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An extension of the Löwner-Heinz inequality. (English) Zbl 1272.47027

An operator \(A\) on a Hilbert space \(\mathcal{H}\) is called positive \((A\geq 0)\) if \(\langle Ax,x\rangle \geq 0\) for all \(x\in \mathcal{H}.\) For positive operators \(A,B\) on a Hilbert space \(\mathcal {H}\) such that \(A\geq B\) and \(A-B\) is invertible, the celebrated Loewner-Heinz inequality says \(A^r\geq B^r\), \(0<r\leq 1\). The authors extend this inequality by proving that \(A^r-B^r\geq ||A||^r-\left (||A||-\frac{1}{||(A-B)^{-1}||}\right )^r>0\). As an application, they obtain the inequality \(\log A-\log B\geq \log ||A||-\log \left (||A||-\frac{1}{||(A-B)^{-1}||}\right )>0.\)

MSC:

47A63 Linear operator inequalities
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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