A converse of Loewner-Heinz inequality and applications to operator means
Let \(f\) be an operator monotone function. Then \(A \leq B\) implies \(f(A) \leq f(B)\), but the converse is not true. In this paper, the authors investigate the converse implication from the viewpoint of operator means. More precisely, they show that, if \(f(t )> 0\) is a non-constant operator monotone function defined on a neighborhood of \(t = 1\) with \(f(1) = 1\), and \(A\) and \(B\) are bounded self-adjoint operators, then the following statements are equivalent: {\parindent=0.8cm\begin{itemize}\item[(i)] \(A \leq B\), \item[(ii)] \(\lambda A + I \leq \lambda B + I\) for every \(\lambda \geq 0\), \item[(iii)] \(f(\lambda A + I)\leq f(\lambda B + I)\) for all sufficiently small \(\lambda \geq 0\), (iv) \(f(\lambda B + I)^{-1}\sharp f(\lambda A + I)\leq I\) for all sufficiently small \(\lambda \geq 0\), where \(\sharp\) denotes the usual operator mean. \end{itemize}} In addition, they prove that, if \(f\) is a non-constant operator monotone function on \((0,\infty)\) with \(f (1) = 1\), and \(A\) and \(B\) be bounded self-adjoint operators acting on a Hilbert space \(H\), then the following are equivalent: {\parindent=0.8cm\begin{itemize}\item[(i)] \(A \leq B\), \item[(ii)] \(\|x\|^2 \leq \|f(\lambda A+I)^{-1/2}x\|\,\|f(-\lambda B+I)^{-1/2}x\|\) for all \(x\in H\) and all sufficiently small \(\lambda \geq 0\), \item[(iii)] \(\|x\|^2 \leq \|e^{-pA}x\|\,\|e^{pB}x\|\) for all \(x\in H\) and all \(p \geq 0\). \end{itemize}} They also extend their results to the case of multi-variable power means and the Karcher mean.
scientific article; zbMATH DE number 6420194