Advertisement

On Extensions of Indefinite Toeplitz-Kreįn-Cotlar triplets

  • Ramón Bruzual
  • Marisela Domínguez
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)

Abstract

We give a definition of к-indefinite Toeplitz-Kreįn-Cotlar triplet of Archimedean type, on an interval of an ordered group G with an Archimedean point. We show that if a group G has the indefinite extension property, then every к-indefinite Toeplitz-Kreįn-Cotlar triplet of Archimedean type on an interval of г, can be extended to a Toeplitz-Kreįn-Cotlar triplet on the whole group г, with the same number of negative squares к.

Keywords

Operator-valued indefinite functions ordered group Archimedean point Toeplitz kernel 

Mathematics Subject Classification (2000)

Primary 47B50 Secondary 46C20 47D03 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. Alpay, T. Constantinescu, A. Dijksma and J. Rovnyak, Notes on interpolation in the generalized Schur class. II. Nudelman’s problem. Trans. Am. Math. Soc. 355 (2003), 813–836.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    R. Arocena, On the Extension Problem for a class of translation invariant positive forms. J. Oper. Theory 21 (1989), 323–347.zbMATHMathSciNetGoogle Scholar
  3. [3]
    J. Ball and J. Helton, A Beurling-Lax theorem for the Lie group U(m; n) which contains most classical interpolation theory, J. Operator Theory 9 (1983), 107–142.zbMATHMathSciNetGoogle Scholar
  4. [4]
    M. Bakonyi and D. Timotin, The intertwining lifting theorem for ordered groups. J. Funct. Anal. 199, No. 2 (2003), 411–426.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    M. Bakonyi and G. Nævdal, The finite subsets of ℤ 2 having the extension property. J. Lond. Math. Soc., II. Ser. 62, No. 3 (2000), 904–916.zbMATHCrossRefGoogle Scholar
  6. [6]
    R. Bruzual and M. Domínguez, Equivalence between the dilation and lifting properties of an ordered group through multiplicative families of isometries. A version of the commutant lifting theorem on some lexicographic groups. Int. Eq. and Op. Theory, 40 No. 1 (2001), 1–15.zbMATHCrossRefGoogle Scholar
  7. [7]
    R. Bruzual and M. Domínguez, Extensions of operator valued positive definite functions and commutant lifting on ordered groups. J. Funct. Anal. 185 No. 2 (2001) 456–473.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    R. Bruzual and M. Domínguez, Extension of locally defined indefinite functions on ordered groups. Int. Eq. and Op. Theory, 50 (2004), 57–81.zbMATHCrossRefGoogle Scholar
  9. [9]
    R. Bruzual and M. Domínguez, A generalization to ordered groups of a Kreℍin theorem. Operator Theory: Advances and Applications 179, (2008) 103–109.CrossRefGoogle Scholar
  10. [10]
    R. Bruzual and M. Domínguez, Dilation of generalized Toeplitz kernels on ordered groups. Journal of Functional Analysis, 238, No. 2, (2006), 405–426.zbMATHMathSciNetGoogle Scholar
  11. [11]
    R. Bruzual and M. Domínguez, On extensions of indefinite functions defined on a rectangle, Complex Anal. Oper. Theory, in press-available on line.Google Scholar
  12. [12]
    R. Bruzual and S.A.M. Marcantognini, Local semigroups of isometries in Π к-spaces and related continuation problems for к-indefinite Toeplitz kernels, Int. Eq. and Op. Theory, 15 (1992), 527–550.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    R. Bruzual and S.A.M. Marcantognini, The Kreįin-Langer problem for Hilbert space operator valued functions on the band, Int. Eq. and Op. Theory, 34 (1999), 396–413.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    M. Cotlar, C. Sadosky, On the Helson-Szegö theorem and a related class of modified Toeplitz kernels. Proc. Symp. Pure Math. AMS. 35-I (1979), 383–407.MathSciNetGoogle Scholar
  15. [15]
    A. Devinatz, On the extensions of positive definite functions. Acta Math. 102, No. 1-2 (1959), 109–134.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    G.I. Eskin, A sufficient condition for the solvability of the moment problem in several dimensions. Dokl. Akad. Nauk. SSSR 113 (1960), 540–543.MathSciNetGoogle Scholar
  17. [17]
    J. Geronimo and H. Woerdeman, The operator valued autoregressive filter problem and the suboptimal Nehari problem in two variables. Int. Eq. and Op. Theory, 53, (2005), 343–361.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    V.I. Gorbachuk (V.I. Plyushceva), On the integral representation of hermitian indefinite kernels with a finite number of negative squares. Dokl. Akad. Nauk. SSRR 145:3 (1962), 534–537.Google Scholar
  19. [19]
    M.G. Kreįn, Sur le problème du prolongement des fonctions hermitiennes positives et continues. Dokl. Akad. Nauk. SSSR 26 (1940), 17–22.Google Scholar
  20. [20]
    M.G. Kreįn and H. Langer, On some continuation problems which are closely related to the theory of operators in spaces Π к. IV. Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight and related continuation problems for some classes of functions, J. Operator Theory 13 (1985), 299–417.zbMATHMathSciNetGoogle Scholar
  21. [21]
    W. Rudin, The extension problem for positive definite functions. Illinois J. Math. 7 (1963), 532–539.zbMATHMathSciNetGoogle Scholar
  22. [22]
    Z. Sasvári, Positive definite and definitizable functions. Akademie Verlag, 1994.Google Scholar
  23. [23]
    Z. Sasvári, The Extension Problem for Positive Definite Functions. A Short Historical Survey. Operator Theory: Advances and Applications, 163, (2005), 365–379.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Ramón Bruzual
    • 1
  • Marisela Domínguez
    • 1
  1. 1.Escuela de Matemática Fac. CienciasUniversidad Central de VenezuelaSpain

Personalised recommendations