## 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., 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 ≤ {pipe}γ{pipe} ≤ 2n + 1}. We will also discuss a similar result in the multivariable complex and polytorus setting.

Original language | English |
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Pages (from-to) | 5393-5430 |

Number of pages | 38 |

Journal | Transactions of the American Mathematical Society |

Volume | 365 |

Issue number | 10 |

DOIs | |

State | Published - 24 Jul 2013 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics

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