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Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc

Lookup NU author(s): Professor Jim Agler, Dr Zinaida LykovaORCiD, Professor Nicholas Young



This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND).


A set V in a domain U in Cn has the norm-preserving extension property ifevery bounded holomorphic function on V has a holomorphic extension to U withthe same supremum norm. We prove that an algebraic subset of the symmetrizedbidiscG def = f(z + w; zw) : jzj < 1; jwj < 1ghas 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 f(2z; z2) : jzj < 1g anda complex geodesic of degree 1 in G. We also prove that the complex geodesics inG coincide with the nontrivial holomorphic retracts in G. Thus, in contrast to thecase of the ball or the bidisc, there are sets in G which have the norm-preservingextension property but are not holomorphic retracts of G. In the course of theproof we obtain a detailed classi cation of the complex geodesics in G moduloautomorphisms of G. We give applications to von Neumann-type inequalities forô€€€-contractions (that is, commuting pairs of operators for which the closure of Gis a spectral set) and for symmetric functions of commuting pairs of contractiveoperators. We nd three other domains that contain sets with the norm-preservingextension property which are not retracts: they are the spectral ball of 22 matrices,the tetrablock and the pentablock. We also identify the subsets of the bidiscwhich have the norm-preserving extension property for symmetric functions

Publication metadata

Author(s): Agler J, Lykova Z, Young N

Publication type: Article

Publication status: Published

Journal: Memoirs of the American Mathematical Society

Year: 2019

Volume: 258

Issue: 1242

Pages: 1-108

Print publication date: 01/03/2019

Online publication date: 22/02/2019

Acceptance date: 21/08/2016

Date deposited: 24/08/2016

ISSN (print): 0065-9266

ISSN (electronic): 1947-6221

Publisher: American Mathematical Society


DOI: 10.1090/memo/1242

Notes: arxiv: 1603.04030 [math.CV] Print ISBN: 9781470435493 Online ISBN: 9781470450755


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