Journal of the Mathematical Society of Japan

Loewner matrices of matrix convex and monotone functions

Fumio HIAI and Takashi SANO

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Abstract

The matrix convexity and the matrix monotony of a real C1 function f on (0,∞) are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with f, tf(t), and t2f(t). Similar characterizations are also obtained for matrix monotone functions on a finite interval (a,b).

Article information

Source
J. Math. Soc. Japan, Volume 64, Number 2 (2012), 343-364.

Dates
First available in Project Euclid: 26 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1335444395

Digital Object Identifier
doi:10.2969/jmsj/06420343

Mathematical Reviews number (MathSciNet)
MR2916071

Zentralblatt MATH identifier
1261.15026

Subjects
Primary: 15A45: Miscellaneous inequalities involving matrices
Secondary: 47A63: Operator inequalities 42A82: Positive definite functions

Keywords
n-convex n-monotone operator monotone operator convex Loewner matrix conditional negative definite conditional positive definite

Citation

HIAI, Fumio; SANO, Takashi. Loewner matrices of matrix convex and monotone functions. J. Math. Soc. Japan 64 (2012), no. 2, 343--364. doi:10.2969/jmsj/06420343. https://projecteuclid.org/euclid.jmsj/1335444395


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References

  • R. B. Bapat and T. E. S. Raghavan, Nonnegative Matrices and Applications, Cambridge University Press, Cambridge, 1997.
  • R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1996.
  • R. Bhatia and T. Sano, Loewner matrices and operator convexity, Math. Ann., 344 (2009), 703–716.
  • R. Bhatia and T. Sano, Positivity and conditional positivity of Loewner matrices, Positivity, 14 (2010), 421–430.
  • A. M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math., 12 (1962), 1203–1215.
  • W. F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer-Verlag, Berlin-Heidelberg-New York, 1974.
  • F. Hansen and G. K. Pedersen, Jensen's inequality for operators and Löwner's theorem, Math. Ann., 258 (1982), 229–241.
  • F. Hansen and J. Tomiyama, Differential analysis of matrix convex functions II, J. Inequal. Pure Appl. Math., 10 (2009), Article 32, 5 pp.
  • F. Hiai, Matrix analysis: matrix monotone functions, matrix means, and majorization, Interdiscip. Inform. Sci., 16 (2010), 139–248.
  • R. A. Horn, Schlicht mappings and infinitely divisible kernels, Pacific J. Math., 38 (1971), 423–430.
  • F. Kraus, Über konvexe Matrixfunktionen, Math. Z., 41 (1936), 18–42.
  • K. Löwner, Über monotone Matrixfunktionen, Math. Z., 38 (1934), 177–216.
  • H. Osaka and J. Tomiyama, Double piling structure of matrix monotone functions and of matrix convex functions, Linear Algebra Appl., 431 (2009), 1825–1832.
  • M. Uchiyama, Operator monotone functions, positive definite kernels and majorization, Proc. Amer. Math. Soc., 138 (2010), 3985–3996.