<p> We investigate the properties of finite tuples of commuting isometries that are constrained by a system of polynomial equations. More precisely, suppose <i>I</i> is an ideal in the ring of complex <i> n</i>-variable polynomials and that <i>I</i> determines an affine algebraic variety of dimension 1. Further, suppose that there are <i> n</i> commuting Hilbert space isometries <i>V</i><sub>1</sub>, . . . ,<i>V<sub>n</sub></i> with the property that <i>p</i>(<i> V<sub>1</sub></i>, . . . ,<i>V<sup>n</sup></i>) = 0 for each <i>p</i> in the ideal <i>I.</i> Because the <i> n</i>-tuple (<i>V</i><sub>1</sub>, . . . ,<i>V<sub>n</sub></i>) can be decomposed as a direct sum of an <i>n</i>-tuple of unitary operators and a completely non-unitary <i>n</i>-tuple, we assume that the unitary summand is trivial. Under these assumptions, we can decompose the <i>n</i>-tuple as a finite direct sum of <i>n</i>-tuples of the form (<i>T</i><sub>1</sub>, . . . ,<i>T<sub>n</sub></i>), where each <i>T<sub>i</sub></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 <i>V</i><sub>1</sub>, . . . ,<i>V<sub>n</sub></i> are generalized shifts of finite multiplicity. In this case we are able to classify such <i>n</i>-tuples up to something we term ‘virtual similarity’ using two pieces of data : the ideal of all polynomials p such that <i>p</i>(<i>V</i><sub> 1</sub>, . . . ,<i>V<sub>n</sub></i>) = 0 and a finite tuple of positive integers.</p><p>
| Identifer | oai:union.ndltd.org:PROQUEST/oai:pqdtoai.proquest.com:10288530 |
| Date | 17 August 2017 |
| Creators | Timko, Edward J. |
| Publisher | Indiana University |
| Source Sets | ProQuest.com |
| Language | English |
| Detected Language | English |
| Type | thesis |
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