A new formula for the weighted Moore-Penrose inverse and its applications

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 dagger(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 dagger(MN )to A, A dagger, M and N is derived, in which A dagger 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 dagger(MN) = A dagger ST for some positive definite operators S and T. This shows that A dagger(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 dagger(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.