On polynomial n-tuples of commuting isometries

2016/04/21 by Timko, Edward J.
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1604.06364

Abstract

We extend some of the results of Agler, Knese, and McCarthy [1] to n-tuples of commuting isometries for n>2. Let \mathbbV=(V₁,…,V_n) be an n-tuple of a commuting isometries on a Hilbert space and let Ann(\mathbbV) denote the set of all n-variable polynomials p such that p(\mathbbV)=0. When Ann(\mathbbV) defines an affine algebraic variety of dimension 1 and \mathbbV is completely non-unitary, we show that \mathbbV decomposes as a direct sum of n-tuples \mathbbW=(W₁,…,W_n) with the property that, for each i=1,…,n, W_i is either a shift or a scalar multiple of the identity. If \mathbbV is a cyclic n-tuple of commuting shifts, then we show that \mathbbV is determined by Ann(\mathbbV) up to near unitary equivalence, as defined in [1].

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