Geodesics, Retracts, and the Norm-Preserving Extension Property in the Symmetrized Bidisc Article Swipe

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Jim Agler , Zinaida A. Lykova , N. J. Young · YOU? · · 2019 · Open Access · · DOI: https://doi.org/10.1090/memo/1242

A set $V$ in a domain $U$ in $\mathbb{C}^n$ has the {\em norm-preserving extension property} if every bounded holomorphic function on $V$ has a holomorphic extension to $U$ with the same supremum norm. We prove that an algebraic subset of the {\em symmetrized bidisc} \[ G := \{(z+w,zw):|z|<1, |w| < 1 \} \] has the norm-preserving extension property if and only if it is either a singleton, $G$ itself, a complex geodesic of $G$, or the union of the set $\{(2z,z^2): |z|<1\}$ and a complex geodesic of degree $1$ in $G$. We also prove that the complex geodesics in $G$ coincide with the nontrivial holomorphic retracts in $G$. Thus, in contrast to the case of the ball or the bidisc, there are sets in $G$ which have the norm-preserving extension property but are not holomorphic retracts of $G$. In the course of the proof we obtain a detailed classification of the complex geodesics in $G$ modulo automorphisms of $G$. We give applications to von Neumann-type inequalities for $Γ$-contractions (that is, commuting pairs of operators for which the closure of $G$ is a spectral set) and for symmetric functions of commuting pairs of contractive operators. We find three other domains that contain sets with the norm-preserving extension property which are not retracts: they are the spectral ball of $2\times 2$ matrices, the tetrablock and the pentablock. We also identify the subsets of the bidisc which have the norm-preserving extension property for symmetric functions.

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preprint

Language

en

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https://doi.org/10.1090/memo/1242

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green

Cited By

5

References

7

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20

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https://openalex.org/W2301999945

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https://openalex.org/W2301999945

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DOI

https://doi.org/10.1090/memo/1242

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Title

Geodesics, Retracts, and the Norm-Preserving Extension Property in the Symmetrized Bidisc

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preprint

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en

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2019

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2019-02-22

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Authors

Jim Agler, Zinaida A. Lykova, N. J. Young

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Landing page

https://doi.org/10.1090/memo/1242

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Yes

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green

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https://arxiv.org/pdf/1603.04030

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Concepts

Mathematics, Holomorphic function, Uniform norm, Norm (philosophy), Geodesic, Pure mathematics, Bounded function, Ball (mathematics), Combinatorics, Mathematical analysis, Law, Political science

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Cited by

5

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2018: 2, 2017: 3

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7

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20

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