In the general setting of the adjointable operators on Hilbert \(C^*\) -modules, this paper deals mainly with the weighted Moore–Penrose inverse (briefly weighted M–P inverse) \(A^\dag _{MN}\) in the case that the weights M and N are self-adjoint invertible operators, which need not to be positive. A new formula linking \(A^\dag _{MN}\) to A, \(A^\dag \) , M and N is derived, in which \(A^\dag \) denotes the M–P inverse of A. Based on this formula, some new results on the weighted M–P inverse are obtained. Firstly, it is shown that \(A^\dag _{MN}=A^\dag _{ST}\) for some positive definite operators S and T. This shows that \(A^\dag _{MN}\) is essentially an ordinary weighted M–P inverse. Secondly, some limit formulas for the ordinary weighted M–P inverse originally known for matrices are generalized and improved. Thirdly, it is shown that when A, M and N act on the same Hilbert \(C^*\) -module, \(A^\dag _{MN}\) belongs to the \(C^*\) -algebra generated by A, M and N. Finally, some characterizations of the continuity of the weighted M–P inverse are provided.
