The Taylor spectrum of pairs of isometries

TL;DR

This paper characterizes the Taylor spectrum of pairs of isometries represented by diagrams, revealing that in most cases, the spectrum is a proper subset of the bidisc with positive Lebesgue measure.

Contribution

It provides a complete description of the Taylor spectrum for pairs of isometries given by diagrams, including cases with non-trivial shift parts.

Findings

The Taylor spectrum is a proper subset of the bidisc in most cases.

The spectrum has positive Lebesgue measure in the interval (0, π^2).

Pairs of isometries with non-trivial shift parts exhibit specific spectral properties.

Abstract

In the paper we fully describe Taylor spectrum of pairs of isometries given by diagrams. In most cases both isometries in such pairs have non-trivial shift part and its Taylor spectrum is a proper subset (of Lebesgue measure in $(0,\pi^2)$) of the closed bidisc.