$\Gamma$-Unitaries, Dilation and a Natural Example

Abstract

This note constructs an explicit normal boundary dilation for a commuting pair $(S,P)$ of bounded operators with the symmetrized bidisk

as a spectral set. Such explicit dilations had hitherto been constructed only in the unit disk [11], the unit bidisk [3] and in the tetrablock [6]. 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.