Journal of the
Korean Mathematical Society JKMS

Necessary and sufficient conditions are provided under which the weighted Moore--Penrose inverse $A^\dag_{MN}$ exists, where $A$ is an adjointable operator between Hilbert $C^*$-modules, and the weights $M$ and $N$ are only self-adjoint and invertible. Relationship between weighted Moore--Penrose inverses $A^\dag_{MN}$ is clarified when $A$ is fixed, whereas $M$ and $N$ are variable. Perturbation analysis for the weighted Moore--Penrose inverse is also provided.

Keywords: Hilbert $C^*$-module, weighted Moore--Penrose inverse, indefinite inner-product space

MSC numbers: Primary 46L08, 15A09, 47A05

Supported by: This work was supported by the National Natural Science Foundation of China (11671261) and a grant from Shanghai Municipal Science and Technology Commission (18590745200).