Operators having the symmetrized bidisc as a spectral set

J. Agler and N. J. Young

Status

Appeared in Proc. Edin. Math. Soc. vol 43 (2000)195-210.

Abstract

We characterise those commuting pairs of operators on Hilbert space which 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 {\em complete} spectral set. A consequence is that every contractive representation of the operator algebra $A(\Gamma)$ of continuous functions on the symmetrized bidisc anlaytic in the interior is completely contractive. The proofs depend on a polynomial identity which is derived with the aid of a realization formula for doubly symmetric hereditary polynomials which are positive on commuting pairs of contractions. \


Nicholas Young
17 May 1999
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