# Annihilating ideals and Agler--McCarthy spectral varieties in the bidisc

## TL;DR

This paper investigates spectral sets for pairs of commuting contractions on the bidisc, identifying conditions for minimal spectral sets related to distinguished varieties and analyzing the connection between annihilating ideals and spectral properties.

## Contribution

It establishes conditions for minimal spectral sets in terms of distinguished varieties and explores the relationship between annihilating ideals and spectral theory for operator pairs.

## Key findings

- Conditions for minimal spectral sets are identified.
- Relationship between annihilating ideals and spectral sets is characterized.
- A natural constrained isometric co-extension is constructed.

## Abstract

The closed unit bidisc $\overline{\mathbb{D}}^2$ is known to be a spectral set for any pair $(T_1,T_2)$ of commuting contractions. When each $T_i$ is pure and has finite defect, the pair admits a much smaller spectral set: the closure of a distinguished variety $V$ inside the bidisc $\mathbb{D}^2$. We find conditions on $(T_1,T_2)$ that guarantee that the closure of $V$ is a minimal spectral set. In addition, we examine the relationship between $V$ and the annihilating ideal $\text{Ann}(T_1,T_2)$ in $H^\infty(\mathbb{D}^2)$. While $V$ is typically strictly larger than the zero set of $\text{Ann}(T_1,T_2)$, we isolate a natural constrained isometric co-extension $(S_1,S_2)$ of $(T_1,T_2)$ whose Taylor spectrum is contained in $V$ and is closely linked to the so-called support of $\text{Ann}(T_1,T_2)$. We also characterize when $\text{Ann}(T_1,T_2)$ is the ideal of functions vanishing on the joint point spectrum of $(S_1^*,S_2^*)$.