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On the structure and the joint spectrum of a pair of commuting isometries

The study of a pair $(V_1,V_2)$ of commuting isometries is a classical theme. We shine new light on it by using the defect operator. In the cases when the defect operator is zero or positive or negative, or the difference of two mutually orthogonal projections with ranges adding up to $\ker (V_1V_2)^*$, we write down structure theorems for $(V_1,V_2)$. The structure theorems allow us to compute the joint spectrum in each of the cases above. Moreover, in each case, we also point out at which stage of the Koszul complex the exactness breaks. A pair of operator valued functions $(φ_1,φ_2)$ is canonically associated by Berger, Coburn and Lebow with $(V_1,V_2)$. If $(V_1,V_2)$ is a pure pair, then in each case above we show that $σ(V_1,V_2)=\bar{\cup_{z\in\D} σ(φ_1(z),φ_2(z))}.$ It has been known that the fundamental pair of commuting isometries with positive defect is the pair of multiplication operators by the coordinate functions on the Hardy space of the bidisc. A major contribution of this note is to figure out the fundamental pair of commuting isometries with negative defect. This pair of commuting isometries is constructed on the Hardy space of the bidisc.