Annals of Functional Analysis

Kwong matrices and operator monotone functions on $(0,1)$

Juri Morishita, Takashi Sano, and Shintaro Tachibana

Full-text: Open access

Abstract

In this paper we study positive operator monotone functions on $(0, 1)$ which have some differences from those on $(0, \infty):$ we show that for a concave operator monotone function $f$ on $(0, 1),$ the Kwong matrices $K_f(s_1, \dots, s_n)$ are positive semidefinite for all $n$ and $s_i \in (0, 1),$ and $f(s^p)^{1/p}$ for $p \in (0,1]$ and $s/f(s)$ are operator monotone. We also give a sufficient condition for the Kwong matrices to be positive semidefinite.

Article information

Source
Ann. Funct. Anal., Volume 5, Number 1 (2014), 121-127.

Dates
First available in Project Euclid: 5 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1391614576

Digital Object Identifier
doi:10.15352/afa/1391614576

Mathematical Reviews number (MathSciNet)
MR3119119

Zentralblatt MATH identifier
1296.47016

Subjects
Primary: 47A63: Operator inequalities
Secondary: 15B48: Positive matrices and their generalizations; cones of matrices

Keywords
Kwong matrix operator monotone function Loewner matrix positive semidefinite

Citation

Morishita, Juri; Sano, Takashi; Tachibana, Shintaro. Kwong matrices and operator monotone functions on $(0,1)$. Ann. Funct. Anal. 5 (2014), no. 1, 121--127. doi:10.15352/afa/1391614576. https://projecteuclid.org/euclid.afa/1391614576


Export citation

References

  • T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26 (1979), 203–241.
  • K.M.R. Audenaert, A characterisation of anti-Löwner functions, Proc. Amer. Math. Soc. 139 (2011), 4217–4223.
  • R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, 169, Springer, 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.
  • W.F. Donoghue, Monotone Matrix Functions and Analytic Continuation, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer, 1974.
  • F. Hansen and G.K. Pedersen, Jensen's inequality for operators and Löwner's theory, Math. Ann. 258 (1982), 229–241.
  • F. Hiai, Matrix Analysis: Matrix monotone functions, matrix means, and majorization, Interdiscip. Inform. Sci. 16 (2010), 139–248.
  • F. Hiai and T. Sano, Loewner matrices of matrix convex and monotone functions, J. Math. Soc. Japan 64 (2012), 343–364.
  • C. Hidaka and T. Sano, Conditional negativity of anti-Loewner matrices, Linear Multilinear Algebra 60 (2012), 1265–1270.
  • M.K. Kwong, Some results on matrix monotone functions, Linear Algebra Appl. 118 (1989), 129–153.
  • K. Löwner, Über monotone Matrixfunctionen, Math. Z. 38 (1934), 177–216.
  • Y. Nakamura, Classes of operator monotone functions and Stieltjes functions, Oper. Theory Adv. Appl. 41, Birkhäuser (1989), 395–404.
  • T. Sano and S. Tachibana, On Loewner and Kwong matrices, Sci. Math. Jpn. 75 (2012), 335–338.