On polynomial $n$-tuples of commuting isometries

TL;DR

This paper extends results on polynomial tuples of commuting isometries to higher dimensions, characterizing their structure when the annihilating polynomial set defines a 1-dimensional algebraic variety and the tuple is non-unitary.

Contribution

It generalizes previous work to n-tuples of commuting isometries for n>2, providing a structural decomposition and uniqueness results for cyclic shift tuples.

Findings

Decomposition of non-unitary n-tuples into shifts and scalar multiples of the identity.

Characterization of cyclic shift n-tuples by their annihilating polynomial set.

Extension of prior results to higher-dimensional commuting isometries.

Abstract

We extend some of the results of Agler, Knese, and McCarthy [1] to $n$-tuples of commuting isometries for $n>2$. Let $\mathbb{V}=(V_1,\dots,V_n)$ be an $n$-tuple of a commuting isometries on a Hilbert space and let Ann$(\mathbb{V})$ denote the set of all $n$-variable polynomials $p$ such that $p(\mathbb{V})=0$. When Ann$(\mathbb{V})$ defines an affine algebraic variety of dimension 1 and $\mathbb{V}$ is completely non-unitary, we show that $\mathbb{V}$ decomposes as a direct sum of $n$-tuples $\mathbb{W}=(W_1,\dots,W_n)$ with the property that, for each $i=1,\dots,n$, $W_i$ is either a shift or a scalar multiple of the identity. If $\mathbb{V}$ is a cyclic $n$-tuple of commuting shifts, then we show that $\mathbb{V}$ is determined by Ann$(\mathbb{V})$ up to near unitary equivalence, as defined in [1].