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.)
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