the classical problem of identifying the structure of $\mathcal {P}^2(\mu )$, 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 [Ann. of Math. (2) 133 (1991), pp. 477–507] 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 $\mathcal {P}^2(\mu )$. It turns out that the space splits according to a certain natural 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.References
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Bibliographic Information
- Bartosz Malman
- Affiliation: Mälardalen University, Västerås, Sweden; and KTH Royal Institute of Technology, Stockholm, Sweden
- MR Author ID: 1231604
- Email: bartosz.malman@mdu.se, malman@kth.se
- Received by editor(s): October 26, 2022
- Received by editor(s) in revised form: June 6, 2023
- Published electronically: August 29, 2023
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 8529-8552
- MSC (2020): Primary 32E30; Secondary 30E10
- DOI: https://doi.org/10.1090/tran/9018
- MathSciNet review: 4669304