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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Norm preserving extensions of bounded holomorphic functions
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by Łukasz Kosiński and John E. McCarthy PDF
Trans. Amer. Math. Soc. 371 (2019), 7243-7257 Request permission

Abstract:

A relatively polynomially convex subset $V$ of a domain $\Omega$ has the extension property if for every polynomial $p$ there is a bounded holomorphic function $\phi$ on $\Omega$ that agrees with $p$ on $V$ and whose $H^\infty$ norm on $\Omega$ equals the sup-norm of $p$ on $V$. We show that if $\Omega$ is either strictly convex or strongly linearly convex in $\mathbb {C}^2$, or the ball in any dimension, then the only sets that have the extension property are retracts. If $\Omega$ is strongly linearly convex in any dimension and $V$ has the extension property, we show that $V$ is a totally geodesic submanifold. We show how the extension property is related to spectral sets.
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Additional Information
  • Łukasz Kosiński
  • Affiliation: Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Krakow, Poland
  • MR Author ID: 825007
  • Email: lukasz.kosinski@uj.edu.pl
  • John E. McCarthy
  • Affiliation: Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, 63130 Missouri
  • MR Author ID: 271733
  • ORCID: 0000-0003-0036-7606
  • Email: mccarthy@wustl.edu
  • Received by editor(s): August 5, 2017
  • Received by editor(s) in revised form: January 18, 2018, and February 28, 2018
  • Published electronically: October 5, 2018
  • Additional Notes: The first author was partially supported by the NCN Grant UMO-2014/15/D/ST1/01972
    The second author was partially supported by National Science Foundation Grant DMS 156243
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 7243-7257
  • MSC (2010): Primary 32D15, 47A57
  • DOI: https://doi.org/10.1090/tran/7597
  • MathSciNet review: 3939576