Generalized inverses of morphisms with kernels
Let \(\phi: X\to Y\) be a morphism with kernel \(\kappa\) : \(K\to X\) in an additive category with involution *. Then \(\phi\) has a Moore-Penrose inverse \(\phi^{\pi}\) with respect to * iff \(\phi\phi^* + \kappa^*\kappa\) is invertible; in this case, \(\phi^{\pi} = \phi^*(\phi\phi^* + \kappa^*\kappa)^{-1}\). If \(X=Y\), then \(\phi\) has a group inverse \(\phi^{\#}\) iff \(\phi\) has a cokernel \(\gamma: X\to K\) and \(\phi^ 2 + \gamma\kappa\) is invertible. In this case, \(\phi^{\#} = \phi(\phi^ 2 + \gamma\kappa)^{-1}\).
scientific article; zbMATH DE number 4029752
scientific article