Original article

Mathematische Annalen

, Volume 315, Issue 4, pp 771-780

First online:

Norm inequalities related to operator monotone functions

  • Tsuyoshi AndoAffiliated withFaculty of Economics, Hokusei Gakuen University, Sapporo 004-8631, Japan (e-mail: ando@hokusei.ac.jp)
  • , Xingzhi ZhanAffiliated withInstitute of Mathematics, Peking University, Beijing 100871, China (e-mail: zhan@sxx0.math.pku.edu.cn)

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Let A,B be positive semidefinite matrices and \(|||\cdot|||\) any unitarily invariant norm on the space of matrices. We show \( |||f(A) + f(B)||| \geq |||f(A + B)|||\) for any non-negative operator monotone function f(t) on \([0,\infty)\), and \( |||g(A) + g(B)||| \leq |||g(A + B)||| \) for non-negative increasing function g(t) on \([0,\infty)\) with g(0) = 0 and \(g(\infty) = \infty\), whose inverse function is operator monotone.

Mathematics Subject Classification (1991):47A30, 47B15, 15A60