Functional Analysis and Its Applications

, Volume 41, Issue 4, pp 314–317 | Cite as

Quadratic operator inequalities and linear-fractional relations

  • V. A. Khatskevich
  • M. I. Ostrovskii
  • V. S. Shulman


We study properties of solution sets of inequalities of the form
$$X^* AX + B^* X + X^* B + C \leqslant 0,$$
, where A, B, and C are bounded Hilbert space operators and A and C are self-adjoint. The following properties are considered: closedness and inferior points in Standard operator topologies, convexity, and nonemptiness.

Key words

Hilbert space bounded linear operator weak operator topology operator inequality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    T. Ya. Azizov and I. S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, John Wiley & Sons, 1989.Google Scholar
  2. [2]
    J. B. Conway, A Course in Operator Theory, Grauate Studies in Math., vol. 21, Amer. Math. Soc., Providence, R.I., 2000.Google Scholar
  3. [3]
    C. C. Cowen, Trans. Amer. Math. Soc., 265:1 (1981), 69–95.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    V. Khatskevich, M. I. Ostrovskii, and V. Shulman, Math. Nachr., 279 (2006), 875–890.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    V. Khatskevich and V. Shulman, Studia Math., 116 (1995), 189–195.MATHMathSciNetGoogle Scholar
  6. [6]
    M. G. Krein and Yu. L. Shmulyan, Amer. Math. Soc. Transl., Ser. 2, 103 (1974), 125–152.Google Scholar
  7. [7]
    G. A. Kurina, J. Comput. Systems Sci. Internat., 32:6 (1994), 30–35.MathSciNetGoogle Scholar
  8. [8]
    M. M. Malamud, in: Recent Advances in Operator Theory (Groningen, 1998), Oper. Theory Adv. Appl., vol. 124, Birkhäuser, Basel, 2001, 401–449.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • V. A. Khatskevich
    • 1
  • M. I. Ostrovskii
    • 2
  • V. S. Shulman
    • 3
  1. 1.Department of Math. And Comp.Sci. St.-John’s UniversityUSA
  2. 2.Department of Math.ORT Braude CollegeIsrael
  3. 3.Department of MathematicsVologda Polytechnical InstituteRussia

Personalised recommendations