[Cancelled] The norm-preserving extension property in the symmetrized bidisc G and von Neumann-type inequalities for G-contractions

Zinaida Lykova (Newcastle University)

Thursday 19th March, 2020 16:00-17:00 Maths 311B

Abstract

A set V in a domain U in â„‚ⁿ has the norm-preserving extension property if every bounded holomorphic function on V has a holomorphic extension to U with the same supremum norm. We describe all algebraic subsets of the symmetrized bidisc

G =^def {(z+w,zw) : |z| < 1, |w| < 1}

which have the norm-preserving extension property. 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. 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.

The talk is based on joint work with Jim Agler and Nicholas Young.

[1] Jim Agler, Zinaida A. Lykova and N. J. Young, Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc, Memoirs of the American Mathematical Society, 2019, v. 258, no. 1242, 108pp. https://doi.org/10.1090/memo/1242

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