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

TL;DR

This paper investigates the structure and joint spectrum of pairs of commuting isometries using defect operators, providing new structure theorems, spectrum calculations, and identifying the fundamental pair with negative defect.

Contribution

It introduces new structure theorems for commuting isometries based on defect operators and characterizes the fundamental pair with negative defect.

Findings

Derived structure theorems for various defect cases.

Computed joint spectra in different defect scenarios.

Identified the fundamental pair with negative defect on the Hardy space.

Abstract

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 $(\varphi_1,\varphi_2)$ 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 $\sigma(V_1,V_2)=\bar{\cup_{z\in\D} \sigma(\varphi_1(z),\varphi_2(z))}.$ 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.