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 = (V1,...,Vn ) 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 th...

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Bibliographic Details
Published in:Journal of operator theory 2017-03, Vol.77 (2), p.391-420
Main Author: TIMKO, EDWARD J.
Format: Article
Language:English
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Summary: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 = (V1,...,Vn ) 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 = (W1,...,Wn ) with the property that, for each i = 1,...,n, Wi 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.
ISSN:0379-4024
1841-7744
DOI:10.7900/jot.2016apr24.2122