Abstract. We consider the classification, up to unitary equivalence, of commuting $n$-tuples $(V_{1},V_{2},\ldots,V_{n})$ of isometries on a Hilbert space. As in earlier work by Berger, Coburn, and Lebow, we start by analyzing the Wold decomposition of $V=V_{1}V_{2}\cdots V_{n}$, but unlike their work, we pay special attention to the case when $\ker V^*$ is of finite dimension. We give a complete classification of $n$-tuples for which $V$ is a pure isometry of multiplicity $n$. It is hoped that deeper analysis will provide a classification whenever $V$ has finite multiplicity. Further, we identify a pivotal operator in the case $n=2$ which captures many of the properties of a bi-isometry.

AMS Subject Classification (1991): 47A13, 47A45

Received June 15, 2006, and in revised form September 7, 2006. (Registered under 5941/2009.)