Trace Positive, Non-commutative Polynomials and the Truncated Moment Problem

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dc.contributor.advisor Klep, I en
dc.contributor.author Bhardwaj, Abhishek en
dc.date.accessioned 2016-09-08T00:13:11Z en
dc.date.issued 2016 en
dc.identifier.uri http://hdl.handle.net/2292/30249 en
dc.description.abstract The (multivariate) truncated moment problem is an important question in analysis with applications to mathematical physics, probability theory, etc. In the 1990’s Curto & Fialkow began their expedition of solving the truncated moment problem. It has since been studied with a variety of motivations, due to its wide ranging applications, such as to multivariate integral computations (in physics, statistics, etc.), option pricing (in finance) and optimization (in mathematics). The tracial moment problem is a non-commutative analogue of the classical moment problem. A sequence of real numbers indexed by words in non-commuting variables, invariant under cyclic permutations is called a tracial sequence. In this work we study conditions for when a tracial sequence is given by the tracial moments of some matrices, focusing on the bivariate problem in low degrees. We present sufficient conditions for Curto and Yoo’s construction of an explicit representing measure on the classical quartic binary moment problem, to hold in the tracial analogue. To each tracial sequence one can associate a multivariate Hankel matrix. If the sequence is given by moments, this matrix is positive semi-definite, but not vice-versa. In the bivariate quartic case this matrix is (7 7). It is known that positive definiteness of this matrix implies the existence of a representing measure. Here we also present a comprehensive analysis on the column relations in the Hankel matrix for lower rank cases in an attempt to find or disprove the existence of a representing measure. en
dc.publisher ResearchSpace@Auckland en
dc.relation.ispartof Masters Thesis - University of Auckland en
dc.relation.isreferencedby UoA99264878797102091 en
dc.rights Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher. en
dc.rights.uri https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm en
dc.rights.uri http://creativecommons.org/licenses/by-nc-sa/3.0/nz/ en
dc.title Trace Positive, Non-commutative Polynomials and the Truncated Moment Problem en
dc.type Thesis en
thesis.degree.discipline Pure Mathematics en
thesis.degree.grantor The University of Auckland en
thesis.degree.level Masters en
dc.rights.holder Copyright: The author en
dc.rights.accessrights http://purl.org/eprint/accessRights/OpenAccess en
pubs.elements-id 540862 en
pubs.record-created-at-source-date 2016-09-08 en


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