Abstract. In this paper we study the relation of Harnack domination for two Hilbert space contractions, defined relative to an operator acting between the two spaces. We construct the sequence of $n$-step intertwining extensions for the underlying operator and we show that this sequence gives, to limit, the intertwining extension of the involved operator. This fact leads to a norm characterization of the relation of Harnack domination, which is used to derive some results in the case when one of the contractions is uniformly stable. As applications, we obtain versions of the commutant dilation theorem of Sz.-Nagy--Foiaş for commuting pairs of contractions, which are related to other similar results.

AMS Subject Classification (1991): 47A20, 47A45

Keyword(s): Harnack domination, dilation, intertwining extension

Received October 25, 2005, and in revised form February 1, 2006. (Registered under 5923/2009.)