An operator-valued generalization of Tchakaloff's theorem

David P. Kimsey

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

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 (t1td) so that 0≤t1+⋯+td≤m, i.e. σ is a representing measure S, then we will show that there exist p×p positive semidefinite matrices P1Pk and x1,xk∈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 languageEnglish
Pages (from-to)1170-1184
Number of pages15
JournalJournal of Functional Analysis
Volume266
Issue number3
DOIs
StatePublished - 1 Feb 2014
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Analysis

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