Norm inequalities for sums and differences of positive operators (Q1826836)

The author gives the following inequality: For positive operators \(A\) and \(B\), \[ \begin{multlined} 2| A\oplus B \oplus 0\oplus 0 | \leq | (A-B)\oplus (A-B)\oplus 0\oplus 0 | + | A\oplus A\oplus B\oplus B| \\ + | A^{\frac{1}{2}} B^{\frac{1}{2}}\oplus A^{\frac{1}{2}} B^{\frac{1}{2}} \oplus A^{\frac{1}{2}} B^{\frac{1}{2}}\oplus A^{\frac{1}{2}}B^{\frac{1}{2}} | ,\end{multlined} \] where \(| \cdot | \) means any unitarily invariant norm. Especially, in the usual operator norm case, it becomes the following inequality: \[ \max(\| A\| , \| B\| )-\| A^{\frac{1}{2}}B^{\frac{1}{2}}\| \leq \| A-B\| . \] Moreover, the author gives some comments on some norm inequalities related to the sum and difference of \(A\) and \(B\).