Diagonal and off-diagonal blocks of positive definite partitioned matrices (Q6141068)

Inequalities for positive (semi-definite) matrices partitioned into \(2\times 2\) blocks constitute a great portion of important theorems in matrix analysis. After reviewing a few remarkable examples of this topic, the authors present something new. One of the featured results states that if \[\begin{bmatrix}A & X\\ X^*& B\end{bmatrix}\] is positive, then \[|X+X^*|\le A+B+\frac{1}{4}V(A+B)V^*\] for some unitary matrix \(V\). The constant \(\frac{1}{4}\) is optimal. Similar results with \(+\) replaced with the Hadamard product \(\circ\), \(-\), and the geometric mean \(\sharp\) are considered as well.