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We investigate the properties of finite tuples of commuting isometries that are constrained by a system of polynomial equations. More precisely, suppose I is an ideal in the ring of complex n-variable polynomials and that I determines an affine algebraic variety of dimension 1. Further, suppose that there are n commuting Hilbert space isometries V_{1}, . . . ,V_{n} with the property that p( V_{1}, . . . ,V^{n}) = 0 for each p in the ideal I. Because the n-tuple (V_{1}, . . . ,V_{n}) can be decomposed as a direct sum of an n-tuple of unitary operators and a completely non-unitary n-tuple, we assume that the unitary summand is trivial. Under these assumptions, we can decompose the n-tuple as a finite direct sum of n-tuples of the form (T_{1}, . . . ,T_{n}), where each T_{i} either is multiplication by a scalar or is unitarily equivalent to a unilaterial shift of some multiplicity. We then focus on the special case in which V_{1}, . . . ,V_{n} are generalized shifts of finite multiplicity. In this case we are able to classify such n-tuples up to something we term ‘virtual similarity’ using two pieces of data : the ideal of all polynomials p such that p(V_{ 1}, . . . ,V_{n}) = 0 and a finite tuple of positive integers.
Advisor: | Bercovici, Hari |
Commitee: | Levenberg, Norman, Snyder, Noah, Torchinsky, Alberto |
School: | Indiana University |
Department: | Mathematics |
School Location: | United States -- Indiana |
Source: | DAI-B 78/12(E), Dissertation Abstracts International |
Source Type: | DISSERTATION |
Subjects: | Mathematics, Theoretical Mathematics |
Keywords: | Commuting isometries, Constrained operator family, Operator theory |
Publication Number: | 10288530 |
ISBN: | 9780355069303 |