Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The extension of positive definite operator-valued functions defined on a symmetric interval of an ordered group

Author: Mihály Bakonyi
Journal: Proc. Amer. Math. Soc. 130 (2002), 1401-1406
MSC (1991): Primary 43A35, 47A57, 42A70, 47A20
Published electronically: October 12, 2001
MathSciNet review: 1879963
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G_1$ be an ordered abelian group and $a\in G_1$. Let $G_2$ be an abelian group and $f$ an operator-valued positive definite function on $(-a,a)\times G_2$. We prove that $f$ admits a positive definite extension to $G_1\times G_2$, generalizing in this way existing results for the case when $G_1=\mathbf{R}$ and $f$ is continuous.

References [Enhancements On Off] (What's this?)

  • 1. Gr. Arsene, Z. Ceausescu, and T. Constantinescu, Schur analysis of some completion problems, Linear Algebra Appl., Vol. 109(1988), 1-35. MR 89k:47010
  • 2. A.P. Artjomenko, Hermitian positive functions and positive functionals, Dissertation, Odessa State University, 1941. Published in Teor. Funkcii, Funk. Anal. i Prilozen, Vol. 41(1983), 1-16; Vol. 42(1984), 1-21 (Russian).
  • 3. W.B. Arveson, Subalegbras of $C^*$-algebras, Acta Math., 123(1969), 141-224. MR 40:6274
  • 4. M. Bakonyi, L. Rodman, I. M. Spitkovsky, and H. J. Woerdeman,
    Positive matrix functions on the bitorus with prescribed Fourier coefficients in a band, J. Fourier Anal. Appl., Vol. 5(1999), 789-812. MR 2001c:42015
  • 5. Y.M. Berezansky and Y.G. Kondratiev, Spectral Methods in Infinite-Dimensional Analysis, Kluwer Academic Publishers, Dortrecht, 1995. MR 96d:46001a; MR 96d:46001b
  • 6. Y.M. Berezansky and I.M. Gali, Positive definite functions of infinitely many variables on a layer, Ukrain. Mat. Zh., Vol. 24(1972), 435-464; English translation in Ukrain. Math. J. Vol. 24(1972).
  • 7. R. Bruzual and M. Dominguez, Extensions of operator valued positive definite functions on an interval of $\mathbf{Z}^2$ with the lexicographic order, Acta Sci. Math. Szeged, Vol. 66(2000), 623-631. CMP 2001:06
  • 8. M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980. MR 81e:68081
  • 9. J. Friedrich and L. Klotz, On extensions of positive definite operator-valued function, Rep. Math. Phys., Vol. 26, No. 1(1988), 45-65. MR 90h:43005
  • 10. M.G. Krein, Sur le probléme de prolongement des functions hermitiniennes positives et continues, Dokl. Akad. Nauk. SSSR, Vol. 26(1940), 17-22.
  • 11. V. Paulsen, Completely Bounded Maps and Dilations, Pitman Research Notes in Mathematics, Vol. 146, New York, 1986. MR 88h:46111
  • 12. V.I. Paulsen, S.C. Power, and R.G. Smith, Schur products and matrix completions, J. Funct. Anal., Vol. 85(1989), 151-178. MR 90j:46051
  • 13. L. Rodman, I. Spitkovsky, and H.J. Woerdeman, Charathéodory-Toeplitz and Nehari problems for matrix valued almost periodic functions, Trans. Amer. Math. Soc., Vol. 350(1998), 2185-2227. MR 98h:47023
  • 14. W. Rudin, Fourier Analysis on Groups, Interscience Publishers, New York, 1962. MR 27:2808
  • 15. Z. Sasvári,
    Positive Definite and Definitizable Functions,
    Akademie Verlag, Berlin, 1994.
  • 16. I. Spitkovsky and H.J. Woerdeman, The Charathéodory-Fejér problem for almost periodic functions, J. Funct. Anal., Vol. 115(1993), 281-293. MR 94f:47020
  • 17. I. Suciu, Function Algebras, Editura Acamediei Române, Bucharest, 1975. MR 51:6428

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 43A35, 47A57, 42A70, 47A20

Retrieve articles in all journals with MSC (1991): 43A35, 47A57, 42A70, 47A20

Additional Information

Mihály Bakonyi
Affiliation: Department of Mathematics, Georgia State University, Atlanta, Georgia 30303

Received by editor(s): November 14, 2000
Published electronically: October 12, 2001
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society