Maps preserving the idempotency of Jordan products

Gordana Radi\'c; Tatjana Petek

REVIEW 3 major objections 6 minor 19 references

Maps preserving the idempotency of Jordan products

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Any map on bounded operators that preserves, in both directions, whether a Jordan product is idempotent must be a similarity by an invertible linear or conjugate-linear operator, up to sign; in finite dimensions only field automorphisms…

desk verdict Solid classification result, but Lemma 21 is false as stated, and the main proof currently depends on it. read the letter →

arxiv 2506.04412 v1 pith:22TFSE32 submitted 2025-06-04 math.FA math.OA

Tatjana Petek , Gordana Radić This is my paper

classification math.FAmath.OA MSC 47B49

keywords JordanproductidempotencypreserverboundedlinearoperatorsBanachspacerank-oneidempotentstripotentsfieldautomorphismsnon-additivemaps

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 determines the structure of arbitrary maps on the algebra of bounded linear operators over a complex Banach space that preserve, in both directions, the statement "the Jordan product of two operators is idempotent". No linearity or additivity is assumed. If the map's range contains all nonzero idempotent and anti-idempotent operators, then in infinite dimensions the map must be a similarity by a bounded invertible linear or conjugate-linear operator, up to a sign, or a similarity through the adjoint when the space is reflexive; in finite dimensions the same rigidity holds up to a field automorphism and transposition. The point is that a purely relational condition, idempotency of $A\circ B$, is rigid enough to recover the full algebraic form of the map.

What carries the argument

The proof is carried by probe sets inside $\mathcal B(X)$: rank-one idempotents, nilpotents of rank one, tripotents (operators with $A^3=A$), and Jordan-block-like operators $T(k;\lambda)$. The core identity is Lemma 16, which says that for nontrivial idempotents $P,Q$ the orthogonality $P\perp Q$ is equivalent to $P\circ Q=0$ and to $-P\circ Q$ being idempotent. This converts the global idempotency condition into orthogonality preservation on rank-one idempotents, where a known classification of bijective maps preserving orthogonality in both directions produces the underlying operator $T$. Lemmas 7 through 10 transfer the relation from rank-one idempotents to nilpotents and tripotents, Lemma 21 identifies Jordan blocks through a Sylvester equation, and Lemma 25 shows that two operators coincide once their Jordan products with every probe in a similarity-invariant test set have the same idempotency and zero pattern.

What would settle it

Search, for instance in $M_3(\mathbb C)$, for a map $\phi$ whose range contains all nonzero idempotent and anti-idempotent matrices and such that $A\circ B$ is idempotent exactly when $\phi(A)\circ\phi(B)$ is idempotent, yet $\phi(P)$ is not a rank-one idempotent for some rank-one idempotent $P$. Step 5 of the proof derives that implication directly, so finding such an example would contradict the theorem; a small matrix search is a concrete way to test the classification.

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Extended reading notes

Core claim

For $X$ of dimension at least three, let $\phi:\mathcal B(X)\to\mathcal B(X)$ have range containing $I^{\pm}(X)$, that is, every nonzero idempotent $P$ with $P^2=P$ and every anti-idempotent $Q\neq0$ with $Q^2=-Q$, and assume that $A\circ B$ is idempotent exactly when $\phi(A)\circ\phi(B)$ is idempotent, where $A\circ B=\frac12(AB+BA)$. Theorem 1 says that in infinite dimensions $\phi$ must be $\phi(X)=\lambda TXT^{-1}$ for a bounded invertible linear or conjugate-linear operator $T$ and $\lambda\in\{1,-1\}$, or $\phi(X)=\lambda TX'T^{-1}$ implemented through the adjoint, the second possibility only when $X$ is reflexive. Theorem 2 says that on $M_n$, $n\ge3$, the same assumption yields $\phi([x_{ij}])=\lambda T[\sigma(x_{ij})]^{\diamond}T^{-1}$, where $\sigma$ is a field automorphism of $\mathbb C$, $\diamond$ is the identity or transposition, and $\lambda=\pm1$. This is the authors' sense of determination: the idempotency equivalence, with no additivity assumed, has only these algebraic solutions.

Load-bearing premise

The load-bearing assumption is that the map's range contains every operator $P$ with $P^2=P$ and every $Q\neq0$ with $Q^2=-Q$; without that supply of test operators the proof cannot establish bijectivity and orthogonality preservation on rank-one idempotents, and the classification argument stops.

Editorial extensions

If this is right

Idempotency preservation for the Jordan product is a complete invariant: the equivalence alone, together with the range condition, forces the map to be a global similarity or its adjoint analogue.
The scaled variants for a fixed nonzero $\alpha$, preserving when $\alpha(AB+BA)$ is idempotent, follow by the change of variables $\psi(X)=\sqrt{2\alpha}\,\phi(X/\sqrt{2\alpha})$, so the same rigidity holds for every rescaling of the Jordan product.
In finite dimensions the only extra freedom is a field automorphism and transposition; for continuous automorphisms of $\mathbb C$, which are only the identity and complex conjugation, the classification reduces to linear or conjugate-linear similarities.
Any such map must send rank-one idempotents bijectively onto rank-one idempotents and preserve orthogonality in both directions, so the whole classification is controlled by the action on one-dimensional idempotents.

Reading between the lines

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

One testable extension is whether the range condition can be weakened to containing only nonzero idempotents; the proof uses anti-idempotents essentially in Step 6 to establish $\phi(-P)=-\phi(P)$, so the symmetric condition $I^{\pm}$ appears necessary for the stated conclusion.
The probe-set strategy seems ready-made for longer Jordan products $A_1\circ\cdots\circ A_k$ or for the Jordan triple product $ABA$; the paper's lemmas already isolate the local information such variants would need.
The theorem implies a practical rigidity statement for any setting that models operations by idempotent transformations: an exact preserver of the idempotency relation is, up to the listed alterations, a change of basis.
Dimensions one and two are excluded; checking whether the conclusion holds there would show whether the dimensional restriction is technical or intrinsic.

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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

Let me get straight to it: the stress-test note is right, and it matters. Lemma 21, as stated, is false. Take n=3, λ=1, A=diag(1,0,0). A has exactly one nonzero spectral point. For k=1, with T_{1,R}=[[1,R],[0,0]], the Jordan product A∘T_{1,R}=2[[1,R/2],[0,0]], which is a nonzero idempotent for every row R. For k=2, T_{2,R}=[[J_2(1)^{-1},R],[0,0]] gives A∘T_{2,R}=[[2,-1,r],[0,0,0],[0,0,0]] (up to halving), again a nonzero idempotent. But A e_2=0, while the conclusion demands A e_2=e_1+e_2. So the lemma's hypothesis does not force the claimed conclusion. The proof step 'single spectral point' is doing work that the stated 'single nonzero spectral point' does not license; zero eigenvalues are allowed.

This is not a cosmetic issue. Lemma 22 cites Lemma 21 to get B11=J_n(λ); Lemma 25 uses Lemma 22 (and 24) to get injectivity of φ at Step 2 and the final identity at Step 14, and Step 13 uses Lemma 22 directly. The main classification therefore currently rests on a false statement. The likely fix is to strengthen Lemma 21 to spectrum(A)={λ}, and to show separately in Lemma 22 that B11 indeed has no zero eigenvalue—the present one-line 'otherwise contradiction' is not enough. The direct-sum situation in Lemma 25 is also only sketched; that is minor compared with Lemma 21 but should be spelled out.

All that said, this is not a crank paper. The classification result is genuinely new, the no-linearity/no-additivity setting is the right one, and the overall strategy—reduce to Semrl's theorem on rank-one idempotents, then extend by technical matrix lemmas—is sensible and mostly executed carefully. Lemmas 7-20 and the Step 1-14 architecture are readable and largely correct. The range condition is strong but explicit and load-bearing. The duplicated reference [16]/[17] is a typo.

My bottom line: this paper deserves a serious referee, but it is not ready as-is. I would send it to review with a specific request to fix Lemma 21 or replace the chain through Lemma 22. If the fix is routine (and I suspect it is), the paper becomes a solid contribution to preserver theory. If the spectral gap cannot be closed, the main theorems are unsupported.

Referee Report

3 major / 6 minor

Summary. The paper characterizes maps φ: B(X) → B(X), for a complex Banach space X of dimension at least three, that preserve in both directions the idempotency of the Jordan product A∘B = (1/2)(AB+BA), with no linearity or additivity assumption on φ. The main results are Theorem 1 (infinite-dimensional case): under the assumption that the range of φ contains all nonzero idempotents and anti-idempotents, φ has the form φ(X)=λT XT^{-1} with T bounded invertible linear or conjugate-linear and λ∈{−1,1}, or, only in the reflexive case, φ(X)=λT X' T^{-1}; and Theorem 2 (finite-dimensional case): the same conclusion up to a field automorphism and transposition. Theorems 4 and 5 are scaled versions obtained by a change of variables. The proof proceeds by first showing φ is injective, reducing the action on rank-one idempotents to a bijective orthogonality-preserving map (hence applying Semrl's classification), then extending the identification to nilpotents, tripotents, and Jordan blocks via a long series of auxiliary lemmas.

Significance. If the proof is repaired, this is a substantial result: it removes linearity and additivity assumptions in a preserver problem for the Jordan product and reduces the classification to Semrl's theorem on orthogonality-preserving maps on rank-one idempotents. The paper is self-contained modulo well-established external results, and the step-by-step reduction is clearly organized. The range condition (range containing I±(X)) is strong and is explicitly used in Steps 5 and 7; the theorems are conditional on it, which is a legitimate and clearly stated assumption. The main obstacle to accepting the paper is that Lemma 21 is false as stated, and the proof chain Lemma 21 → Lemma 22 → Lemma 25 → Theorems 1/2 therefore contains a load-bearing error. The error appears to be locally repairable by strengthening the lemma's hypothesis, but repair requires also filling a gap in Lemma 22.

major comments (3)

[Section 2, Lemma 21] Lemma 21 is false as stated. The hypothesis allows A to have 0 as an additional spectral point, but the proof uses the stronger assumption that A has a single spectral point. A concrete counterexample is A=diag(1,0,0) ∈ M_3 with λ=1. For every k=1,2 and every rank-one R, A∘T_{k,R} is a nonzero idempotent, as can be checked directly from the formula for T_{k,R}; however the conclusion would require Ae_2=e_1+e_2, whereas Ae_2=0. The failure occurs at the line "Since A is assumed to have a single spectral point, we obtain that A11 is an upper-triangular matrix with all diagonal entries equal to λ": when the additional spectral point is 0, the (k+1,k+1) entry of A11 can be 0, and A11∘J_{k+1}(λ)^{-1} can be a nonzero idempotent without forcing A11=J_{k+1}(λ). This lemma is load-bearing because Lemma 22 cites it to conclude B11=J_n(λ), and Lemma 22 is used in Step 13 of the proof of Theorems 1/2 and in Lemma 25. The statement should be corrected to require spectrum(A)={λ}, and the proof should be aligned with that hypothesis.
[Section 2, Lemma 22] In the proof of Lemma 22, the sentence "Since the only eigenvalue of B11 is λ, otherwise we are in a contradiction with Corollary 18 or Lemma 19, it follows that B11 = Jn(λ) from Lemma 21" is not a complete argument. To apply a corrected Lemma 21 one must prove that spectrum(B11)={λ}; the current text merely asserts this, and the referenced contradictions with Corollary 18 or Lemma 19 are not spelled out. This is not a cosmetic gap: if B11 had an additional spectral point 0, the (false) form of Lemma 21 would be the only available tool, and it cannot yield the conclusion. The authors need to provide a detailed proof that B11 has no eigenvalues other than λ, using the equivalences for F1(X) and T(n−1;λ^{−1}), before invoking Lemma 21. This is load-bearing for Step 13 and hence for Theorems 1 and 2.
[Section 2, Lemma 25] The final reduction in Lemma 25 — "apply Lemma 22 if λ_k≠0 and Lemma 24 otherwise" to a direct sum of distinct Jordan blocks — is only sketched. Lemmas 22 and 24 are stated for a single Jordan block J_n(λ) in the top-left corner of a 2×2 block decomposition, not for a direct sum of several blocks with distinct eigenvalues. The intended argument is plausible: one reorders the decomposition so that each Jordan block in turn occupies the distinguished subspace Y and applies the corresponding lemma, using the block-diagonal structure of A11. However, this needs to be written out, because Lemma 25 is used in Step 2 to prove injectivity of φ and again in Step 14 to complete the proof of Theorems 1/2. As it stands, the proof of Lemma 25 relies on an unstated generalization of Lemmas 22 and 24.

minor comments (6)

[Theorem 5 statement] The phrase "and cα, c2 α = σ(α)/α" is malformed; it should assert the existence of a scalar cα satisfying cα² = σ(α)/α.
[Proof of Theorems 4 and 5] In the finite-dimensional part of the proof, "by Theorem 5" should read "by Theorem 2", since Theorem 5 is the statement being proved.
[Step 13 of the proof of Theorems 1 and 2] The sentence "Lemma 16 provides that φ(T)=B11⊕0" is a mis-citation: the conclusion that φ(T) has the block-diagonal form B11⊕0 follows from Lemma 14, not Lemma 16.
[References] References [16] and [17] are identical (both list Semrl, "Non-linear commutativity preserving maps", Acta Sci. Math. (Szeged) 71 (2005), 781–819); the duplicate entry should be removed.
[References] In reference [9], "2th ed." should be "2nd ed.".
[Page 13, Proposition 23] The word "sclalars" should be "scalars".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof reduces to an external Semrl classification and internal lemmas; the sole self-citation is contextual.

full rationale

The paper's derivation chain is self-contained modulo independent published results. Step 7 reduces the problem to Semrl's classification [16, Theorems 2.3 and 2.4] of bijective orthogonality-preserving maps on rank-one idempotents, but only after Steps 4-6 prove, from the hypotheses and internal lemmas, that the restriction of phi to rank-one idempotents is bijective and preserves orthogonality in both directions. This is a genuine reduction to an external theorem, not an import of the target result. The remaining steps (8-14) build the full form of phi from this classification using internally proved lemmas: Lemma 21/22/24/25 and Proposition 23 are derived inside the paper, and the scaling theorems are obtained as explicit corollaries via a change of variables rather than by renaming the assumptions. No parameter is fitted, no quantity is defined in terms of the conclusion, and no load-bearing claim rests on a self-citation. The only self-citation, [14], appears in the introduction as contextual literature and is not used in any proof. The skeptic's counterexample to Lemma 21 is a mathematical-correctness concern about an intermediate lemma, not a circularity concern, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the published orthogonality-preserver classification [16] and standard operator-theoretic facts; no new entities or fitted parameters are introduced.

assumptions (5)

domain assumption Semrl's theorems [16, Thm 2.4 and Thm 2.3] classify bijective maps on rank-one idempotents preserving orthogonality in both directions.
Invoked in Step 7 of the proof of Theorems 1 and 2 to obtain the similarity form on rank-one idempotents. This is an external published theorem.
standard math Every algebraic operator has finite spectrum consisting of eigenvalues and shares it with its adjoint; each spectral value is an eigenvalue.
Used in Lemma 19 and Lemma 25 to handle algebraic operators.
standard math The Sylvester equation AX + XB = C has a unique solution when the spectra of A and -B are disjoint (Horn-Johnson [9, Thm 2.4.4.1]).
Used in Lemma 21 to identify A11 = J_{k+1}(lambda).
standard math Tripotent operators decompose as P - Q with unique orthogonal idempotents P, Q ([2, Proposition 1]).
Used in Step 9 and Lemma 24 to construct tripotents.
domain assumption The Banach space X has dimension at least 3; the finite-dimensional case is C^n with n >= 3.
Stated in the introduction; the theorems exclude small dimensions, as is common in preserver problems.

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Cite this review

Pith. "Pith review of Maps preserving the idempotency of Jordan products." pith.science (2026). https://pith.science/paper/22TFSE32

@misc{pith2026250604412,
  author       = {Pith},
  title        = {Pith review of: Maps preserving the idempotency of Jordan products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/22TFSE32}},
  note         = {Machine review of arXiv:2506.04412}
}

read the original abstract

Let B(X) be the algebra of all bounded linear operators on a complex Banach space X of dimension at least three. For an arbitrary nonzero complex number t we determine the form of mappings f: B(X)-->B(X) with sufficiently large range such that t(AB+BA) is idempotent if and only if t(f(A)f(B)+f(B)f(A)) is idempotent, for all A, B in B(X). Note that f is not assumed to be linear or additive.

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[17]

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Pith tools

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

[1] [17]

ˇSemrl, Non-linear commutativity preserving maps , Acta Sci

P. ˇSemrl, Non-linear commutativity preserving maps , Acta Sci. Math. (Szeged) 71 (2005), no. 3-4, 781–819

work page 2005

[2] [1]

Alaminos, M

J. Alaminos, M. Breˇ sar, J. Extremera and A. R. Villena, Maps preserving zero products, Studia Math. 193 (2009), no. 2, 131–159

work page 2009

[3] [2]

A. M. Bikchentaev and R. S. Yakushev, Representation of tripotents and repre- sentations via tripotents , Linea Algebra Appl. 435 (2011), no. 9, 2156–2165

work page 2011

[4] [3]

Breˇ sar and P

M. Breˇ sar and P. ˇSemrl, Zero product preserving maps on matrix rings over division rings , Linear and multilinear algebra and function spaces (2020), 195– 213

work page 2020

[5] [4]

M. A. Chebotar, W.-F. Ke and P.-H. Lee, Maps preserving zero Jordan products on Hermitian operators , Illinois J. Math. 49 (2005), no. 2, 445–452

work page 2005

[6] [5]

M. A. Chebotar, W.-F. Ke, P.-H. Lee and R. Zhang, On maps preserving zero Jordan product, Monatsh. Math. 149 (2006), no. 2, 91–101

work page 2006

[7] [6]

Fang, Linear maps preserving the idempotency of Jordan products of operators, Electron

L. Fang, Linear maps preserving the idempotency of Jordan products of operators, Electron. J. Linear Algebra 22 (2011), 767–779. 22

work page 2011

[8] [7]

Fang and W.-Z

L. Fang and W.-Z. Bai, Maps preserving the idempotency of products of triple Jordan products of idempotent operators, J. Shandong Univ. Nat. Sci. 45 (2010), no. 12, 98–105

work page 2010