A norm compression inequality for block partitioned positive semidefinite matrices

Let A be a positive semidefinite matrix, block partitioned asA=BCC*D,where B and D are square blocks. We prove the following inequalities for the Schatten q-norm ∥·∥q:‖A‖qq⩽(2q-2)‖C‖qq+‖B‖qq+‖D‖qq,1⩽q⩽2,and‖A‖qq⩾(2q-2)‖C‖qq+‖B‖qq+‖D‖qq,2⩽q.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],‖A‖qq⩽∑i‖Aii‖qq+(2q-2)∑i