# Journal of Operator Theory

Volume 77, Issue 2, Spring 2017 pp. 391-420.

On polynomial $n$-tuples of commuting isometries

**Authors**:
Edward J. Timko

**Author institution:** Department of Mathematics, Indiana University,
Bloomington, IN, 47401, U.S.A.

**Summary: ** We extend some of the results of Agler, Knese, and
McCarthy in
\textit{J. Operator Theory} \textbf{67}(2012), 215--236, 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
$\mathrm{Ann}(\VV)$ denote the set of all $n$-variable polynomials $p$ such
that $p(\mathbb{V})=0$. When $\mathrm{Ann}(\mathbb{V})$ 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 $\mathrm{Ann}(\mathbb{V})$ up to near unitary equivalence, as
defined in \textit{J. Operator Theory} \textbf{67}(2012),
$\mbox{215--236}$.

**DOI: **http://dx.doi.org/10.7900/jot.2016apr24.2122

**Keywords: ** polynomial, commuting, isometries

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