Page 94 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 94
COMPUTATIONAL ASPECTS OF COMMUTATIVE AND NONCOMMUTATIVE
POSITIVE POLYNOMIALS (MS-77)
The tracial moment problem on quadratic varieties
Aljaž Zalar, aljaz.zalar@fri.uni-lj.si
University of Ljubljana, Slovenia
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. The solution of the bivariate quartic tracial moment problem with
a nonsingular 7 × 7 moment matrix M2 whose columns are indexed by words of degree 2 was
established by Burgdorf and Klep. In the talk we will present the solution of the singular bivari-
ate tracial moment problem on quadratic varieties. In case the moment matrix Mn satisfies two
quadratic column relations the existence of the measure can be completely characterized by cer-
tain numerical conditions, while in the presence of one quadratic column relation the existence
of a representing measure is equivalent to the feasibility of certain linear matrix inequalities.
The atoms in the representing measure are always pairs of 2 × 2 matrices. This is joint work
with Abhishek Bhardwaj.
References
[1] A. Bhardwaj, A. Zalar, The singular bivariate quartic tracial moment problem. Complex
Anal. Oper. Theory 12:4 (2018), 1057–1142.
[2] A. Bhardwaj, A. Zalar, The tracial moment problem on quadratic varieties. J. Math. Anal.
Appl. 498 (2021).
92
POSITIVE POLYNOMIALS (MS-77)
The tracial moment problem on quadratic varieties
Aljaž Zalar, aljaz.zalar@fri.uni-lj.si
University of Ljubljana, Slovenia
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. The solution of the bivariate quartic tracial moment problem with
a nonsingular 7 × 7 moment matrix M2 whose columns are indexed by words of degree 2 was
established by Burgdorf and Klep. In the talk we will present the solution of the singular bivari-
ate tracial moment problem on quadratic varieties. In case the moment matrix Mn satisfies two
quadratic column relations the existence of the measure can be completely characterized by cer-
tain numerical conditions, while in the presence of one quadratic column relation the existence
of a representing measure is equivalent to the feasibility of certain linear matrix inequalities.
The atoms in the representing measure are always pairs of 2 × 2 matrices. This is joint work
with Abhishek Bhardwaj.
References
[1] A. Bhardwaj, A. Zalar, The singular bivariate quartic tracial moment problem. Complex
Anal. Oper. Theory 12:4 (2018), 1057–1142.
[2] A. Bhardwaj, A. Zalar, The tracial moment problem on quadratic varieties. J. Math. Anal.
Appl. 498 (2021).
92
