A Classification of ${\varvec{n}}$-Tuples of Commuting Shifts of Finite Multiplicity

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 ${{\mathrm{Ann}}}(\mathbb {V})$.