The Singular Bivariate Quartic Tracial Moment Problem

Abstract

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 on . 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 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 moment matrix 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 is studied. For 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 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 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.