TY - JOUR

T1 - The tracial moment problem on quadratic varieties

AU - Bhardwaj, Abhishek

AU - Zalar, Aljaž

N1 - Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/6/1

Y1 - 2021/6/1

N2 - 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 7×7 moment matrix M2 whose columns are indexed by words of degree 2 was established by Burgdorf and Klep, while in our previous work we completely solved all cases with M2 of rank at most 5, split M2 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 M2 satisfying the third possible column relation, i.e., Y2=1+X2. 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 M2 satisfying Y2=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 M2 of rank 5 to general Mn, n≥2, 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.

AB - 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 7×7 moment matrix M2 whose columns are indexed by words of degree 2 was established by Burgdorf and Klep, while in our previous work we completely solved all cases with M2 of rank at most 5, split M2 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 M2 satisfying the third possible column relation, i.e., Y2=1+X2. 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 M2 satisfying Y2=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 M2 of rank 5 to general Mn, n≥2, 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.

KW - Affine linear transformations

KW - Flat extensions

KW - Moment matrix

KW - Noncommutative polynomial

KW - Truncated moment problem

UR - http://www.scopus.com/inward/record.url?scp=85099433077&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2021.124936

DO - 10.1016/j.jmaa.2021.124936

M3 - Article

SN - 0022-247X

VL - 498

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

M1 - 124936

ER -