Relations between two operator inequalities motivated by the theory of operator means (Q813291)

The author shows several results on operator inequalities motivated by the theory of operator means. The main result is the following Theorem. Let \(A\) and \(B\) be positive operators, and let \(f\) and \(g\) be non-negative continuous functions on \([0,\infty)\) satisfying \(f(t)g(t)=t\). Then the following hold: (i) \(f(B^\frac{1}{2}AB^\frac{1}{2})\geq B\) ensures that \(A-g(A^\frac{1}{2}BA^\frac{1}{2})\geq A^\frac{1}{2}E_BA^\frac{1}{2}-g(0)E_{A^\frac{1}{2} BA^\frac{1}{2}},\) (ii) \(B\geq f(B^\frac{1}{2}AB^\frac{1}{2})\) ensures that \(g(A^\frac{1}{2}BA^\frac{1}{2})-A\geq g(0))E_{A^\frac{1}{2} BA^\frac{1}{2}}-A^\frac{1}{2}E_BA^\frac{1}{2},\) where \(E_B\) and \(E_{A^\frac{1}{2} BA^\frac{1}{2}}\) are the orthoprojections to the kernels of \(B\) and \(A^\frac{1}{2}BA^\frac{1}{2}\), respectively.