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

Tirthankar Bhattacharyya, Shubham Rastogi, Vijaya Kumar U

Published 2021-05-05Version 1

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 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 spectra in each of the cases above. Moreover, in each case, we also point out at which stage of the Koszul complex the exactness breaks. With $(V_1, V_2)$, a pair of operator valued functions $\phi_1,\phi_2$ can be canonically associated. In certain cases, $\overline{\cup_{z\in\D} \sigma(\phi_1(z),\phi_2(z))}$ is the same as $\sigma(V_1,V_2)$ and in certain cases, it is smaller. All such cases are described. A major contribution of this note is to figure out the fundamental pair of commuting isometries with negative defect. We construct this pair of commuting isometries on the Hardy space of the bidisc (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).