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Transactions of the American Mathematical Society

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

 
 

 

Norm preserving extensionsof bounded holomorphic functions


Authors: Łukasz Kosiński and John E. McCarthy
Journal: Trans. Amer. Math. Soc. 371 (2019), 7243-7257
MSC (2010): Primary 32D15, 47A57
DOI: https://doi.org/10.1090/tran/7597
Published electronically: October 5, 2018
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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
Email: lukasz.kosinski@uj.edu.pl

John E. McCarthy
Affiliation: Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, 63130 Missouri
Email: mccarthy@wustl.edu

DOI: https://doi.org/10.1090/tran/7597
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
Article copyright: © Copyright 2018 American Mathematical Society