REVIEW 3 major objections 2 minor 15 references
Factorizations of linear relations by idempotents
T0 review · 3 major / 2 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The product of two multivalued projections always equals a multivalued projection times a single-valued projection.
desk verdict Clean extension of Douglas factorization to linear relations with idempotent factors; the headline Mp² = Mp Q result checks out, though the abstract oversells the scope and the proofs lean heavily on the authors' earlier paper. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machine that drives the proofs is the canonical decomposition of idempotent relations: every idempotent $Q$ equals $P_{\operatorname{ran} Q \cap \operatorname{dom} Q, \ker Q} + (\{0\} \times \operatorname{mul} Q)$, a multivalued projection with that range and kernel together with a purely multivalued vertical piece (Proposition 2.5). Around this, the authors use linear selections — single-valued subrelations that keep the same domain — to split each multivalued projection into an operator part and a vertical part. That separation is what lets them move the product's range and multivalued part into the first factor and its domain and kernel into the second, so that the second factor can finally be replaced by a single-valued projection.
What would settle it
Produce two multivalued projections $P$ and $Q$ on a finite-dimensional linear space whose product $T = P Q$ cannot be written as $P' Q_0$ with $P'$ a multivalued projection and $Q_0$ a single-valued projection preserving $\operatorname{ran} P' = \operatorname{ran} T$, $\operatorname{mul} P' = \operatorname{mul} T$, $\operatorname{dom} Q_0 = \operatorname{dom} T$, and $\ker Q_0 = \ker T$; such a $T$ would be an explicit element of $\mathrm{Mp}^2$ outside the class $\mathrm{Mp}\,\mathcal{Q}$, directly contradicting Theorem 6.2.
Extended reading notes
Core claim
The central discovery is stated as Theorem 6.2: on any linear space $X$, if $P$ and $Q$ are multivalued projections and $T = P Q$, then there exist a multivalued projection $P'$ and a single-valued projection $Q_0$ such that $T = P' Q_0$, with $\operatorname{ran} P' = \operatorname{ran} T$, $\operatorname{mul} P' = \operatorname{mul} T$, $\operatorname{dom} Q_0 = \operatorname{dom} T$, and $\ker Q_0 = \ker T$. Equivalently, $\mathrm{Mp}^2 = \mathrm{Mp}\,\mathcal{Q}$: the product of two multivalued projections never really needs a second multivalued factor. The proof first uses Lemma 6.1 to adjust the given factors so that the first carries the range and multivalued part of $T$ and the second carries the domain and kernel of $T$, and then applies an operator-solution construction to replace the second factor by a single-valued projection while keeping the product unchanged. Direct corollaries are that every element of $\mathrm{Mp}^2$ has the normal form $P_0 Q_0 \oplus (\{0\}\times S)$ with $P_0$ and $Q_0$ single-valued projections and $S \subseteq \ker P_0$ (Corollary 6.4), and that in finite-dimensional spaces any such $T$ satisfies $\dim \operatorname{ran}(T-I) \le 2 \dim \ker T + \dim \operatorname{mul} T$ (Proposition 6.6).
Load-bearing premise
The main theorems assume that the previously established structural characterization of idempotent linear relations — that every idempotent decomposes as a multivalued projection plus a purely multivalued piece — is correct; if that characterization fails, the factorization proofs in Sections 5 and 6 lose their foundation.
Editorial extensions
If this is right
- The identity $\mathrm{Mp}^2 = \mathrm{Mp}\,\mathcal{Q}$ means every product of two multivalued projections has a representation with one single-valued projection, so the two-factor products are no larger as a class than the one-multivalued-one-single-valued products.
- Every $T \in \mathrm{Mp}^2$ admits the normal form $T = P_0 Q_0 \oplus (\{0\}\times S)$ with $P_0, Q_0$ single-valued projections and $S \subseteq \ker P_0$, separating the single-valued product from the multivalued part.
- Factorization of the form $R = S Q$ with $Q$ a projection holds exactly when $\operatorname{mul} R = \operatorname{mul} S$ and $\operatorname{dom} R = \{x : S(x)=R(x)\} + \ker R$ (Theorem 5.4).
- For the reverse order, $R = Q S$ with $Q$ a projection exists (with $Q$ sending the multivalued part of $S$ onto that of $R$) exactly when the same holds for some linear selections of $R$ and $S$ (Theorem 5.6).
- In finite-dimensional spaces, membership in $\mathrm{Mp}^2$ forces the dimension inequality $\dim \operatorname{ran}(T-I) \le 2 \dim \ker T + \dim \operatorname{mul} T$, extending the classical matrix criterion for products of idempotent matrices.
Reading between the lines
- The two-factor reduction in Theorem 6.2 suggests testing whether a product of three or more multivalued projections can also be collapsed to at most one multivalued factor; if the mechanism extends, finite products would have a simple normal form.
- The necessary condition $\dim \operatorname{ran}(T-I) \le 2 \dim \ker T + \dim \operatorname{mul} T$ from Proposition 6.6 is a natural candidate for a full characterization of $\mathrm{Mp}^2$ in finite dimensions; checking its converse would make the class easy to test with matrices.
- The agreement set $\{x : S(x)=R(x)\}$ appearing in Theorem 5.4 could be used as a quantitative defect in Hilbert-space factorization theorems, linking these algebraic characterizations to operator-range conditions.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies factorizations of linear relations by idempotents and multivalued projections. Section 5 gives Douglas-type criteria for when a linear relation R can be written as R = SQ or R = QS with Q a single-valued projection; Section 6 analyzes products of two multivalued projections. The central result, Theorem 6.2, states that every product T = PQ with P, Q multivalued projections can be rewritten as T = P' Q0, where P' is a multivalued projection with ran P' = ran T and mul P' = mul T, and Q0 is a single-valued projection with dom Q0 = dom T and ker Q0 = ker T. Equivalently, Mp^2 = Mp Q. The proofs rely on the structural description of idempotent relations from the authors' earlier paper [5], on linear selections, and on the operator factorization lemma of Sandovici and Sebestyén.
Significance. If the main results hold, the paper gives a clean canonical form for products of multivalued projections, extending known Hilbert-space results about products of projections to the linear-relation setting. The strongest claim, Theorem 6.2, is genuinely useful and is proved by a careful accounting of domains, kernels, and multivalued parts. The paper also provides useful characterizations of when a linear relation factors through a projection, and it includes a finite-dimensional dimension bound. The central proof of Theorem 6.2 appears mathematically sound, and I found no counterexample to its statement. The main reservations concern several places where the written proof asserts or invokes statements that are not justified as stated; these are fixable but need attention.
major comments (3)
- [§5, Proposition 5.3] In the proof of the forward implication, the line 'ran Q ⊆ ker(S−R)' is false as stated when S and R are not everywhere defined. Since dom(S−R) need not contain ran Q, membership in ker(S−R) may be undefined. For example, take X = K^2, S = 0 with dom S = span e1, Q = P_{span e2, span e1}, and R = SQ = 0 on span e1; then ran Q = span e2 lies outside dom(S−R). The intended argument can be repaired by observing that for every x in dom R, one has Qx in dom S ∩ dom R and S(Qx) = R(Qx), so x = Qx + (I−Q)x lies in ker(S−R) + ker R. The proof should be rewritten with this corrected step.
- [§6, Corollary 6.4] In the converse direction, the sentence 'By Lemma 6.1, we can assume that ker Q0 = ker P0 Q0' is not justified, because Lemma 6.1 applies only to relations already known to belong to Mp^2, whereas the converse is precisely trying to establish that membership. The conclusion does not need this assumption: since ker Q0 × {0} ⊆ P0 Q0, we have P0 Q0 + (ker Q0 × mul T) = P0 Q0 + ({0} × mul T), so T = (P0 ⊕ ({0} × mul T)) Q0 follows directly. The proof should be corrected accordingly.
- [§6, Theorem 6.2] The statement that Q0 := EQ satisfies dom Q0 = dom T = dom Q and ker Q0 = ker T is asserted immediately after invoking Remark 2, but Remark 2 only proves that Q0 is an operator with T = P Q0; it does not prove the domain and kernel prescriptions. These facts are true and have short verifications: for x in dom Q, the existence of z in Q(x) ∩ dom P follows from T = P Q and dom T = dom Q, and then z is in dom E, giving dom EQ = dom Q; and Q0 ⊆ Q together with the fact that (x,0) ∈ EQ whenever (x,0) ∈ Q gives ker Q0 = ker Q. Please add these justifications, as the prescribed domain and kernel are load-bearing parts of the theorem.
minor comments (2)
- [§6, Proposition 6.6] The display 'ran(T−I) = ran(P0Q0−I) ∔ mul T' is not generally a direct sum; for instance, if P0 = Q0 = 0 and mul T is nonzero, then ran(P0Q0−I) = X intersects mul T nontrivially. The subsequent dimension inequality only needs the ordinary sum, so the symbol ∔ should be replaced by +.
- [§4, Proposition 4.2] In the proof of (4.3), the symbol T is used both for the newly constructed linear relation and for the subspace T appearing in the decomposition, which makes the argument hard to follow. Please rename one of these objects.
Circularity Check
No significant circularity: the main factorization theorems are proved by explicit construction, and the cited prior results serve as transparent external support rather than as restatements of the target claims.
full rationale
I walked the derivation chain from Proposition 2.5 through Sections 3–6. The structural formula for idempotents is imported from the authors' earlier paper [5], but it is a published, externally checkable result whose statement does not include the target conclusion Mp^2 = Mp Q; it is used as a tool inside proofs, not as a substitute for them. Lemma 6.1 starts from an arbitrary factorization T = EF with E,F multivalued projections and constructs new factors P,Q with the prescribed invariants ran P = ran T, mul P = mul T, dom Q = dom T, and ker Q = ker T; this is a genuine construction verified through Propositions 3.3 and 3.7 and Corollary 3.4. Theorem 6.2 then builds a projection E with dom E = ran Q ∩ dom P and ker E = mul Q ∩ dom P, sets Q0 = E Q, and proves Q0 is a single-valued projection with the required domain and kernel. The inclusion Mp Q ⊆ Mp^2 is immediate because every projection is a multivalued projection, while the reverse inclusion is exactly what the construction shows; no assumption of the theorem's conclusion enters the argument. The use of Theorem 5.2 and Remark 2 relies on the external result [15], and the finite-dimensional corollary uses Ballantine's theorem [7]. The self-citations to [5] and [6] are normal scholarly dependencies, and I found no step where a quantity is defined in terms of itself, a fitted input is relabeled as a prediction, or a known result is merely renamed. The derivation is therefore self-contained modulo standard cited theorems, and the appropriate circularity score is 0.
Assumptions & free parameters
assumptions (4)
- standard math Zorn's lemma / axiom of choice: every linear relation admits a linear selection and algebraic complements of subspaces exist.
- domain assumption The structural theorem for idempotent relations from [5, Corollaries 4.3, 4.4] restated as Proposition 2.5: every Q in Id(X) equals P_{ran Q ∩ dom Q, ker Q + ({0} × mul Q)}.
- domain assumption The super-idempotent and idempotent characterizations from [5, Corollary 3.14 and Proposition 3.17] used in Proposition 4.2 to characterize {T: T ⊆ T²}, Id(X), and Mp(X).
- standard math Ballantine's theorem (Theorem 6.5): an n x n matrix S is a product of k idempotent matrices iff dim ran(I-S) <= k dim ker S.
Cite this review
Pith. "Pith review of Factorizations of linear relations by idempotents." pith.science (2026). https://pith.science/paper/JACCPVBV
@misc{pith2026250521123,
author = {Pith},
title = {Pith review of: Factorizations of linear relations by idempotents},
year = {2026},
howpublished = {\url{https://pith.science/paper/JACCPVBV}},
note = {Machine review of arXiv:2505.21123}
}
read the original abstract
We study the class of those linear relations that can be factorized as products of idempotent relations. We provide several characterizations of this class, extending known factorization results for operators to the more general setting of linear relations.
Reference graph
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Reviewed August 7, 2026 · model on record in the stance chip above.
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