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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