On a generalization of the classical moment problem (Q1101639)

Let \(N^ n\) denote the set of n-tuples of non-negative integers and let the multisequence \(\{S_{\alpha}:\alpha \in N^ n\}\) be a collection of \(k\times k\) matrices having complex entries \(s_{ij}(\alpha)\), \(i,j=1,...,k\). A multisequence \(\{S_{\alpha}\}\) is a k-moment sequence represented by a \(k\times k\) matrix \(\Lambda =(\lambda_{ij})\) of complex Borel measures on \({\mathbb{R}}^ n\) if for all \(\alpha =(\alpha_ 1,...,\alpha_ n)\in {\mathbb{R}}^ n\) \(s_{ij}(\alpha)=\int_{{\mathbb{R}}^ n}x^{\alpha} d\lambda_{ij},\quad x^{\alpha}=x_ 1^{\alpha_ 1},x_ 2^{\alpha_ 2},...,x_ n^{\alpha_ n},\) and provided \(\Lambda (m)=(\lambda_{ij}(m))\) is non-negative for each Borel set \(M\subset {\mathbb{R}}^ n\) and \(| \lambda_{ij}|\) is a positive Borel measure having moment of all orders. Given a multisequence \(\{S_{\alpha}\}\), define \({\mathcal S}\) from the set \({\mathcal P}_ n\) of n-variate polynomials to the set of \(k\times k\) complex matrices by \({\mathcal S}(p)=\sum a_{\alpha}S_{\alpha}\) where \(p(x)=\sum a_{\alpha}x^{\alpha}.\) \({\mathcal S}\) is k-positive if \(\sum^{k}_{k,j=1}({\mathcal S}(p_{ij})c_ i,c_ j)\geq 0\) for all \(c_ 1,...,c_ k\in {\mathbb{C}}^ k\) and all \(k\times k\) matrices \((p_{ij})\) of non-negative n-variate polynomials (this class is denoted by (\({\mathcal P}_ n\otimes M_ k)^+)\). Then Theorem. Let \(\{S_{\alpha}\}\) and \({\mathcal S}\) be as above. The following are equivalent: (i) \(\{S_{\alpha}\}\) is a k-moment sequence. (ii) \({\mathcal S}\) is k-positive. (iii) \(\sum^{k}_{k,j=1}s_{ij}(p_{ij})\geq 0\) for all \((p_{ij})\in ({\mathcal P}_ n\otimes M_ k)^+\) where \(s_{ij}(p_{ij})\) are the matrix components of \({\mathcal S}(p_{ij}).\) The paper also has several uniqueness or determinancy results.