Extension of Löwner-Heinz Inequality Via Analytic Continuation

  • Mitsuru Uchiyama
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 9)

Abstract

That t α (0 < α < 1) is operator monotone on [0, ∞) means, by definition, 0 ≤ ABA α B α , which is called a Löwner-Heinz inequality. We consider the converse of this statement. We systematically construct a family of operator monotone functions which includes t α . Moreover, we give operator inqualities which are extensions of those by Furuta and Ando.

Keywords

Analytic Continuation Monotone Function Operator Monotone Selfadjoint Operator Dimensional Hilbert Space 
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.

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References

  1. 1.
    T. Ando, Topics on operator inequalities, Lecture note, Sapporo, 1978.MATHGoogle Scholar
  2. 2.
    T. Ando, On some operator inequalities, Math. Ann. 279 (1987), 157–159.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    R. Bhatia, Matrix Analysis, Springer-Verlag, New York (1996).MATHGoogle Scholar
  4. 4.
    W. Donoghue, Monotone matrix functions and analytic continuation, Springer, 1974.Google Scholar
  5. 5.
    M. Fujii, E. Kamei, Furuta’s inequality and a generalization of Ando’s theorem, Proc. A. M. S. 115 (1992), 409–413.MathSciNetMATHGoogle Scholar
  6. 6.
    T. Furuta, A B 0 assures (BrApBr)I/Q B(p+2r)/q for r 0, p 0, q 1 with (1 + 2r)q p + 2r, Proc. A.M.S. 101 (1987), 85–88.MathSciNetMATHGoogle Scholar
  7. 7.
    T. Furuta, An elementary proof of an order preserving inequality, Proc. Japan Acad. 65 ser. A (1989), 126.Google Scholar
  8. 8.
    F. Hansen, G. K. Pedersen, Jensen’s inequality for operators and Löwner’s theorem. Math. Ann. 258 (1982), 229–241MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann. 123 (1951), 415–438.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    R. Horn, The Hadamard product, Matrix theory and applications, Amer. Math. Soc. Proc. Symposia in Applied Math. 40 (1990), 87–169.MathSciNetCrossRefGoogle Scholar
  11. 11.
    R. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge, (1991).Google Scholar
  12. 12.
    K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177–216.MathSciNetCrossRefGoogle Scholar
  13. 13.
    R. Mathias, The equivalence of two partial orders on a convex cone of positive semidefinite matrices, Linear Alg. Apl. 151 (1991), 27–55.MathSciNetMATHGoogle Scholar
  14. 14.
    M. Rosenblum, J. Rovnyak, Handy classes and operator theory, Oxford University Press (1985).Google Scholar
  15. 15.
    K. Tanahashi, S. Yamagami, Exponential ordering on bounded self-adjoint operators, Jour. Operator Th.Google Scholar
  16. 16.
    K. Tanahashi, Best possibility of the Furuta inequality, Proc. A. M. S. 124 (1996), 141–146.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    M. Uchiyama, Commutativity of selfadjoint opeators, Pacific J. Math. 161 (1993), 385–392.MathSciNetMATHGoogle Scholar
  18. 18.
    M. Uchiyama, Some exponential operator inequalities, Math. Inequal. Appl. 2 (1999), 469–471.MathSciNetMATHGoogle Scholar
  19. 19.
    M. Uchiyama, Strong monotonicity of operator functions, Integral Eq. Operator Theory, 37(2000)95–105.Google Scholar
  20. 20.
    M. Uchiyama, Operator monotone functions which are defined implicitly and operator inequalities, J. Functional Analysis 175 (2000) 330–347.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Mitsuru Uchiyama
    • 1
  1. 1.Department of MathematicsFukuoka University of EducationMunakata,FukuokaJapan

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