# Edward Timko

## A Classification of $$n$$-tuples of Commuting Isometries

Date Tuesday, November 21, 2017

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