A new formula for the weighted Moore-Penrose inverse and its applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929715401457664 |
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| author | Xu, Qingxiang |
| author_facet | Xu, Qingxiang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new formula for the weighted Moore-Penrose inverse and its applications Xu, Qingxiang Functional Analysis 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. |
| title | A new formula for the weighted Moore-Penrose inverse and its applications |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2502.09991 |