Edward Timko

(Department of Mathematics, University of Manitoba)

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