The Singular Bivariate Quartic Tracial Moment Problem
The (classical) truncated moment problem, extensively studied by Curto and Fialkow, asks to characterize when a finite sequence of real numbers indexes by words in commuting variables can be represented with moments of a positive Borel measure \(\mu \) on \(\mathbb R^n\). Burgdorf and Klep (J Oper Theory 68:141–163, 2012) introduced its tracial analog, the truncated tracial moment problem, which replaces commuting variables with non-commuting ones and moments of \(\mu \) with tracial moments of matrices. In the bivariate quartic case, where indices run over words in two variables of degree at most four, every sequence with a positive definite \(7\times 7\) moment matrix \(\mathcal M_2\) can be represented with tracial moments (Burgdorf and Klep in C R Math Acad Sci Paris 348:721–726, 2010, 2012). In this article the case of singular \(\mathcal M_2\) is studied. For \(\mathcal M_2\) of rank at most 5 the problem is solved completely; namely, concrete measures are obtained whenever they exist and the uniqueness question of the minimal measures is answered. For \(\mathcal M_2\) of rank 6 the problem splits into four cases, in two of which it is equivalent to the feasibility problem of certain linear matrix inequalities. Finally, the question of a flat extension of the moment matrix \(\mathcal M_2\) is addressed. While this is the most powerful tool for solving the classical case, it is shown here by examples that, while sufficient, flat extensions are mostly not a necessary condition for the existence of a measure in the tracial case.
KeywordsTruncated moment problem Non-commutative polynomial Moment matrix Affine linear transformations Flat extensions
Mathematics Subject ClassificationPrimary 47A57 15A45 13J30 Secondary 11E25 44A60 15-04
Part of this paper was written at The University of Auckland under the supervision of Igor Klep who was the MSc supervisor of the first author and the PhD co-supervisor of the second author. Both authors wish to thank him for introducing us to this topic, the many insightful and inspiring discussions and support throughout the research. We are also thankful to two anonymous referees for useful comments and suggestions for improvements of the paper.
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