Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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THE FLAT EXTENSION OF NONSINGULAR EMBRY MOMENT MATRICES E(3)

  • Li, Chunji (Department of Mathematics Northeastern University) ;
  • Liang, Hongkai (Department of Mathematics Northeastern University)
  • Received : 2018.10.17
  • Accepted : 2019.03.11
  • Published : 2020.01.31
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Abstract

Let γ(n) ≡ {γij} (0 ≤ i+j ≤ 2n, |i-j| ≤ n) be a sequence in the complex number set ℂ and let E (n) be the Embry truncated moment matrices corresponding from γ(n). For an odd number n, it is known that γ(n) has a rank E (n)-atomic representing measure if and only if E(n) ≥ 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 (3), and also its sufficient conditions.

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Acknowledgement

The authors would like to thank the referee for the helpful suggestions

References

  1. R. E. Curto and L. A. Fialkow, Solution of the truncated complex moment problem for flat data, Mem. Amer. Math. Soc. 119 (1996), no. 568, x+52 pp. https://doi.org/10.1090/memo/0568
  2. R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: recursively generated relations, Mem. Amer. Math. Soc. 136 (1998), no. 648, x+56 pp. https://doi.org/10.1090/memo/0648
  3. R. E. Curto and L. A. Fialkow, Solution of the singular quartic moment problem, J. Operator Theory 48 (2002), no. 2, 315-354.
  4. R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: relations in analytic or conjugate terms, in Nonselfadjoint operator algebras, operator theory, and related topics, 59-82, Oper. Theory Adv. Appl., 104, Birkhauser, Basel, 1998.
  5. L. Fialkow, Positivity, extensions and the truncated complex moment problem, in Multivariable operator theory (Seattle, WA, 1993), 133-150, Contemp. Math., 185, Amer. Math. Soc., Providence, RI, 1995. https://doi.org/10.1090/conm/185/02152
  6. I. B. Jung, E. Ko, C. Li, and S. Park, Embry truncated complex moment problem, Linear Algebra Appl. 375 (2003), 95-114. https://doi.org/10.1016/S0024-3795(03)00617-7
  7. C. Li, The singular Embry quartic moment problem, Hokkaido Math. J. 34 (2005), no. 3, 655-666. https://doi.org/10.14492/hokmj/1285766290
  8. C. Li and M. Cho, The quadratic moment matrix E(1), Sci. Math. Jpn. 57 (2003), no. 3, 559-567.
  9. C. Li, I. B. Jung, and S. S. Park, Complex moment matrices via Halmos-Bram and Embry conditions, J. Korean Math. Soc. 44 (2007), no. 4, 949-970. https://doi.org/10.4134/JKMS.2007.44.4.949
  10. C. Li and X. Sun, On nonsingular Embry quartic moment problem, Bull. Korean Math. Soc. 44 (2007), no. 2, 337-350. https://doi.org/10.4134/BKMS.2007.44.2.337
  11. MacKichan Software Inc., Scientific WorkPlace, Version 4.0. MacKichan Software, Inc., 2002.