A classification of $n$-tuples of commuting shifts of finite multiplicity

TL;DR

This paper classifies $n$-tuples of commuting shifts of finite multiplicity using algebraic invariants like ideals and their zero sets, extending the understanding of their equivalence classes.

Contribution

It introduces a classification framework for $n$-tuples of shifts based on their annihilator ideals and geometric properties of their zero sets.

Findings

Equivalence classes are determined by annihilator ideals and positive integers.

Classification reduces to prime ideal cases and their irreducible components.

Provides a complete invariant for the classification problem.

Abstract

Let $\mathbb{V}$ denote an $n$-tuple of shifts of finite multiplicity, and denote by $\mathrm{Ann}(\mathbb{V})$ the ideal consisting of polynomials $p$ in $n$ complex variables such that $p(\mathbb{V})=0$. If $\mathbb{W}$ on $\mathfrak{K}$ is another $n$-tuple of shifts of finite multiplicity, and there is a $\mathbb{W}$-invariant subspace $\mathfrak{K}'$ of finite codimension in $\mathfrak{K}$ so that $\mathbb{W}|\mathfrak{K}'$ is similar to $\mathbb{V}$, then we write $\mathbb{V}\lesssim \mathbb{W}$. If $\mathbb{W}\lesssim \mathbb{V}$ as well, then we write $\mathbb{W}\approx \mathbb{V}$. In the case that $\mathrm{Ann}(\mathbb{V})$ is a prime ideal we show that the equivalence class of $\mathbb{V}$ is determined by $\mathrm{Ann}(\mathbb{V})$ and a positive integer $k$. More generally, the equivalence class of $\mathbb{V}$ is determined by $\mathrm{Ann}(\mathbb{V})$ and an $m$-tuple of positive integers, where $m$ is the number of irreducible components of the zero set of $\mathrm{Ann}(\mathbb{V})$.