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

TL;DR

This paper introduces a new formula for the weighted Moore-Penrose inverse in the context of Hilbert C*-modules, generalizing existing results and exploring its properties, limits, and continuity.

Contribution

It derives a novel formula linking the weighted Moore-Penrose inverse to the standard inverse and weights, extending known matrix results to Hilbert C*-modules.

Findings

A new formula for the weighted Moore-Penrose inverse is established.

The inverse is shown to be equivalent to an ordinary weighted inverse with positive definite weights.

Limit formulas for the inverse are generalized and improved.

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 $A^\dag_{MN}$ 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.