A norm compression inequality for block partitioned positive semidefinite matrices

Let A be a positive semidefinite matrix, block partitioned as where B and D are square blocks. We prove the following inequalities for the Schatten q-norm • q: and We show that these bounds obey a strong sharpness condition when the blocks are of size at least 2 × 2, and B q, D q C q. Finally, our bounds can be extended to symmetric partitionings into larger numbers of blocks: for A = [Aij], while for 2 q the inequality is reversed.