A new formula for the weighted Moore-Penrose inverse and its applications
Qingxiang Xu
TL;DR
The paper develops a new, robust representation for the weighted Moore-Penrose inverse $A^ abla_{MN}$ on Hilbert $C^*$-modules with self-adjoint invertible weights, showing $A^ abla_{MN} = R_{A,N}^{-1} A^ abla L_{A,M^{-1}}^{-1}$ and $A^ abla_{MN}=A^ abla_{S_{A,M}T_{A,N}}$ where $T_{A,N}$ and $S_{A,M}$ are positive definite. This reveals that, under suitable conditions, $A^ abla_{MN}$ essentially corresponds to an ordinary weighted MP inverse and connects to the $C^*$-algebra generated by $A,M,N$, with the latter proving that the weighted inverse lies in $C^*\{A,M,N\}$. The work extends limit formulas to the Hilbert $C^*$-module setting, offers generalized continuity results for the weighted inverse, and establishes a faithful representation principle linking existence and values of weighted inverses across representations. A key contribution is the counterexample illustrating the necessity of the stated assumptions for continuity and limit results. Overall, the paper provides foundational tools for weighted least-squares and inverse problems in noncommutative settings.
Abstract
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 $A^†_{MN}$ in the case that the weights $M$ and $N$ are self-adjoint invertible operators, which need not to be positive. A new formula linking $A^†_{MN}$ to $A$, $A^†$, $M$ and $N$ is derived, in which $A^†$ 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 $A^†_{MN}=A^†_{ST}$ for some positive definite operators $S$ and $T$. This shows that $A^†_{MN}$ 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, $A^†_{MN}$ 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.