A norm inequality for positive block matrices

Abstract Any positive matrix M = ( M i , j ) i , j = 1 m with each block M i , j square satisfies the symmetric norm inequality ‖ M ‖ ≤ ‖ ∑ i = 1 m M i , i + ∑ i = 1 m − 1 ω i I ‖ , where ω i ( i = 1 , … , m − 1 ) are quantities involving the width of numerical ranges. This extends the main theorem of Bourin and Mhanna (2017) [4] to a higher number of blocks.