Papers › The singular bivariate quartic tracial moment problem
The singular bivariate quartic tracial moment problem
Abhishek Bhardwaj, Aljaž Zalar
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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 μ on ℝⁿ. In \cite{BK12} Burgdorf and Klep introduced its tracial analog, the truncated tracial moment problem, which replaces commuting variables with non-commuting ones and moments of μ 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 7×7 moment matrix ℳ₂ can be represented with tracial moments \cite{BK10,BK12}. In this article the case of singular ℳ₂ is studied. For ℳ₂ 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 ℳ₂ 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 ℳ₂ 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.
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