Diagonal and off-diagonal blocks of positive definite partitioned matrices

Abstract

We obtain new relations for the blocks of a positive semidefinite matrix [AXX⁎B] partitioned into four blocks in Mn. A consequence is(0.1)|X+X⁎|≤A+B+14V(A+B)V⁎ for some unitary V∈Mn. Here, for n≥2, the constant 1/4 cannot be replaced by any smaller one. Several eigenvalue inequalities for general matrices follow from our result. We can also derive from (0.1) some triangle type inequalities, for instance for three contractions,(0.2)|S+T+R|≤34I+|S|+|T|+|R| where I stands for the identity. We conjecture that the constant 3/4 is optimal.