On sums of range symmetric matrices with reference to indefinite inner product (Q2283217)

A complex square matrix \(A\) is called \textit{range-Hermitian} if \(R(A)=R(A^*)\), where \(R(\cdot)\) stands for the range space (or the column space) of \(A\). There are manifold characterizations for a matrix to be range-Hermitian. One is that the Moore-Penrose inverse and the group inverse of \(A\) coincide. The authors characterize range-Hermitian matrices, viewed as linear operators over the space of vectors with complex entries, which is endowed with an indefinite inner product and not the usual inner product. Results for sums of range-Hermitian matrices to be range-Hermitian, as well as the parallel sum of parallel summable matrices to be range-Hermitian, are presented.