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We characterize those commuting pairs of operators on Hilbert 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(Gamma) 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.
Author(s): Agler J, Young NJ
Publication type: Article
Publication status: Published
Journal: Proceedings of the Edinburgh Mathematical Society
Year: 2000
Volume: 43
Pages: 195-210
ISSN (print): 0308-2105
ISSN (electronic): 1473-7124
Publisher: The RSE Scotland Foundation
URL: http://dx.doi.org/10.1017/S0013091500020812
DOI: 10.1017/S0013091500020812
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