$\Gamma$-unitaries, dilation and a natural example

This note constructs an explicit normal boundary dilation for a commuting pair $(S,P)$ of bounded operators with the symmetrized bidisk $$\Gamma=\{(z_1+z_2,z_1z_2):|z_1|,|z_2| \leq 1\}$$ as a spectral set. Such explicit dilations had hitherto been constructed only in the unit disk \cite{sfr}, the unit bidisk \cite{ando} and in the tetrablock \cite{sir and me1}. The dilation is minimal and unique under a suitable condition. This paper also contains a natural example of a $\Gamma$-isometry. We compute its associated fundamental operator.