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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The truncated matrix-valued $ K$-moment problem on $ \mathbb{R}^d$, $ \mathbb{C}^d$, and $ \mathbb{T}^d$


Authors: David P. Kimsey and Hugo J. Woerdeman
Journal: Trans. Amer. Math. Soc. 365 (2013), 5393-5430
MSC (2010): Primary 47A57; Secondary 30E05, 42A70, 44A60
Published electronically: May 16, 2013
MathSciNet review: 3074378
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Abstract: The truncated matrix-valued $ K$-moment problem on $ \mathbb{R}^d$, $ \mathbb{C}^d$, and $ \mathbb{T}^d$ will be considered. The truncated matrix-valued $ K$-moment problem on $ \mathbb{R}^d$ requires necessary and sufficient conditions for a multisequence of Hermitian matrices $ \{ S_{\gamma } \}_{\gamma \in \Gamma }$ (where $ \Gamma $ is a finite subset of $ \mathbb{N}_0^d$) to be the corresponding moments of a positive Hermitian matrix-valued Borel measure $ \sigma $, and also the support of $ \sigma $ must be contained in some given non-empty set $ K \subseteq \mathbb{R}^d$, i.e.,

$\displaystyle S_{\gamma } = \int _{\mathbb{R}^d} \xi ^{\gamma } d\sigma (\xi ),\;\;{\rm for}\;{\rm all}\; \gamma \in \Gamma ,$ (0.1)

and

$\displaystyle {\rm supp} \; \sigma \subseteq K.$ (0.2)

Given a non-empty set $ K \subseteq \mathbb{R}^d$ and a finite multisequence, indexed by a certain family of finite subsets of $ \mathbb{N}_0^d$, 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 $ \Gamma = \{ \gamma \in \mathbb{N}_0^d \colon 0 \leq \vert\gamma \vert \leq 2n+1 \}$. We will also discuss a similar result in the multivariable complex and polytorus setting.

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Additional Information

David P. Kimsey
Affiliation: Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104
Address at time of publication: Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, POB 26, Rehovot 76100, Israel
Email: kimsey@drexel.edu, david.kimsey@weizmann.ac.il

Hugo J. Woerdeman
Affiliation: Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104
Email: hugo@math.drexel.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05812-6
PII: S 0002-9947(2013)05812-6
Received by editor(s): October 5, 2010
Received by editor(s) in revised form: February 19, 2012
Published electronically: May 16, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.