## Abstract

Let S:={St1, td}0≤t1+⋯+td≤m be a finite multisequence of p×p Hermitian matrices. If there exists a p×p positive semidefinite matrix-valued Borel measure σ on Rd so thatSt1, td=∫⋯∫Rdx1t1⋯xdtddσ(x1,xd), for all d-tuples of non-negative integers (t_{1}t_{d}) so that 0≤t_{1}+⋯+t_{d}≤m, i.e. σ is a representing measure S, then we will show that there exist p×p positive semidefinite matrices P_{1}P_{k} and x_{1},x_{k}∈suppσ so that ∑q=1kδxqPq is also a representing measure for S, where ∑q=1krankPq≤p2(m+d)!/(m!d!). We will pose a necessary and sufficient condition on a given sequence S, of bounded linear operators on a separable Hilbert space, so that an operator-valued generalization of Tchakaloff's theorem holds. We will make use of an operator-valued generalization of Tchakaloff's theorem on the unit circle to obtain a solution to the operator-valued Carathéodory interpolation problem.

Original language | English |
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Pages (from-to) | 1170-1184 |

Number of pages | 15 |

Journal | Journal of Functional Analysis |

Volume | 266 |

Issue number | 3 |

DOIs | |

State | Published - 1 Feb 2014 |

Externally published | Yes |

## Keywords

- Operator-valued Carathéodory interpolation problem
- Tchakaloff's theorem
- Truncated matrix-valued moment problem
- Truncated operator-valued moment problem

## ASJC Scopus subject areas

- Analysis