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The Tracial Moment Problem on Quadratic Varieties

The truncated moment problem asks to characterize finite sequences of real numbers that are the moments of a positive Borel measure on Rn. Its tracial analog is obtained by integrating traces of symmetric matrices and is the main topic of this article. The solution of the bivariate quartic tracial moment problem with a nonsingular (7x7) moment matrix M(2) whose columns are indexed by words of degree 2 was established by Burgdorf and Klep, while in our previos work we completely solved all cases with M(2) of rank at most 5, split M(2) of rank 6 into four possible cases according to the column relation satisfied and solved two of them. Our first main result in this article is the solution for M(2) satisfying the third possible column relation, i.e., Y^2 = 1 + X^2. Namely, the existence of a representing measure is equivalent to the feasibility problem of certain linear matrix inequalities. The second main result is a thorough analysis of the atoms in the measure for M(2) satisfying Y^2 = 1, the most demanding column relation. We prove that size 3 atoms are not needed in the representing measure, a fact proved to be true in all other cases. The third main result extends the solution for M(2) of rank 5 to general M(n), n>1, with two quadratic column relations. The main technique is the reduction of the problem to the classical univariate truncated moment problem, an approach which applies also in the classical truncated moment problem. Finally, our last main result, which demonstrates this approach, is a simplification of the proof for the solution of the degenerate truncated hyperbolic moment problem first obtained by Curto and Fialkow.