The truncated matrix-valued \(K\)-moment problem on \(\mathbb R^d\), \(\mathbb C^d\), and \(\mathbb T^d\) (Q2847200)

The paper under review studies the truncated matrix-valued \(K\)-moment problem on \(\mathbb{R}^{d}\), \(\mathbb{C}^{d}\), and \(\mathbb{T}^{d}\).NEWLINENEWLINELet \(\mathcal{H}_{p}\subseteq\mathbb{C}^{p\times p}\) be the real Hilbert space of \(p\times p\) Hermitian matrices. A set \(\Gamma\subseteq\mathbb{N}_{0}^{d}\) is a lattice set if, for every \(\gamma=(\gamma_{1},\dots ,\gamma_{d})\in\Gamma\), there exist \(\gamma_{1}=(0,\dots ,0)\in\mathbb{N}_{0}^{d}\), \(\gamma_{2},\dots, \gamma_{k}\in\Gamma\) and \(j_{1}, \dots , j_{k}\in\{1,\dots ,d\}\) so that \(\gamma_{2}=\gamma_{1}+e_{j_{1}}\), \dots, \(\gamma=\gamma_{k}+e_{j_{k}}\), where \(e_{m}=(0,\dots ,\underbrace{1}_{m},\dots ,0)\in\mathbb{N}_{0}^{d}\) and \(k=|\gamma|=\gamma_1+\dots +\gamma_d\).NEWLINENEWLINEGiven \(K\subseteq\mathbb{R}^{d}\) (resp., \(K\subseteq\mathbb{C}^{d}\), \(K\subseteq\mathbb{T}^{d}\)) and the \(\mathcal{H}_{p}\)-valued sequence \((S_{\gamma})_{\gamma\in\Gamma}\) for some lattice set \(\Gamma\in\mathbb{N}_{0}^{d}\), the truncated matrix-valued \(K\)-moment problem studies the existence of a positive \(\mathcal{H}_{p}\)-valued measure \(\sigma\) on \(\mathbb{R}^{d}\) (resp., \(\mathbb{C}^{d}\), \(\mathbb{T}^{d}\)) so that \(\int\xi^{m}d\sigma(\xi)\) exists for every \(m\in\mathbb{N}_{0}^{d}\), under the conditions that for every \(m\in\Gamma\), the previous integral coincides with \(S_{\gamma}\) for every \(\gamma\in\Gamma\), and, moreover, the support of \(\sigma\) is contained in \(K\).NEWLINENEWLINEThe main result in the present paper states for given \(K\subseteq \mathbb{R}^{d}\) that, given a lattice set \(\Lambda\subseteq\mathbb{N}^{d}_{0}\) and a \(\mathcal{H}_{p}\)-valued sequence \((S_\gamma)_{\gamma\in\Gamma}\), where \(\Gamma\) is the set consisting of the sum of two elements in \(\Lambda\) and the elements of the form \(\lambda+\mu+e_{m}\) for \(\lambda,\mu\in \Lambda\) and \(m\in\{1,\dots ,d\}\), then the truncated matrix-valued \(K\)-moment problem on \(\mathbb{R}^{d}\) is solvable if and only ifNEWLINENEWLINE\((1)\) the matrix \(\Phi:=(S_{\lambda+\mu})_{\lambda,\mu\in\Lambda}\) is positive semidefinite;NEWLINENEWLINE\((2)\) there exist \(\Theta_1,\dots ,\Theta_d\) so that \(\Phi\Theta_{j}=(S_{\lambda+\mu+e_{j}})_{\lambda,\mu\in\Lambda}\) for \(j=1,\dots ,d\) and \(\Theta_1,\dots ,\Theta_d\) satisfy a property concerning the location of certain eigenvalues of associated matrices with respect to the set \(\text{Ran} \Phi\).NEWLINENEWLINEIn that case, \(\sigma\) is of the form \(\sum_{q=1}^{k}T_{q}\delta_{w_{q}}\) for some positive semidefinite matrices \(T_{1},\dots ,T_{k}\) and different points \(w_1,\dots ,w_{k}\) in \(K\). In addition to this, \(\sigma\) is minimal. The converse statement is also valid.NEWLINENEWLINEThe third section is devoted to provide analogous conditions on a given set of given square matrices, indexed by an adequate family of lattice sets, to admit a minimal \(K\)-representing measure on \(\mathbb{C}^{d}\). This result also leans on some matrix factorization results. The last section deals with the problem in \(\mathbb{T}^{d}\).NEWLINENEWLINEThe paper begins with a brief survey on different known moment problems. Along the way, some of these are solved as a corollary of the main result.NEWLINENEWLINEMoreover, the authors provide several examples throughout the text which make the different results easier to follow.