The truncated matrix-valued $K$-moment problem on $\mathbb {R}^d$, $\mathbb {C}^d$, and $\mathbb {T}^d$

Abstract

The truncated matrix-valued K-moment problem on R d , C d , and T d will be considered. The truncated matrix-valued K-moment problem on R d requires necessary and sufficient conditions for a multisequence of Hermitian matrices {S γ } γ∈Γ (where Γ is a finite subset of N d 0) to be the corresponding moments of a positive Hermitian matrix-valued Borel measure σ, and also the support of σ must be contained in some given non-empty set K ⊆ R d , i.e., (0.1) S γ = R d ξ γ dσ(ξ), for all γ ∈ Γ, and (0.2) supp σ ⊆ K. Given a non-empty set K ⊆ R d and a finite multisequence, indexed by a certain family of finite subsets of N d 0 , of Hermitian matrices we obtain necessary and sufficient conditions for the existence of a minimal finitely atomic measure which satisfies (0.1) and (0.2). In particular, our result can handle the case when Γ = {γ ∈ N d 0 : 0 ≤ |γ| ≤ 2n + 1}. We will also discuss a similar result in the multivariable complex and polytorus setting.

The Cubic Complex Moment Problem

“…If Φ(n) 0 and there exist Θ z (n) and Θz(n) such that Proof. This is a particular case of Theorem 3.4 in [15] for K = C and Λ = {(j, k) ∈ N 2 0 : 0 ≤ j + k ≤ n}. Here N 2 0 denotes the set of all 2-tuples of nonnegative integers.…”

“…Theorem 2.6. [15] Let s = {s jk } 0≤j+k≤2n+1 be given. If Φ(n) 0 and there exist Θ z (n) and Θz(n) such that Proof.…”

“…360 D. P. Kimsey IEOT Remark 2.7. Let Λ ⊂ N 2 0 be a finite set which is path connected to the origin (a lattice set in the terminology of [15]) and put…”

“…In addition, the conditions Φ 0 and s 00 ≤ s 11 are necessary and sufficient for the existence of a (rank Φ)-atomic representing measure σ which satisfies supp σ ⊂ D, where D = {z ∈ C : |z| ≤ 1}. When m = 2n + 1, [14] and [15] provided a solution to a matricial analog of the truncated complex moment problem (and also matricial moment problems on C p and R p ) when one desires a minimal representing measure. The solution is based on commutativity conditions of matrices determined by {s jk } 0≤j+k≤2n+1 .…”

Sextic Moment Problems on 3 Parallel Lines

“…When m = 2n + 1, partial solutions can be seen in [14] and [16] as well as a solution to the truncated matrix moment problem; in particular, a solution to the cubic complex moment problem (when n = 1) was given in [15]. However, the problem is still often for n ≥ 2.…”