Abstract. It is proved that every semigroup of contractions with parameter on ${\msbm Q}_{+} \times{\msbm Q}_{+}$ or ${\msbm Q}_{+} \times{\msbm N}$ has a unitary dilation. The dilation result about ${\msbm Q}_{+} \times{\msbm Q}_{+}$ is used to obtain a new proof of the Slociński dilation theorem, which says that every strongly continuous semigroup of contractions, with parameter on ${\msbm R}_{+} \times{\msbm R}_{+}$, has a strongly continuous unitary dilation. The result about ${\msbm Q}_{+} \times{\msbm N}$ is used to obtain a new proof of the continuous version of the commutant lifting theorem.

AMS Subject Classification (1991): 47A20, 47D03

Keyword(s): unitary dilation, semigroup of contractions, commutant lifting

Received November 29, 2009, and in revised form April 13, 2010. (Registered under 6233/2009.)