Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2020; 35(1): 137-149

Online first article August 29, 2019      Printed January 31, 2020

https://doi.org/10.4134/CKMS.c180432

Copyright © The Korean Mathematical Society.

The flat extension of nonsingular Embry moment matrices $E(3)$

Chunji Li, Hongkai Liang

Northeastern University; Dalian University of Technology

Abstract

Let $\gamma ^{\left( n\right) }\equiv \{\gamma _{ij}\}\,(0\leq i+j\leq 2n,\,|i-j|\leq n)$ be a sequence in the complex number set $\mathbb{C}$ and let $E\left( n\right) $ be the Embry truncated moment matrices corresponding from $\gamma ^{\left( n\right) }$. For an odd number $n$, it is known that $ \gamma ^{\left( n\right) }$ has a rank $E\left( n\right) $\textit{-}atomic representing measure if and only if $E(n)\geq 0$ and $E(n)$ admits a flat extension $E(n+1)$. In this paper we suggest a related problem: if $E(n)$ is positive and nonsingular, does $E(n)$ have a flat extension $E(n+1)$? and give a negative answer in the case of $E(3)$. And we obtain some necessary conditions for positive and nonsingular matrix $E\left( 3\right)$, and also its sufficient conditions.

Keywords: Embry truncated complex moment problem, representing measure, flat extension

MSC numbers: Primary 47A57, 44A60; Secondary 15A57, 47A20

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