Abstract. As shown by Berger, Coburn and Lebow [1] and recently rediscovered by Douglas and Foiaş [3] every c.n.u. bi-isometry is unitarily equivalent with a certain isometric pair on $H^2({\msbm T},{\cal E})$ ($\cal E$ is a Hilbert space) defined in terms of two operators on ${\cal E}$, $U$ unitary and $P$ orthogonal projection. It is our aim in this paper to characterize the structure of a double commuting c.n.u. bi-isometry related to the Wold-Słociński decomposition [10] in terms of a representative $\{U,P\} $ of its complete unitary invariant. Some results concerning the minimal unitary extension are also given.

AMS Subject Classification (1991): 47A45

Received October 18, 1999. (Registered under 2763/2009.)