EP matrices in indefinite inner product spaces (Q2867600)

Let \(J_n\) be a fixed \(n\times n\) complex matrix such that \(J_n=J_n^*=J_n^{-1}\). The indefinite matrix product of complex matrices \(A,B\) (with formats \(m\times n\) and \(n\times l\), respectively) is defined by \(A\circ B=AJ_nB\). For \(A\in \mathbb C^{m\times n}\), a matrix \(X\in \mathbb C^{n\times m}\) is called the Moore-Penrose inverse and denoted by \(A^{[\dagger]}\) if \(A\circ X\circ A=A\), \(X\circ A\circ X=X\), \((A\circ X)^{[\dagger]}=A\circ X\) and \((X\circ A)^{[\dagger]}=X\circ A\). The author calls a matrix \(A\in \mathbb C^{n\times n}\) a \(J\)-\(EP\) matrix if \(A\circ A^{[\dagger]}=A^{[\dagger]}\circ A\). Several properties of such matrices are proved and compared with the respective properties of \(EP\)-matrices in the sense of \textit{K. Kamaraj, K. Ramanathan} and \textit{K. C. Sivakumar} [J. Anal. 12, 135--142 (2004; Zbl 1097.47002)]. In particular, the inverse order law with respect to the Moore-Penrose inverse in the indefinite setting is discussed. Some appropriate examples are presented.