From Stinespring dilation to Sz.-Nagy dilation on the symmetrized bidisc and operator models

Abstract

We provide an explicit normal distinguished boundary dilation to a pair of commuting operators $(S,P)$ having the closed symmetrized bidisc $\Gamma$ as a spectral set. This is called Sz.-Nagy dilation of $(S,P)$. The operator pair that dilates $(S,P)$ is obtained by an application of Stinespring dilation of $(S,P)$ given by Agler and Young. We further prove that the dilation is minimal and the dilation space is no bigger than the dilation space of the minimal unitary dilation of the contraction $P$. We also describe model space and model operators for such a pair $(S,P)$.