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Mathematische Annalen

, Volume 344, Issue 3, pp 703–716 | Cite as

Loewner matrices and operator convexity

  • Rajendra BhatiaEmail author
  • Takashi Sano
Article

Abstract

Let f be a function from \({\mathbb{R}_{+}}\) into itself. A classic theorem of K. Löwner says that f is operator monotone if and only if all matrices of the form \({\left [\frac{f(p_i) - f(p_j)}{p_i-p_j}\right ]_{\vphantom {X_{X_1}}}}\) are positive semidefinite. We show that f is operator convex if and only if all such matrices are conditionally negative definite and that f (t) = t g(t) for some operator convex function g if and only if these matrices are conditionally positive definite. Elementary proofs are given for the most interesting special cases f (t) = t r , and f (t) = t log t. Several consequences are derived.

Mathematics Subject Classification (2000)

15A48 47A63 42A82 

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References

  1. 1.
    Ando T.: Topics on Operator Inequalities. Hokkaido University, Sapporo (1978)zbMATHGoogle Scholar
  2. 2.
    Ando T.: Comparison of norms ||| f (A)  −  f (B)||| and ||| f (|A  −  B|))|||. Math. Z. 197, 403–409 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ando T., Zhan X.: Norm inequalities related to operator monotone functions. Math. Ann. 315, 771–780 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bapat R.B.: Multinomial probabilities, permanents and a conjecture of Karlin and Rinott. Proc. Am. Math. Soc. 102, 467–472 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bapat R.B., Raghavan T.E.S.: Nonnegative Matrices and Applications. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  6. 6.
    Baxter B.J.C.: Conditionally positive functions and p-norm distance matrices. Constr. Approx. 7, 427–440 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bendat J., Sherman S.: Monotone and convex operator functions. Trans. Am. Math. Soc. 79, 58–71 (1955)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bhatia R.: Matrix Analysis. Springer, Berlin (1996)zbMATHGoogle Scholar
  9. 9.
    Bhatia R.: Infinitely divisible matrices. Am. Math. Monthly 113, 221–235 (2006)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Bhatia R.: Positive Definite Matrices. Princeton University Press, Princeton (2007)Google Scholar
  11. 11.
    Bhatia R., Holbrook J.A.: Frechet derivatives of the power function. Indiana Univ. Math. J. 49, 1155–1173 (2003)MathSciNetGoogle Scholar
  12. 12.
    Bhatia R., Kosaki H.: Mean matrices and infinite divisibility. Linear Algebra Appl. 424, 36–54 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bhatia R., Parthasarathy K.R.: Positive definite functions and operator inequalities. Bull. Lond. Math. Soc. 32, 214–228 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Bhatia R., Sinha K.B.: Variation of real powers of positive operators. Indiana Univ. Math. J. 43, 913–925 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Davis, C.: Notions generalizing convexity for functions defined on spaces of matrices. In: Proc. Sympos. Pure Math., vol. VII, Convexity, pp. 187–201. American Mathematical Society, Providence (1963)Google Scholar
  16. 16.
    Donoghue W.F.: Monotone Matrix Functions and Analytic Continuation. Springer, Berlin (1974)zbMATHGoogle Scholar
  17. 17.
    Hansen F., Pedersen G.K.: Jensen’s inequality for operators and Löwner’s theorem. Math. Ann. 258, 229–241 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Horn R.A.: Schlicht mappings and infinitely divisible kernels. Pac. J. Math. 38, 423–430 (1971)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Horn R.A., Johnson C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)zbMATHGoogle Scholar
  20. 20.
    Kraus F.: Über konvexe Matrixfunktionen. Math. Z. 41, 18–42 (1936)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Kwong M.K.: Some results on matrix monotone functions. Linear Algebra Appl. 118, 129–153 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Löwner K.: Über monotone Matrixfunctionen. Math. Z. 38, 177–216 (1934)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Schoenberg I.J.: Metric spaces and completely monotone functions. Ann. Math. 39, 811–841 (1938)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Indian Statistical InstituteNew DelhiIndia
  2. 2.Department of Mathematical Sciences, Faculty of ScienceYamagata UniversityYamagataJapan

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