Dissertation/Thesis Abstract

Polynomial Tuples of Commuting Isometries Constrained by 1-Dimensional Varieties
by Timko, Edward J., Ph.D., Indiana University, 2017, 70; 10288530
Abstract (Summary)

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 V1, . . . ,Vn with the property that p( V1, . . . ,Vn) = 0 for each p in the ideal I. Because the n-tuple (V1, . . . ,Vn) 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 (T1, . . . ,Tn), where each Ti 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 V1, . . . ,Vn 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, . . . ,Vn) = 0 and a finite tuple of positive integers.

Indexing (document details)
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
Subjects: Mathematics, Theoretical Mathematics
Keywords: Commuting isometries, Constrained operator family, Operator theory
Publication Number: 10288530
ISBN: 9780355069303
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