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 (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.