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.
|Commitee:||Levenberg, Norman, Snyder, Noah, Torchinsky, Alberto|
|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|
Copyright in each Dissertation and Thesis is retained by the author. All Rights Reserved
The supplemental file or files you are about to download were provided to ProQuest by the author as part of a
dissertation or thesis. The supplemental files are provided "AS IS" without warranty. ProQuest is not responsible for the
content, format or impact on the supplemental file(s) on our system. in some cases, the file type may be unknown or
may be a .exe file. We recommend caution as you open such files.
Copyright of the original materials contained in the supplemental file is retained by the author and your access to the
supplemental files is subject to the ProQuest Terms and Conditions of use.
Depending on the size of the file(s) you are downloading, the system may take some time to download them. Please be