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.
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Nicholas Young
17 May 1999.