Eigenvalue majorization inequalities for positive semidefinite block matrices and their blocks

Let H=(MKK⁎N) be a positive semidefinite block matrix with square matrices M and N of the same order and denote i=−1. The main results are the following eigenvalue majorization inequalities: for any complex number z of modulus 1,λ(H)≺12λ([M+N+i(zK⁎−z¯K)]⊕O)+12λ([M+N+i(z¯K−zK⁎)]⊕O). If, in addition, K is Hermitian, then for any real number r∈[−2,2],λ(H)≺12λ((M+N+rK)⊕O)+12λ((M+N−rK)⊕O), while if K is skew-Hermitian, then for any real number r∈[−2,2],λ(H)≺12λ((M+N+riK)⊕O)+12λ((M+N−riK)⊕O), where O is the zero matrix of compatible size. These majorization inequalities generalize some results due to Furuichi and Lin, Turkmen, Paksoy and Zhang, Lin and Wolkowicz.