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

Abstract

This note constructs an explicit normal boundary dilation for a commuting pair (S,P)(S,P) of bounded operators with the symmetrized bidisk Γ={(z1+z2,z1z2):∣z1∣,∣z2∣≤1} Γ={(z1​+z2​,z1​z2​):∣z1​∣,∣z2​∣≤1} 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 ΓΓ-isometry. We compute its associated fundamental operator.