A Norm Compression Inequality for Block Partitioned Positive Semidefinite Matrices

Let $A$ be a positive semidefinite matrix, block partitioned as $$A=\text{twomat}BC{C}^{\ast }D,$$ where $B$ and $D$ are square blocks. We prove the following inequalities for the Schatten $q$-norm $\mathrm{\mid }\mathrm{\mid }\mathrm{.}\mathrm{\mid }{\mathrm{\mid }}_q$, which are sharp when the blocks are of size at least $2\times 2$: $$\mathrm{\mid }\mathrm{\mid }A\mathrm{\mid }{\mathrm{\mid }}_q^q\le (2^q-2)\mathrm{\mid }\mathrm{\mid }C\mathrm{\mid }{\mathrm{\mid }}_q^q+\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }{\mathrm{\mid }}_q^q+\mathrm{\mid }\mathrm{\mid }D\mathrm{\mid }{\mathrm{\mid }}_q^q,1\le q\le 2,$$ and $$\mathrm{\mid }\mathrm{\mid }A\mathrm{\mid }{\mathrm{\mid }}_q^q\ge (2^q-2)\mathrm{\mid }\mathrm{\mid }C\mathrm{\mid }{\mathrm{\mid }}_q^q+\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }{\mathrm{\mid }}_q^q+\mathrm{\mid }\mathrm{\mid }D\mathrm{\mid }{\mathrm{\mid }}_q^q,2\le q\mathrm{.}$$ These bounds can be extended to symmetric partitionings into larger numbers of blocks, at the expense of no longer being sharp: $$\mathrm{\mid }\mathrm{\mid }A\mathrm{\mid }{\mathrm{\mid }}_q^q\le \sum _i\mathrm{\mid }\mathrm{\mid }{A}_{ii}\mathrm{\mid }{\mathrm{\mid }}_q^q+(2^q-2)\sum _{i<j}\mathrm{\mid }\mathrm{\mid }{A}_{ij}\mathrm{\mid }{\mathrm{\mid }}_q^q,1\le q\le 2,$$ and $$\mathrm{\mid }\mathrm{\mid }A\mathrm{\mid }{\mathrm{\mid }}_q^q\ge \sum _i\mathrm{\mid }\mathrm{\mid }{A}_{ii}\mathrm{\mid }{\mathrm{\mid }}_q^q+(2^q-2)\sum _{i<j}\mathrm{\mid }\mathrm{\mid }{A}_{ij}\mathrm{\mid }{\mathrm{\mid }}_q^q,2\le q\mathrm{.}$$