Diagonal and off-diagonal blocks of positive definite partitioned matrices

We obtain new relations for the blocks of a positive semidefi-[(A) (X)(X* ) (B) ]partitioned into four blocks in M-n. A consequence is |X + X*| <= A + B +(1)(4) V(A + B)V* (0.1) for some unitary V is an element of M-n. 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, |S+T + R | <= I-3(4)+|S|+|T|+|R| (0.2) where I stands for the identity. We conjecture that the con-stant 3/4 is optimal. (c) 2023 Elsevier Inc. All rights reserved.