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 µ on R n . In [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 M 2 can be represented with tracial moments [BK10,BK12]. In this article the case of singular M 2 is studied. For M 2 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 M 2 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 M 2 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. INTRODUCTION1.1. Context. The Moment problem (MP) is a classical question in analysis and concerns the existence of a positive Borel measure µ supported on a subset K of R n , representing a given sequence of real or complex numbers indexed by monomials as the integration of the corresponding monomials w.r.t. µ; nice expositions on the MP are [Akh65, KN77]. The solution to the MP on R n is given by Haviland's theorem [Hav35], which establishes the duality with positive polynomials and relates the MP to real algebraic geometry (RAG). One of the cornerstones of RAG is the celebrated Schmüdgen theorem [Sch91], which solves the problem on compact basic closed semialgebraic sets and is the beginning of extensive research of the MP in RAG; we refer the reader to [Put93, PV99, DP01, PS01, KM02, Lau05, Lau09, Mar08, Las09] and the references therein for further details. Another important aspect of the MP is uniqueness of the representing measures. For compact sets the measure is unique (see e.g., [Mar08]), while for noncompact sets, the question of uniqueness is highly nontrivial (see [PS06,PS08]). There are various generalizations of the MP. Functional analysis studies various versions of matrix and operator MPs; see [Kre49, Kov83, AV03, Vas03, BW11, CZ12, KW13] and references therein. The quantum MP from quantum physics is considered in [DLTW08]. The rational MP, which extends Schmüdgen theorem from the polynomial algebra to its localizations, is solved in [CMN11], while [GKM16] investigates the MP for the polynomial algebra in infinitely many variables. The MP on semialgebraic sets of generalized function is considered in [IKR14]. The beginning of free RAG is the sol...

The Tracial Moment Problem on Quadratic Varieties

“…Theorem 3.1 (along with the others from [BZ18]) provides with a new computational method for testing the existence of a measure. While searching for a flat extension from M 2 to M 3 is reasonable, this approach quickly becomes intractable if M 2 admits positive extensions M k , for a large k, which then admits a flat extension to M k+1 .…”

“…Inspired by the work of Burgdorf and Klep and Curto and Fialkow, we studied the bivariate quartic TTMP having a singular (7 × 7) tracial moment matrix M 2 in [BZ18]. Following the approach of Curto and Fialkow, we analyzed the moment matrix based on its rank, giving a complete classification when the rank is at most five.…”

“…is of rank at most 5 and hence admits a measure if type (m, 1), m ∈ {1, 2, 3}, by [BZ18,§6]. In the fourth relation of (1.3) the nc atoms need not be of the form (1.4), making this case particularly difficult.…”

Nonnegative Polynomials, Sums of Squares and the Moment Problem

“…Two words v, w ∈ X, Y are cyclically equivalent (w ∼ v) if there are words u 1 , u 2 ∈ X, Y such that w = u 1 u 2 and v = u 2 u 1 . For a word w ∈ X, Y , let w * be the reverse of w. A sequence of numbers (λ w ) indexed by words w ∈ X, Y satisfying λ w = λ v whenever w ∼ v, λ w = λ w * for all w [3] The moment problem 339 and λ φ = 1 is called a (normalised) tracial sequence. As an example, given n ∈ N and a positive probability measure µ on (SR n×n ) 2 , the sequence given by λ w := tr(w(A, B)) dµ(A, B)…”

“…The next theorem demonstrates this for one of the cases (see [1,Ch. 4] or [3] for a more detailed exposition).…”