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

Qingxiang Xu

REVIEW 1 major objections 5 minor 23 references

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

T0 review · 1 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A new formula gives the weighted Moore-Penrose inverse from the ordinary one.

desk verdict Genuinely new formula for weighted M-P inverses on Hilbert C*-modules; well-proved, but the referee should verify the imported existence criterion from [15]. read the letter →

arxiv 2502.09991 v1 pith:XBZFHT4G submitted 2025-02-14 math.FA

Qingxiang Xu This is my paper · ORCID

classification math.FA MSC 46L0847A0515A09

keywords weightedMoore-PenroseinverseHilbertC*-moduleadjointableoperatorindefiniteweightslimitformulasC*-algebrageneratedbyoperatorscontinuityofgeneralized

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that for adjointable operators on Hilbert $C^*$-modules, the weighted Moore-Penrose inverse with self-adjoint invertible weights exists exactly when the ordinary Moore-Penrose inverse exists and two operator factors are invertible. The weighted inverse is then $R^{-1}_{A,N} A^\dagger L^{-1}_{A,M^{-1}}$. This matters because it turns a previously opaque existence question into a check on two explicit operators and makes weighted inverses computable from ordinary inverses. The paper then uses this identity to show the weighted inverse is actually an ordinary weighted inverse with positive definite weights, to generalize limit formulas from matrices, to locate the weighted inverse in a $C^*$-algebra generated by $A$, $M$, and $N$, and to characterize continuity.

What carries the argument

The load-bearing object is the pair of factor operators $R_{A,X}=A^\dagger A+(I-A^\dagger A)X$ on the domain module and $L_{A,Y}=AA^\dagger+Y(I-AA^\dagger)$ on the codomain module, defined once $A$ is Moore-Penrose invertible. In the identity $A^\dagger_{MN}=R^{-1}_{A,N}A^\dagger L^{-1}_{A,M^{-1}}$, the left factor carries the weight $N$, the right factor carries the weight $M$ through $M^{-1}$, and invertibility of exactly these two operators is the extra condition beyond closed range that makes the weighted inverse exist. This factorization is the engine of the paper: the applications in Sections 3 through 5 are obtained by manipulating it into limit formulas, $C^*$-algebra membership, and continuity criteria.

What would settle it

For a Hilbert $C^*$-module, choose any adjointable $A$ with closed range and self-adjoint invertible weights $M,N$ such that $R_{A,N}$ and $L_{A,M^{-1}}$ are invertible, then check directly whether $X=R^{-1}_{A,N}A^\dagger L^{-1}_{A,M^{-1}}$ satisfies the four weighted Penrose equations; a single failure would refute Theorem 2.10, and testing Hermitian invertible matrices $M,N$ would be a concrete trial.

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Extended reading notes

Core claim

Theorem 2.10 is the central claim: for weights $M$ and $N$ that are self-adjoint and invertible on Hilbert $C^*$-modules, and for an adjointable operator $A$, the weighted Moore-Penrose inverse $A^\dagger_{MN}$ exists if and only if the ordinary Moore-Penrose inverse $A^\dagger$ exists and both $R_{A,N}=A^\dagger A+(I-A^\dagger A)N$ and $L_{A,M^{-1}}=AA^\dagger+M^{-1}(I-AA^\dagger)$ are invertible, and then $A^\dagger_{MN}=R^{-1}_{A,N}A^\dagger L^{-1}_{A,M^{-1}}$. The proof moves through the one-sided weighted inverses $A^\dagger_{I_K,N}$ and $A^\dagger_{M,I_H}$, using the weight-change lemmas of the cited existence theory. A second theorem shows that whenever $A^\dagger_{MN}$ exists it equals $A^\dagger_{S,T}$ for positive definite operators $S$ and $T$, so indefinite-weight inverses still fall inside the classical positive-definite weighted inverse framework.

Load-bearing premise

The paper relies on a published characterization: the weighted inverse exists exactly when the range of $A$ is closed and the two range conditions $R(AN^{-1}A^*)=R(A)$ and $R(A^*MA)=R(A^*)$ hold; if that characterization fails in some Hilbert $C^*$-module setting, the stated domain of the new formula would be misdescribed.

Editorial extensions

If this is right

Existence of the weighted Moore-Penrose inverse for self-adjoint invertible weights is reduced to two invertibility checks, so any method that computes ordinary Moore-Penrose inverses can be adapted to the weighted case.
The equality $A^\dagger_{MN}=A^\dagger_{S,T}$ with positive definite $S,T$ means the indefinite-weight case is not a separate species; it is an ordinary weighted inverse with carefully chosen positive definite weights.
The limit formula $\lim_{t\to 0^+}(A^*VA+tB^*WB)^\dagger A^*V = A^\dagger_{VU}$ holds under the weaker assumption that $R(A^*)+R(B^*)$ is closed, rather than the whole space, and formula (3.24) is new even for matrices.
When $A$, $M$, and $N$ act on the same Hilbert $C^*$-module and $A^\dagger_{MN}$ exists, it belongs to the $C^*$-algebra generated by $A$, $M$, and $N$, so faithful unital representations preserve both existence and value.
The weighted Moore-Penrose inverse is continuous in operator norm under perturbations exactly when the ordinary Moore-Penrose inverse is continuous, and small norm perturbations of the weights of an existing weighted inverse preserve existence and convergence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the factorization holds at operator level, the same identity should transfer directly to numerical linear algebra: any routine that forms $A^\dagger$ can compute $A^\dagger_{MN}$ for Hermitian invertible weights by solving two linear equations, with no dedicated weighted-inverse solver needed.
The freedom described in Remark 2.4, where many positive definite weights give the same weighted inverse, suggests a practical choice of well-conditioned replacement weights in computations, though the paper does not address conditioning.
If the imported existence characterization from [15] were ever found to require additional range hypotheses in exotic Hilbert $C^*$-modules, the equivalence in Theorem 2.10 would need repair, but the formula itself would likely remain valid on the smaller domain where the characterization holds.
The continuity results could be sharpened by asking for explicit convergence rates or bounds in terms of the perturbations of $A$, $M$, and $N$; the paper establishes norm convergence but does not quantify the rate.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Main take: this is a solid paper with a genuinely new formula. Theorem 2.10/2.13 is the centerpiece: for self-adjoint invertible weights M,N on Hilbert C*-modules, the weighted M-P inverse exists if and only if A† exists and the two operators R_{A,N} and L_{A,M^{-1}} are invertible, and then A†_{MN} = R^{-1}_{A,N} A† L^{-1}_{A,M^{-1}}. The reduction to positive definite weights (Theorem 2.11) and the limit formulas in Section 3 are real extensions of Ward's matrix results, under the weaker condition that R(A*)+R(B*) is closed rather than the whole space; formula (3.24) appears to be new even for matrices.

The proofs are careful, and the algebraic steps after the imported lemmas check out. The block-matrix computations in Lemma 3.12 and Theorem 2.11 are standard but executed well. The paper also correctly flags where it is generalizing known results, and I found no circularity: the heavy self-citations to [15] are to published, proved theorems, and (2.13) is a new consequence of them, not an assumption.

The one soft spot is exactly the one your stress-test names. Lemma 2.2, quoted from [15, Theorem 2.4], is the existence criterion on which the equivalence in Theorem 2.10 rests. It is stated without proof, and the paper does not discuss its scope. That said, I see no evidence that [15] was only proved for Hilbert spaces; it appears to be in the same indefinite inner-product operator setting. So I would treat this as a referee checklist item, not a demonstrated flaw. There are also minor typos ('lim_{t→0†}' in Theorem 3.24, some garbled words in the abstract). Cosmetic.

Bottom line: the paper is for specialists in generalized inverses and Hilbert C*-module operator theory. It deserves a serious referee, and I would accept after minor proofreading plus a referee check of [15, Theorem 2.4]'s scope.

Referee Report

1 major / 5 minor

Summary. The paper studies the weighted Moore-Penrose inverse A†_{MN} for adjointable operators on Hilbert C*-modules, where M and N are self-adjoint invertible weights that need not be positive. The main result is a new formula, A†_{MN} = R^{-1}_{A,N} A† L^{-1}_{A,M^{-1}} (Theorem 2.10), proved to hold if and only if the ordinary Moore-Penrose inverse A† exists and the two operators R_{A,N} and L_{A,M^{-1}} are invertible. The paper then derives several consequences: every such weighted M-P inverse is in fact an ordinary weighted M-P inverse with positive definite weights (Theorem 2.11); limit formulas for ordinary weighted M-P inverses are generalized from the matrix case to Hilbert C*-modules (Theorems 3.14, 3.15, 3.20, 3.24); a C*-algebra generation result is proved (Theorem 4.3); and continuity characterizations for the weighted M-P inverse are given (Theorems 5.2 and 5.3). The proofs are detailed and appear algebraically correct, with a numerical example illustrating the main formula.

Significance. If the central formula is correct, it provides a direct and conceptually clean way to compute weighted M-P inverses with indefinite weights from the ordinary M-P inverse and two explicitly defined factors. The paper also shows that such weighted inverses are essentially ordinary positive-definite weighted inverses, and it yields new limit formulas that were previously known only for matrices under stronger hypotheses. The C*-algebra generation result and the continuity theorems are natural and useful applications. The proofs are rigorous, and the paper is clearly written apart from numerous typographical errors. The main relevance is to the theory of generalized inverses on Hilbert C*-modules and to researchers working with indefinite weights.

major comments (1)

[Section 2, Lemma 2.2] The central equivalence in Theorem 2.10 and all later applications rest on Lemma 2.2, which is quoted verbatim from [15, Theorem 2.4]. Since the paper's central claim is formula (2.13) in the full Hilbert C*-module setting, the author should either include a proof of Lemma 2.2 or explicitly state that [15, Theorem 2.4] is proved in exactly this setting. In particular, please confirm that the two range equalities R(AN^{-1}A*) = R(A) and R(A*MA) = R(A*) do not need to be supplemented by additional closedness or complementedness conditions on these ranges in Hilbert C*-modules. If the original theorem in [15] was proved under more restrictive hypotheses, the statement of Theorem 2.10 and all subsequent results that rely on it would need to be narrowed accordingly.

minor comments (5)

[Proof of Theorem 3.24] The expressions 'lim_{t→0†}' appear twice and should read 'lim_{t→0+}'.
[Abstract and title page] There are multiple typos, including 'Hilbe rt', 'Sh anghai', and 'br ieﬂy'.
[Proof of Theorem 5.2] The line 'L_{A_n,M_n} → L_{A,M^{-1}}' should be 'L_{A_n,M_n^{-1}} → L_{A,M^{-1}}'.
[Proof of Theorem 2.11] The word 'Combing' should be 'Combining'.
[Remark 5.1] The phrase 'there exist certain Hilbert space H' should read 'there exist a Hilbert space H'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the new weighted Moore-Penrose formula is derived from published lemmas, not assumed as its own input, though the main existence criterion is imported from the author's prior work.

full rationale

The paper's central formula (2.13), A†_{MN} = R^{-1}_{A,N} A† L^{-1}_{A,M^{-1}}, is obtained by algebraic combination of Lemma 2.2 and Lemmas 2.5–2.9, which are quoted from the published paper [15]. Lemma 2.2 is the only load-bearing imported premise; it gives a necessary-and-sufficient existence criterion in terms of range conditions, and its conclusion is not the paper's target formula. Theorem 2.10 then proves that those range conditions imply the invertibility of R_{A,N} and L_{A,M^{-1}}, and uses Lemmas 2.8–2.9 together with uniqueness of the weighted M-P inverse to establish (2.13). The later applications in Theorems 2.11, 3.14, 3.15, 3.20, 4.3, and 5.2 apply that formula rather than assuming its conclusion. There are no fitted parameters, no data subsets, and no prediction that is statistically forced by a previous fit. Although [15] is co-authored by the present author and is cited heavily, the cited theorem is parameter-free, has stated assumptions that do not include the target equality, and is externally checkable mathematical support; under the stated rules this does not constitute circularity. The score of 2 reflects the substantial but non-circular reliance on the author's earlier lemmas, not any reduction of the derivation to its own inputs. The only serious caveat is correctness-related: if [15, Theorem 2.4] had a narrower domain than full Hilbert C*-modules, Theorem 2.10 would overstate its scope, but that would be an imported-premise error, not circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. It relies on standard Hilbert C*-module theory plus several published theorems from the same research group (especially [15]), which are treated as black boxes. The most load-bearing imported premise is the existence characterization of the weighted M-P inverse.

assumptions (7)

domain assumption Existence criteria for weighted M-P inverse (Lemma 2.2 from [15, Theorem 2.4]): A†_{MN} exists iff R(A) is closed, R(AN^{-1}A*) = R(A), and R(A*MA) = R(A*).
Imported from a prior paper co-authored by the current author; used as the foundation for Theorem 2.10.
domain assumption M-P inverse exists iff range is closed (Lemma 2.3 from [21]).
Standard in Hilbert C*-module theory; used throughout the paper.
standard math Closedness of R(A), R(A*), R(AA*), R(A*A) are equivalent (Lemma 2.4 from [7,21]).
Classical result used to reduce existence conditions and orthogonal decompositions.
domain assumption Parallel sum results for adjointable operators (Lemma 3.18 from [10]).
Used in Lemma 3.18 to connect closedness of R(A*A)+R(B*B) to closedness of R(A*) and R(B*).
domain assumption Two-projections theorem for Hilbert C*-modules (Lemma 3.17 from [9]).
Used to characterize separated pairs and the invertibility of 2I - P - Q.
domain assumption Continuity of M-P inverse in C*-algebras (Theorem 1.6 from [6]).
Used as the base equivalence in Theorem 5.2 after reduction to a C*-algebra setting.
standard math Basic C*-algebra theory: invertible elements in a generated C*-algebra and norm limits of elements in a C*-algebra remain in it (e.g., [12]).
Used in Section 4 for C*-algebra membership of the weighted M-P inverse.

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Cite this review

Pith. "Pith review of A new formula for the weighted Moore-Penrose inverse and its applications." pith.science (2026). https://pith.science/paper/XBZFHT4G

@misc{pith2026250209991,
  author       = {Pith},
  title        = {Pith review of: A new formula for the weighted Moore-Penrose inverse and its applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XBZFHT4G}},
  note         = {Machine review of arXiv:2502.09991}
}

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.

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [15]

M. Qin, Q. Xu and A. Zamani, Weighted Moore-Penrose inve rses of ad- jointable operators on indeﬁnite inner-product spaces, J. Korean Math. Soc. 57 (2020), no. 3, 691–706

work page 2020

[2] [1]

Corach and A

G. Corach and A. Maestripieri, Weighted generalized inv erses, oblique projections, and least-squares problems, Numer. Funct. An al. Optim. 26 (2005), no. 6, 659–673

work page 2005

[3] [2]

Eld´ en, Perturbation theory for the least squares pro blem with linear equality constraints, SIAM J

L. Eld´ en, Perturbation theory for the least squares pro blem with linear equality constraints, SIAM J. Numer. Anal. 17 (1980), no. 3, 338–350

work page 1980

[4] [3]

Eld´ en, A weighted pseudoinverse, generalized singu lar values, and constrained least squares problems, BIT 22 (1982), no

L. Eld´ en, A weighted pseudoinverse, generalized singu lar values, and constrained least squares problems, BIT 22 (1982), no. 4, 48 7–502

work page 1982

[5] [4]

Eskandari, W

R. Eskandari, W. Luo, M. S. Moslehian, Q. Xu and H. Zhang, S eparated pairs of submodules in Hilbert C ∗-modules, J. Math Anal. Appl. 545 (2025), no. 1, Paper No. 129102, 18 pp

work page 2025

[6] [5]

Kamaraj and K

K. Kamaraj and K. C. Sivakumar, Moore–Penrose inverse in an indeﬁ- nite innner product space, J. Appl. Math. Comput. 19 (2005), no. 1–2, 297–310

work page 2005

[7] [6]

J. J. Koliha, Continuity and diﬀerentiability of the Moor e-Penrose in- verse in C ∗-algebras, Math. Scand. 88 (2001), no. 1, 154–160. 38

work page 2001

[8] [7]

E. C. Lance, Hilbert C ∗-modules–A toolkit for operator algebraists (London Mathematical Society Lecture Note Series 210), Cam bridge University Press, Cambridge, 1995

work page 1995