The generalized and pseudo $n$-strong Drazin inverse of the sum of elements in Banach algebras

Falguni Roy; Rounak Biswas

REVIEW 3 major objections 3 minor 19 references

The generalized and pseudo $n$-strong Drazin inverse of the sum of elements in Banach algebras

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In a Banach algebra, an element is generalized n-strong Drazin invertible exactly when $a^n-a^{2n}$ is quasinilpotent, and pseudo n-strong Drazin invertible exactly when $a-a^{n+1}$ lies in the radical closure $p\sqrt{J(A)}$.

desk verdict Useful characterizations with a repairable additive gap: the main theorem pair is solid, but Theorem 3.8 needs a real proof fix before the additive claims are accepted. read the letter →

arxiv 2506.03845 v1 pith:SX2R5A57 submitted 2025-06-04 math.FA

Rounak Biswas , Falguni Roy This is my paper

classification math.FA MSC 15A0916N20

keywords generalizedn-strongDrazininversepseudoadditivepropertyBanachalgebraJacobsonradicalquasinilpotentweighted

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

This paper establishes a single spectral test for two flavours of n-strong Drazin invertibility in a Banach algebra: an element $a$ is generalized n-strong Drazin invertible precisely when $a^n-a^{2n}$ is quasinilpotent, equivalently when $a-a^{n+1}$ is quasinilpotent, and pseudo n-strong Drazin invertible precisely when $a^n-a^{2n}$, equivalently $a-a^{n+1}$, lies in the radical closure $p\sqrt{J(A)}$. With these characterizations the authors prove additive results: conditions under which $a+b$ is gns- or pns-invertible, including an equivalence theorem tying $a+b$, $a\cdot a^{\mathrm{nsd}}\cdot(a+b)$, $(a+b)\cdot b^{\mathrm{nsd}}$, and the two-sided product. The paper then defines weighted gns and pns inverses and shows they reduce to ordinary ones by multiplying on either side by the weight.

What carries the argument

The main machinery is the polynomial $p(x)=x-x^{n+1}$, used with the spectral mapping theorem to force the spectrum of a gns-invertible element into the roots of unity together with zero, making the spectral point $0$ isolated and the element g-Drazin invertible. The additive proofs rely on block-matrix decompositions relative to spectral idempotents, on lemmas saying that a sum of quasinilpotent (or radical-closure) elements remains quasinilpotent (or radical-closure) under weak commutation conditions, and on the identity $(ab)-(ab)^{n+1}=a^{n+1}(b-b^{n+1})+(a-a^{n+1})b$, which was derived in Lemma 2.7 under the hypothesis $a^2b=aba$.

What would settle it

Compute both sides of the identity $(ab)-(ab)^{n+1}=a^{n+1}(b-b^{n+1})+(a-a^{n+1})b$ in a Banach algebra under the hypothesis $ab^2=bab$. In $M_2(\mathbb{C})$ with $a=e_{12}$, $b=e_{21}$, and $n=2$, the left side is $0$ and the right side is $e_{11}$, so the identity is not a consequence of the stated hypothesis. To falsify Theorem 3.8 as a statement, one would need a pair of pns-invertible elements satisfying $ab^2=bab$ for which $ab$ is not pns-invertible.

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Extended reading notes

Core claim

The central claim is that generalized n-strong Drazin invertibility and pseudo n-strong Drazin invertibility are each characterized by one condition on powers of the element, with no separate Drazin-invertibility hypothesis needed. For sums, Theorem 2.20 states a list of equivalences under the mixed commutation conditions $a^2b=aba$ and $ab^2=bab$, and Theorem 3.17 is the pseudo analogue. The weighted version asserts that $a$ is wgns-invertible if and only if $wa$ and $aw$ are gns-invertible, with the explicit formula $a^{\mathrm{nsd},w}=((aw)^{\mathrm{nsd}})^2 a$, and similarly for the pseudo weighted inverse.

Load-bearing premise

The product proof for pseudo n-strong Drazin inverses relies on an identity previously proved under the condition $a^2b=aba$, while the theorem's stated hypothesis is only $ab^2=bab$; that identity can fail under the weaker condition.

Editorial extensions

If this is right

Checking gns or pns invertibility reduces to a single quasinilpotence or radical-closure test on powers of the element, instead of verifying the full Drazin equations.
The additive equivalence theorems give four interchangeable formulations for $a+b$, so establishing any one of them certifies the other three.
Weighted gns and pns inverses are explicitly constructible from ordinary ones via $((aw)^{\mathrm{nsd}})^2 a$ and $((aw)^{\mathrm{pnsD}})^2 a$.
The results extend earlier sum theorems for generalized Drazin and generalized strong Drazin inverses to the n-strong setting, including the pseudo case.
Examples show that $a+b$ can be n-strong Drazin invertible without $ab$ being quasinilpotent, so the additive characterizations are genuinely broader than a simple product condition.

Reading between the lines

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

The pseudo product theorem (Theorem 3.8) is not established as written: its proof invokes the identity from Lemma 2.7, which was proved under $a^2b=aba$, while the theorem assumes only $ab^2=bab$.
A concrete check shows the identity is not a consequence of the weaker hypothesis: in $M_2(\mathbb{C})$ with $a=e_{12}$, $b=e_{21}$, and $n=2$, the left side $(ab)-(ab)^{n+1}$ is $0$ while the right side $a^{n+1}(b-b^{n+1})+(a-a^{n+1})b$ equals $e_{11}$.
Repairing the pseudo product theorem likely requires adding $a^2b=aba$ to its hypotheses or finding a different identity that holds under $ab^2=bab$ alone.
The weighted characterization suggests that in operator-algebra settings, weighted n-strong Drazin invertibility can be tested by ordinary invertibility after one-sided multiplication by the weight, which may simplify computations.

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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

Rounak and Falguni's paper is a serious contribution to the generalized-inverse literature. The central results—Theorem 2.4 and Theorem 3.5—are clean characterizations: a is gns-Drazin invertible iff a^n − a^{2n} is quasinilpotent, and pns-invertible iff the same expression lies in √J(A). These are new, they unify earlier special cases, and the proofs check out. That alone is worth something.

The additive sections are where I have problems. The sign errors in Theorem 2.9 and Theorem 3.9 are real but minor: the expansion of (a+b)−(a+b)^{n+1} should have a minus sign on the mixed term. With ab=0 the corrected term is still quasinilpotent/radical, so the theorem survives a simple fix.

Theorem 3.8 is more serious. The reader's report claims equation (1) fails under ab^2=bab, with an e12/e21 example. That example does not satisfy ab^2=bab, and the identity is actually correct under that hypothesis—it follows from (ab)^m = a^m b^m. So the stress-test note is right to dismiss that counterexample. But the stress-test also identifies the real gap: after writing ab−(ab)^{n+1}=X+Y, the proof gives no reason why X=a^{n+1}(b−b^{n+1}) and Y=(a−a^{n+1})b lie in √J(A). The radical √J(A) is not closed under multiplication by arbitrary elements, and the hypothesis ab^2=bab is not used to show those specific products remain in √J. The one-line proof "It follows from equation 1 and Theorem 3.5" is not enough. As a result, Theorem 3.8 is not established as written, and Theorem 3.17 inherits the gap because it uses the same decomposition pattern.

None of this is fatal to the main characterizations. The paper is honest, the bibliography is standard, and the weighted section in Section 4 is a reasonable extension. The right outcome is a revise-and-resubmit, not a rejection. I would send it to a referee who knows the area and will insist on a full proof of Theorem 3.8 and the two sign fixes. It belongs in a specialized journal. I would bring it to our reading group, if only to talk about what a correct proof of Theorem 3.8 needs.

Referee Report

3 major / 3 minor

Summary. The paper introduces and studies generalized n-strong Drazin (gns) and pseudo n-strong Drazin (pns) invertibility in Banach algebras. The main structural results are the characterizations of Theorem 2.4 (a is gns-invertible if and only if a^n - a^{2n} is quasinilpotent) and Theorem 3.5 (a is pns-invertible if and only if a^n - a^{2n} lies in the radical closure psqrt(J(A))), together with the corollaries a - a^{n+1} quasinilpotent or in psqrt(J(A)). These results are then applied to prove product and additive theorems, including the equivalence statements in Theorems 2.20 and 3.17, and weighted analogues in Section 4.

Significance. The main characterizations are clean, potentially useful, and appear to be correct; they extend earlier results of Chen-Sheibani and Mosic in a natural direction. The manuscript contains no fitted parameters or circular definitions, and the proofs of the characterizations rely on standard spectral mapping and radical arguments. However, several load-bearing additive and product theorems are not established as written because of missing justifications and a sign error. If the gaps are repaired, the paper would make a solid contribution to the generalized Drazin inverse literature.

major comments (3)

[Section 3, Theorem 3.8] The proof of Theorem 3.8 is the single sentence "It follows from equation 1 and Theorem 3.5." This is not sufficient. Corollary 3.6 gives a - a^{n+1} and b - b^{n+1} in psqrt(J(A)), but equation (1) expresses (ab) - (ab)^{n+1} as a^{n+1}(b - b^{n+1}) + (a - a^{n+1})b. The proof never shows that these two summands lie in psqrt(J(A)) or satisfy the hypotheses of Lemma 3.2 or Lemma 3.3. psqrt(J(A)) is not closed under multiplication by arbitrary elements, and no argument is given that the particular multipliers a^{n+1} and b preserve it under the hypothesis ab^2 = bab. The concrete example a = e12, b = e21 does not satisfy the hypothesis ab^2 = bab, so it is not a counterexample to the theorem; nevertheless the proof gap is real. Consequently Theorem 3.8, and any result that depends on it, is not established as written.
[Theorems 2.9 and 3.9] The expansion in the proof of Theorem 2.9 has the wrong sign. The displayed identity reads (a+b) - (a+b)^{n+1} = (a - a^{n+1}) + (b - b^{n+1}) + sum_{i=1}^n b^i a^{n+1-i}, but under ab = 0 the mixed term should be subtracted, not added. The subsequent appeal to Lemma 2.2 is therefore not justified for the expression as written. The same sign error appears in the proof of Theorem 3.9. Since Theorem 2.9 is used in later arguments, this error propagates and must be corrected.
[Theorem 2.20 and Theorem 3.17] The proof of Theorem 2.20 contains unsupported steps that need to be supplied. In particular, the claim that b4 is in Ansd_2 is inferred from a a^nsd b being in Ansd, although that product only sees the (1,1) block of b; an additional spectral or block-matrix argument is needed. Later, the step "by Lemma 2.7" from p1 + (b41)^{-1}a21 in Ansd to a21 + b41 in Ansd requires both factors of the product to be gns-invertible, but invertibility alone does not imply gns-invertibility. The statement of Theorem 3.17 says it follows "as Theorem 2.20" and therefore inherits these unresolved points. These are gaps in the proof of the paper's main additive equivalence results, not merely presentation issues.

minor comments (3)

[Lemma 3.11(2) and Lemma 3.12] There are evident subscript typos: in Lemma 3.11(2) the conclusion should refer to ApnsD_1 and ApnsD_2 in the correct order, and in Lemma 3.12 the statement that "b1 is in ApnsD_2" should say ApnsD_1, since b1 is an element of the corner A1.
[Introduction, definition of g-Drazin inverse] The displayed definition of the g-Drazin inverse uses the condition b = ab^2. This is nonstandard and may confuse readers; it is equivalent to the usual b = bab only when combined with ab = ba, so the equivalence should be stated explicitly.
[Throughout] The notation "gns" and "pns" is typeset inconsistently (e.g., "g ns-invertibility" in the abstract and "gns-Drazin invertible" in the body). The paper would benefit from consistent formatting.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the characterizations and additive theorems are derived from the definitions together with standard spectral, radical, and block-decomposition facts, and the apparent gap in Theorem 3.8 is a proof issue rather than a circular reduction.

full rationale

This paper contains no fitted parameters, no empirical predictions, and no definitional identifications that smuggle in the target result. The main characterizations, such as Theorem 2.4 and Corollary 2.5 for generalized n-strong Drazin invertibility and Theorem 3.5 and Corollary 3.6 for pseudo n-strong Drazin invertibility, are proved from the definitions together with standard spectral mapping, Jacobson radical properties, and previously established lemmas; the equivalence between a^n - a^{2n} and a - a^{n+1} belonging to the relevant class is proved in Proposition 2.3 and Proposition 3.4, not assumed. The additive theorems are obtained by explicit block decompositions relative to spectral idempotents and by reducing the sum to diagonal blocks whose invertibility is handled by earlier lemmas. The weighted section explicitly observes that wgns and wpns invertibility is ordinary gns and pns invertibility in the Banach algebra A_w, so the transfer of additive results is a direct deduction rather than a renaming of a known result. The only concern visible in the manuscript is the one-line proof of Theorem 3.8, which invokes Equation (1) from Lemma 2.7 even though that identity was derived under the hypothesis a^2 b = a b a while Theorem 3.8 assumes only a b^2 = b a b. That is a possible derivation gap or missing hypothesis, not a circular step: the proof does not assume the conclusion, does not define a quantity in terms of the target result, and does not rely on a self-citation. There are no load-bearing self-citations: the cited results from Mosic, Chen, Sheibani, Zou, and others are external prior work. Accordingly, there is no significant circularity and the score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new mathematical objects beyond standard definitions from the cited literature. It relies on standard spectral and radical properties, and on block decomposition facts for Banach algebras with idempotents.

assumptions (3)

standard math Spectral mapping theorem for Banach algebra elements
Used in Theorem 2.4 to infer σ(a) ⊂ {0} ∪ {λ : λ^n = 1} from σ(a - a^(n+1)) = {0}.
standard math Properties of the Jacobson radical J(A) and the set p√J(A)
Used throughout Section 3, with results cited from [18] for closure under sums and products.
domain assumption Spectrum of a block triangular element equals the union of spectra of diagonal blocks
Used in Lemmas 2.12 and 3.11 for Peirce decompositions relative to idempotents; standard but not proved in this paper.

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Pith. "Pith review of The generalized and pseudo $n$-strong Drazin inverse of the sum of elements in Banach algebras." pith.science (2026). https://pith.science/paper/SX2R5A57

@misc{pith2026250603845,
  author       = {Pith},
  title        = {Pith review of: The generalized and pseudo $n$-strong Drazin inverse of the sum of elements in Banach algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SX2R5A57}},
  note         = {Machine review of arXiv:2506.03845}
}

abstract

In this paper, we begin by introducing some necessary and sufficient conditions for generalized $n$-strong Drazin invertibility (g$n$s-invertibility) and pseudo $n$-strong Drazin invertibility (p$n$s-invertibility) of an element in a Banach algebra for $n\in\mathbb{N}$. Subsequently, these results are utilized to prove some additive properties of g$n$s (p$n$s)-Drazin inverse in a Banach algebra. This process produces a generalization of some recent results of H Chen, M Sheibani (Linear and Multilinear Algebra \textbf{70.1} (2022): 53-65) for g$n$s and p$n$s-Drazin inverse. Furthermore, we define and characterize weighted g$n$s and weighted p$n$s-Drazin inverse in a Banach algebra.

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Works this paper leans on

19 extracted references · 19 canonical work pages

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Pith tools

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

[1] [1]

Castro-Gonz´ alez, N and Mart ´ ınez-Serrano, MF, Expressions for the g-Drazin inverse of additive perturbed elements in a Banach algebra, Linear algebra and its applications 432 (2010) 1885–1895

work page 2010

[2] [2]

Chen, Huanyin and Sheibani, Marjan, Generalized Hirano inverses in rings, Communications in Algebra47 (2019) 2967–2978

work page 2019

[3] [3]

Chen, Huanyin and Sheibani, Marjan, The g-Drazin inverse of the sum in Banach algebras, Linear and Multilinear Algebra70 (2022) 53–65

work page 2022

[4] [4]

Cvetkovi´ c-Ili´ c, Dragana S and Djordjevi´ c, Dragan S and Wei, Yimin, Additive results for the generalized Drazin inverse in a Banach algebra, Linear algebra and its applications418 (2006) 53–61

work page 2006

[5] [5]

Cvetkovi´ c-Ili´ c, Dragana S, The generalized Drazin inverse with commutativity up to a factor in a Banach algebra,Linear algebra and its applications431 (2009) 783–791

work page 2009

[6] [6]

Daji´ c, A and Koliha, JJ, The weighted g-Drazin inverse for operators, Journal of the Australian Mathematical Society82 (2007) 163–181

work page 2007

[7] [7]

Deng, Chun-yuan and Wei, Yi-min, New additive results for the generalized Drazin inverse, Journal of mathematical analysis and applications370 (2010) 313–321

work page 2010

[8] [8]

Diesl, Alexander J, Nil clean rings, Journal of Algebra383 (2013) 197–211

work page 2013