, Volume 315, Issue 4, pp 771-780

Norm inequalities related to operator monotone functions

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Abstract.

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.

Received: 1 February 1999