Positivity and the Existence of Unitary Dilations of Commuting Contractions

  • J. Robert Archer
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 171)


The central result of this paper is a method of characterizing those commuting tuples of operators that have a unitary dilation, in terms of the existence of a positive map with certain properties. Although this positivity condition is not necessarily easy to check given a concrete example, it can be used to find practical tests in some circumstances. As an application, we extend a dilation theorem of Sz.-Nagy and Foiaş concerning regular dilations to a more general setting


Hilbert Space Trigonometric Polynomial Linear Manifold Defect Operator Positive Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    T. Andô. On a pair of commutative contractions. Acta Sci. Math. (Szeged), 24:88–90, 1963.MATHMathSciNetGoogle Scholar
  2. [2]
    S. Brehmer. Über vetauschbare Kontraktionen des Hilbertschen Raumes. Acta Sci. Math. Szeged, 22:106–111, 1961.MATHMathSciNetGoogle Scholar
  3. [3]
    R.G. Douglas. On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Amer. Math. Soc., 17:413–415, 1966.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Michael A. Dritschel. On factorization of trigonometric polynomials. Integral Equations Operator Theory, 49(1):11–42, 2004.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Dumitru Gaşpar and Nicolae Suciu. On the intertwinings of regular dilations. Ann. Polon. Math., 66:105–121, 1997. Volume dedicated to the memory of Włodzimierz Mlak.MathSciNetGoogle Scholar
  6. [6]
    M. Naimark. Positive definite operator functions on a commutative group. Bull. Acad. Sci. URSS Sér. Math. [Izvestia Akad. Nauk SSSR], 7:237–244, 1943.MATHGoogle Scholar
  7. [7]
    Stephen Parrott. Unitary dilations for commuting contractions. Pacific J. Math., 34:481–490, 1970.MATHMathSciNetGoogle Scholar
  8. [8]
    Vern Paulsen. Completely bounded maps and operator algebras, volume 78 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2002.Google Scholar
  9. [9]
    Donald Sarason. On spectral sets having connected complement. Acta Sci. Math. (Szeged ), 26:289–299, 1965.MATHMathSciNetGoogle Scholar
  10. [10]
    J. J. Schäffer. On unitary dilations of contractions. Proc. Amer. Math. Soc., 6:322, 1955.CrossRefMathSciNetGoogle Scholar
  11. [11]
    W. Forrest Stinespring. Positive functions on C*-algebras. Proc. Amer. Math. Soc., 6:211–216, 1955.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Béla Sz.-Nagy. Sur les contractions de l’espace de Hilbert. Acta Sci. Math. Szeged, 15:87–92, 1953.MATHMathSciNetGoogle Scholar
  13. [13]
    Béla Sz.-Nagy. Transformations de l’espace de Hilbert, fonctions de type positif sur un groupe. Acta Sci. Math. Szeged, 15:104–114, 1954.MathSciNetGoogle Scholar
  14. [14]
    Béla Sz.-Nagy and Ciprian Foiaş. Harmonic analysis of operators on Hilbert space. Translated from the French and revised. North-Holland Publishing Co., Amsterdam, 1970.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • J. Robert Archer
    • 1
  1. 1.Department of MathematicsUniversity of GlasgowGlasgowUK

Personalised recommendations