Eigenvalue majorization inequalities for positive semidefinite block matrices and their blocks

Let H = ((M)(K* N) (K)) be a positive semidefinite block matrix with square matrices M and N of the same order and denote i = root-1. The main results are the following eigenvalue majorization inequalities: for any complex number z of modulus 1, lambda(H) < 1/2 lambda ([M + N + i(zK* - <(z)over bar>K)] circle plus O) 1, + 1/2 lambda ([M + N + i((z) over barK - zK*)] circle plus O). If, in addition, K is Hermitian, then for any real number r is an element of [-2, 2], lambda(H) < 1/2 lambda ((M + N + rK) circle plus O) + 1/2 lambda ((M + N - rK) circle plus O) , while if K is skew-Hermitian, then for any real number r is an element of [-2, 2], lambda(H) < 1/2 lambda ((M + N + riK) circle plus O) + 1/2 lambda ((M + N - riK) circle plus 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. (C) 2013 Elsevier Inc. All rights reserved.