## 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.## References

- Jim Agler, Zinaida Lykova, and Nicholas J. Young,
*Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc*, arXiv:1603.04030 (2016). - Jim Agler and John E. McCarthy,
*Pick interpolation and Hilbert function spaces*, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR**1882259**, DOI 10.1090/gsm/044 - Jim Agler and John E. McCarthy,
*Distinguished varieties*, Acta Math.**194**(2005), no. 2, 133–153. MR**2231339**, DOI 10.1007/BF02393219 - Jim Agler and John E. McCarthy,
*Norm preserving extensions of holomorphic functions from subvarieties of the bidisk*, Ann. of Math. (2)**157**(2003), no. 1, 289–312. MR**1954268**, DOI 10.4007/annals.2003.157.289 - J. Agler and N. J. Young,
*The complex geodesics of the symmetrized bidisc*, Internat. J. Math.**17**(2006), no. 4, 375–391. MR**2220650**, DOI 10.1142/S0129167X06003564 - E. Amar,
*Ensembles d’interpolation dans le spectre d’une algèbre d’operateurs*, 1977. Thesis (Ph.D.)–University of Paris. - E. Amar,
*On the Toëplitz corona problem*, Publ. Mat.**47**(2003), no. 2, 489–496. MR**2006496**, DOI 10.5565/PUBLMAT_{4}7203_{1}1 - Mats Andersson, Mikael Passare, and Ragnar Sigurdsson,
*Complex convexity and analytic functionals*, Progress in Mathematics, vol. 225, Birkhäuser Verlag, Basel, 2004. MR**2060426**, DOI 10.1007/978-3-0348-7871-5 - Joseph A. Ball, Israel Gohberg, and Leiba Rodman,
*Interpolation of rational matrix functions*, Operator Theory: Advances and Applications, vol. 45, Birkhäuser Verlag, Basel, 1990. MR**1083145**, DOI 10.1007/978-3-0348-7709-1 - H. Cartan,
*Séminaire Henri Cartan 1951/2*, W. A. Benjamin, Inc., New York, 1967. - E. M. Chirka, P. Dolbeault, G. M. Khenkin, and A. G. Vitushkin,
*Introduction to complex analysis*, Springer-Verlag, Berlin, 1997. A translation of*Current problems in mathematics. Fundamental directions. Vol. 7*(Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985 [ MR0850489 (87f:32003)]; Translated by P. M. Gauthier; Translation edited by Vitushkin; Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [*Several complex variables. I*, Encyclopaedia Math. Sci., 7, Springer, Berlin, 1990; MR1043689 (90j:32003)]. MR**1734390**, DOI 10.1007/978-3-642-61525-2 - Brian Cole, Keith Lewis, and John Wermer,
*Pick conditions on a uniform algebra and von Neumann inequalities*, J. Funct. Anal.**107**(1992), no. 2, 235–254. MR**1172022**, DOI 10.1016/0022-1236(92)90105-R - Jean-Pierre Demailly,
*Analytic methods in algebraic geometry*, 2011, https://www-fourier.ujf-grenoble.fr/~demailly/books.html. - Robert C. Gunning,
*Introduction to holomorphic functions of several variables. Vol. I*, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. Function theory. MR**1052649** - Kunyu Guo, Hansong Huang, and Kai Wang,
*Retracts in polydisk and analytic varieties with the $H^\infty$-extension property*, J. Geom. Anal.**18**(2008), no. 1, 148–171. MR**2365671**, DOI 10.1007/s12220-007-9005-8 - Marek Jarnicki and Peter Pflug,
*Invariant distances and metrics in complex analysis*, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. MR**1242120**, DOI 10.1515/9783110870312 - Łukasz Kosiński,
*Three-point Nevanlinna-Pick problem in the polydisc*, Proc. Lond. Math. Soc. (3)**111**(2015), no. 4, 887–910. MR**3407188**, DOI 10.1112/plms/pdv045 - Łukasz Kosiński and Tomasz Warszawski,
*Lempert theorem for strongly linearly convex domains*, Ann. Polon. Math.**107**(2013), no. 2, 167–216. MR**3007944**, DOI 10.4064/ap107-2-5 - Łukasz Kosiński and Włodzimierz Zwonek,
*Nevanlinna-Pick problem and uniqueness of left inverses in convex domains, symmetrized bidisc and tetrablock*, J. Geom. Anal.**26**(2016), no. 3, 1863–1890. MR**3511461**, DOI 10.1007/s12220-015-9611-9 - László Lempert,
*La métrique de Kobayashi et la représentation des domaines sur la boule*, Bull. Soc. Math. France**109**(1981), no. 4, 427–474 (French, with English summary). MR**660145** - László Lempert,
*Intrinsic distances and holomorphic retracts*, Complex analysis and applications ’81 (Varna, 1981) Publ. House Bulgar. Acad. Sci., Sofia, 1984, pp. 341–364. MR**883254** - Takahiko Nakazi,
*Commuting dilations and uniform algebras*, Canad. J. Math.**42**(1990), no. 5, 776–789. MR**1080996**, DOI 10.4153/CJM-1990-041-1 - Peter Pflug and Włodzimierz Zwonek,
*Exhausting domains of the symmetrized bidisc*, Ark. Mat.**50**(2012), no. 2, 397–402. MR**2961329**, DOI 10.1007/s11512-011-0153-5 - Walter Rudin,
*Function theory in polydiscs*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0255841** - Joseph L. Taylor,
*The analytic-functional calculus for several commuting operators*, Acta Math.**125**(1970), 1–38. MR**271741**, DOI 10.1007/BF02392329 - P. J. Thomas,
*Appendix to norm preserving extensions of holomorphic functions from subvarieties of the bidisk*, Ann. of Math.**157**(2003), no. 1, 310–311. - Tavan T. Trent and Brett D. Wick,
*Toeplitz corona theorems for the polydisk and the unit ball*, Complex Anal. Oper. Theory**3**(2009), no. 3, 729–738. MR**2551635**, DOI 10.1007/s11785-008-0083-9

## 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