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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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

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