A Norm Compression Inequality for Block Partitioned Positive Semidefinite Matrices

Let $A$ be a positive semidefinite matrix, block partitioned as $$A=\twomat{B}{C}{C^*}{D},$$ where $B$ and $D$ are square blocks. We prove the following inequalities for the Schatten $q$-norm $||.||_q$, which are sharp when the blocks are of size at least $2\times2$: $$||A||_q^q \le (2^q-2) ||C||_q^q + ||B||_q^q+||D||_q^q, \quad 1\le q\le 2,$$ and $$||A||_q^q \ge (2^q-2) ||C||_q^q + ||B||_q^q+||D||_q^q, \quad 2\le q.$$ These bounds can be extended to symmetric partitionings into larger numbers of blocks, at the expense of no longer being sharp: $$||A||_q^q \le \sum_{i} ||A_{ii}||_q^q + (2^q-2) \sum_{i<j} ||A_{ij}||_q^q, \quad 1\le q\le 2,$$ and $$||A||_q^q \ge \sum_{i} ||A_{ii}||_q^q + (2^q-2) \sum_{i<j} ||A_{ij}||_q^q, \quad 2\le q.$$