HLPK and Sherman-type theorems for Schur complement of matrices

Abstract

In this paper, by using majorization relations for tuples of matrices and of linear operations, some operator versions of Hardy-Littlewood-Pólya-Karamata and Sherman theorems for Schur complements of positive definite matrices are established. The stochasticity of the used operation matrices is partially reduced, and the commutativity of the involved matrices is engaged.