Commuting and Doubly Commuting Pairs of Isometries

Abstract

We review the model theory for pairs of commuting isometries developed by Berger, Coburn and Lebow in 1978. We exhibit three ways to arrive at the models. Using this model theory, we present a new proof of Słociński’s Wold-type decomposition for doubly commuting pairs of isometries. Characterizations of joint invariant subspaces of commuting pairs of isometries are also presented. We also show how the recent development on operators associated with the tetrablock can be used to derive the Berger-Coburn-Lebow model for commuting isometries. Several examples are considered to illustrate the model theory.