ON POLYNOMIAL n-TUPLES OF COMMUTING ISOMETRIES

We extend some of the results of Agler, Knese, and McCarthy in J. Operator Theory 67(2012), 215-236, to n-tuples of commuting isometries for n > 2. Let V = (V-1,...,V-n) be an n-tuple of a commuting isometries on a Hilbert space and let Ann (V) denote the set of all n-variable polynomials p such that p (V) = 0. When Ann (V) defines an affine algebraic variety of dimension 1 and V is completely non-unitary, we show that V decomposes as a direct sum of n-tuples W = (W-1,...,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 V is a cyclic n-tuple of commuting shifts, then we show that V is determined by Ann (V) up to near unitary equivalence, as defined in J. Operator Theory 67(2012), 215-236.