A norm compression inequality for block partitioned positive semidefinite matrices (Q817644)

The paper deals with Schatten-norm inequalities of block matrices. More explicitly, the author shows that if \(A\) is a positive semi-definite block matrix \(A=\left[\begin{smallmatrix} B&C\\ C^*&D\end{smallmatrix}\right]\) with \(B\), \(D\) square, then for \(1\leq q\leq2\), \[ \| A\| _q^q\leq(2^q-2)\,\| C\| _q^q+\| B\| _q^q+\| D\| _q^q, \tag{1} \] with the inequality reversed for \(q\geq2\). Equality occurs not only in the \(q=2\) case, but also in the \(q=1\) case. The inequality (1) admits a straightforward extension to \(n\times n\) positive semi-definite block matrices.