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Operators having the symmetrized bidisc as a spectral set

Published online by Cambridge University Press:  20 January 2009

J. Agler  and
N. J. Young
J. Agler
Affiliation:
Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA
N. J. Young
Affiliation:
Department of Mathematics, University of Newcastle, Newcastle-upon-Tyne NE1 7RU, UK
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Abstract

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We characterize those commuting pairs of operators on Hubert space that have the symmetrized bidisc as a spectral set in terms of the positivity of a hermitian operator pencil (without any assumption about the joint spectrum of the pair). Further equivalent conditions are that the pair has a normal dilation to the distinguished boundary of the symmetrized bidisc, and that the pair has the symmetrized bidisc as a complete spectral set. A consequence is that every contractive representation of the operator algebra A(Γ) of continuous functions on the symmetrized bidisc analytic in the interior is completely contractive. The proofs depend on a polynomial identity that is derived with the aid of a realization formula for doubly symmetric hereditary polynomials, which are positive on commuting pairs of contractions.

Keywords

spectral setcommuting operatorshereditary functions interpolationcommutant lifting theorem
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 1 , February 2000 , pp. 195 - 210
Copyright
Copyright © Edinburgh Mathematical Society 2000

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