REVIEW 5 major objections 4 minor 21 references
Convergence of Complementable Operators
T0 review · 5 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read The paper proves that a norm limit of $(M,N)$-complementable operators is again $(M,N)$-complementable whenever $\lambda_n\|D-D_n\|\to 0$, where $\lambda_n$ is the complementability constant and $D_n$ is the $D$-block of the approximants.
desk verdict Worth refereeing: new and mostly correct results on convergence of complementable operators, but Theorem 3.5 has a repairable proof gap, Lemma 3.10 is false as stated, and Section 4 needs rewriting. 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 central object is the $(M,N,\lambda)$-complementability pair of unit-ball inclusions $C(B_M)\subset \lambda D(B_{M^\perp})$ and $B^*(B_N)\subset \lambda D^*(B_{N^\perp})$. These inclusions turn the algebraic range-inclusion conditions $R(C)\subseteq R(D)$ and $R(B^*)\subseteq R(D^*)$ into quantitative bounds controlled by a single constant $\lambda$, which is what makes convergence arguments possible. The class $B_{CD}(M,N)$ enters through the reduced minimum modulus criterion $\gamma(D)>0$ for $R(D)$ closed. Douglas's factorization lemma supplies the bounded factorizations and Douglas reduced solutions used to construct Schur complements, while Theorem 3.2 converts norm convergence of the block operators into norm convergence of each component, allowing the limiting inclusions to be derived block by block.
What would settle it
On $H=K=\ell^2\oplus\ell^2$ with $M=\ell^2\oplus\{0\}$, set $B_n=0$, choose diagonal $D_n\to D$ with $R(D)$ closed and diagonal $C_n\to C\neq 0$, and compute the minimal $\lambda_n$ for which $C_n(B_M)\subset \lambda_nD_n(B_{M^\perp})$. If $\lambda_n\|D-D_n\|\to 0$ yet $R(C)\not\subseteq R(D)$, Theorem 3.5 is false. Example 3.4 shows the difficulty: there $\lambda_n$ must grow like $1/\|D_n\|$, leaving $\lambda_n\|D-D_n\|$ bounded away from zero.
Extended reading notes
Core claim
Theorem 3.5 is the paper's central assertion: for $\{T_n\}\subseteq \psi(M,N,\lambda_n)$ and $T\in B_{CD}(M,N)$ with $T_n\to T$ in norm and $\lambda_n\|D-D_n\|\to 0$, one obtains $T\in\psi(M,N)$. The proof uses the characterization of complementability by the inclusions $C(B_M)\subset \lambda D(B_{M^\perp})$ and $B^*(B_N)\subset \lambda D^*(B_{N^\perp})$ and shows that the approximating inclusions pass to the limit once the $D$-block error is controlled. Corollaries give the same conclusion when the $\lambda_n$ are bounded or when $\|D-D_n\|/\gamma(D_n)\to 0$. Theorems 3.12 and 3.13 show that every $(M,N)$-complementable operator with infinite-dimensional $R(C)$, or with finite-dimensional $R(C)$ and infinite-dimensional $R(D)$, is a strong limit of non-$(M,N)$-complementable operators, hence lies on the boundary of the non-complementable set; thus complementability is not preserved by strong approximation in general. Finally, Section 4 proves that $\psi(M,N,\lambda)$ is closed inside the class of operators with closed-range $D$-block, and closed in $B(H,K)$ when $K$ is finite-dimensional.
Load-bearing premise
The load-bearing premise is a characterization quoted without proof from the authors' companion paper: an operator is $(M,N)$-complementable exactly when one $\lambda$ makes both unit-ball inclusions hold, and the convergence theorems do not check this characterization independently.
Editorial extensions
If this is right
- With bounded $\lambda_n$, the limit lies in $\psi(M,N,\lambda)$ with $\lambda=\sup_n\lambda_n$, so fixed-constant complementability is closed under uniform limits inside $B_{CD}(M,N)$.
- If $\|D-D_n\|/\gamma(D_n)\to 0$, the limit is again complementable, giving a closed-range relative-error version of the theorem.
- For $T\in\varphi(M,N,\lambda)$, every scalar multiple of a power $\alpha_nT^n$, every partial sum of the power series, and under the geometric growth condition the full series $\sum\alpha_nT^n$ stays in $\psi(M,N,\lambda)$ whenever its limit has closed-range $D$-block.
- Every complementable operator with infinite-dimensional $R(C)$, and every one with finite-dimensional $R(C)$ and infinite-dimensional $R(D)$, lies on the boundary of the non-complementable set in the strong operator topology.
- $\psi(M,N,\lambda)$ is closed in $B_{CD}(M,N)$, and when $K$ is finite-dimensional it is closed in $B(H,K)$.
Reading between the lines
- The printed proof of Theorem 3.5 contains a step where $\lambda_n\|D-D_n\|\to 0$ is used to conclude $\beta_n\|D-D_n\|\to 0$ for the running maximum $\beta_n$; this implication is not valid as written, though the conclusion appears recoverable by comparing $C_nx=\lambda_nD_ny_n$ directly with $Dy_n$, possibly with an extra monotonicity assumption on $\lambda_n$.
- A likely extension is a weak-complementability version of Theorem 3.5: because $(M,N,\lambda)$-complementability implies weak complementability and the two notions coincide when $R(D)$ is closed, the same rate condition should force the limit to be at least weakly $(M,N)$-complementable, and fully complementable whenever the limit $D$-block has closed range.
- Read as a stability margin, the condition $\lambda_n\|D-D_n\|\to 0$ says that an iterative approximation preserves complementability exactly when the growth of the complementability constant is certified against the $D$-block error; without that certificate, Example 3.4 shows complementability can be lost in the limit.
- The strong-topology boundary results suggest that complementability is generically edge behavior: in infinite dimensions, a typical complementable operator is surrounded by non-complementable operators, so preservation theorems must involve norm convergence or extra range regularity, not strong convergence alone.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops convergence and topological results for (M,N)-complementable operators between Hilbert spaces. After recalling the block-operator framework and the characterization of complementability via unit-ball inclusions (Theorem 2.5, quoted from the authors' previous work), it proves in Theorem 3.5 that a norm limit of (M,N,λ_n)-complementable operators with closed-range D block is again (M,N)-complementable provided λ_n‖D−D_n‖→0. This result is then used to derive corollaries on bounded λ_n, on convergence relative to reduced minimum modulus, and on subsequences/series of powers. The paper also constructs strong-operator approximations showing that complementable operators with infinite-dimensional R(C), or with finite-dimensional R(C) and infinite-dimensional R(D), lie on the boundary of the non-complementable set (Theorems 3.12 and 3.13), and it closes with a characterization of closed subsets of ψ(M,N,λ) inside the closed-range class.
Significance. If the results are correct, the main closure criterion identifies a sharp stability threshold for infinite-dimensional Schur complement decompositions, and the boundary theorems give a concrete topological picture of where complementable operators sit among all bounded operators. The paper is constructive: Example 3.4 is a correct counterexample to unqualified closure, and the boundary proofs explicitly build approximating sequences that fail complementability. The intended proof of Theorem 3.5 has a transparent repair, and the statements of the boundary results are credible. However, the manuscript in its current form contains a false implication in the proof of the central theorem and a false lemma on power series, so the technical content needs substantive revision before the results can be accepted.
major comments (5)
- [Theorem 3.5 proof] The step 'Since λ_n‖D−D_n‖ → 0, it follows that β_n‖D−D_n‖ → 0' is false for the running maximum β_n = sup_{i≤n} λ_i. For instance, take λ_{2k}=2k, λ_{2k+1}=1, ε_{2k}=(2k)^{-2}, ε_{2k+1}=(2k)^{-1}; then λ_n ε_n→0 but β_n ε_n = 1 along odd indices. The subsequent Cauchy argument for {β_n D y_n} depends on this implication, so the proof of Theorem 3.5 as printed is invalid. The theorem can be repaired: for x∈B_M write C_nx=λ_nD_nz_n with ‖z_n‖≤1; since C_nx→Cx and λ_n(D_n−D)z_n→0 by the hypothesis, D(λ_nz_n)→Cx, and closedness of R(D) gives Cx∈R(D), with a symmetric argument for B*. This repair should be incorporated explicitly, as Corollaries 3.6–3.8 and Theorem 4.1 rely on Theorem 3.5.
- [Lemma 3.10] Lemma 3.10 is false as stated. The 'only if' direction fails: take T=I on a Hilbert space, α_0=0, α_n=1/n^2 for n≥1. The series ∑ α_nT^n converges in norm, but |α_n|^{1/n}‖T‖ = n^{-2/n} → 1, so no β<1 satisfies |α_n|^{1/n}‖T‖ ≤ β for all n. The 'if' direction is correct and is the only part used in Theorem 3.11(3), but the equivalence asserted in the lemma is wrong and should be replaced by a statement of the valid one-way implication (or by a corrected criterion, e.g., limsup |α_n|^{1/n}‖T‖ < 1).
- [Theorem 2.5 / Section 2] All λ-indexed results (ψ(M,N,λ), Theorems 3.5, 3.6, 3.8, 3.11, Section 4) rest on Theorem 2.5, which is quoted from the authors' forthcoming paper [19] without proof. Since the characterization in Theorem 2.5 is the foundation of the closure argument, the present paper is not self-contained: an error or a missing hypothesis in [19] would propagate into every main theorem. I recommend including a proof or a detailed sketch of Theorem 2.5, or citing a publicly available version of [19], so that the foundation is verifiable.
- [Corollary 3.8 proof] The proof of Corollary 3.8 applies Theorem 2.5 to write inclusions with the limit operators' norms (‖C‖ and ‖B‖) rather than with ‖C_n‖ and ‖B_n‖. As written, this does not follow. The intended argument is to apply the characterization to each T_n with λ_n = ‖C_n‖/γ(D_n) (and similarly for the adjoint block), then check λ_n‖D−D_n‖→0 using the fact that ‖C_n‖ is bounded. The displayed inclusions should be corrected accordingly.
- [Corollary 4.2 proof] The proof of Corollary 4.2 does not correctly establish closure of ψ(M,N,λ) inside B_CD(M,N). The displayed inclusions show only trivial subset relations; the needed argument is that any T in the closure of ψ(M,N,λ) inside B_CD(M,N) is the limit of a sequence in ψ(M,N,λ), and then Theorem 4.1 gives T∈ψ(M,N,λ). The present text should be rewritten to present that argument.
minor comments (4)
- [Corollary 3.14 proof] The proof line 'we have T ∈∂((ψ(M,N))∁), if R(D)' is incomplete; it should read 'if R(D) is infinite-dimensional' (or, more precisely, combine Theorems 3.12 and 3.13 to cover all cases with R(D) infinite-dimensional).
- [Theorem 3.11 proof] In part (2), the expression 'Sn = (∑_{i=0}^n α_{i+1}T^n)T' is a typo: the index of T inside the sum should be i, not n, and the sum should run over i=0,…,n−1 (or equivalently Sn = (∑_{i=1}^n α_iT^i)).
- [Abstract and Introduction] There are several typos and duplicated words, e.g., 't hereby', 'pap er', and 'is said to be is said to be strongly convergent'. A careful copyedit is needed.
- [Corollary 3.6 proof] In the proof of Corollary 3.6, the notation β is introduced without definition, and the line 'Cx = βDy = λD(β/λ y)' is unclear. Since this corollary depends on Theorem 3.5, the repair of Theorem 3.5 should be mirrored here to make the proof coherent.
Circularity Check
No significant circularity. Theorem 3.5 derives range inclusions from unit-ball inclusions by internal norm estimates (Example 3.4 shows the conclusion is not forced), and the only self-cited input, Theorem 2.5 from [19], is a parameter-free characterization that does not assume the target results.
full rationale
No circular reduction is present. Theorem 3.5 takes the unit-ball inclusions Cn(BM) ⊆ λnDn(BM⊥), Bn*(BN) ⊆ λnDn*(BN⊥) (the hypothesis Tn ∈ ψ(M,N,λn)) and, via block-wise norm convergence (Theorem 3.2) and closedness of R(D), derives Cx ∈ R(D) for x ∈ BM and R(B*) ⊆ R(D*); complementability then follows from the external range-inclusion criterion Proposition 2.3 ([2]). The conclusion is not contained in the hypotheses: Example 3.4 exhibits a norm limit of M-complementable operators that is not complementable, so the growth condition λn‖D−Dn‖→0 carries real content. The only same-author citation used as a premise is Theorem 2.5, quoted without proof from the authors' 'To appear' paper [19], identifying (M,N)-complementability with the unit-ball inclusions (2) and underwriting ψ(M,N) = ⋃λ ψ(M,N,λ); it is load-bearing for every λ-indexed statement, and an error there would propagate. Under the rubric, however, this citation counts as independent support: it is parameter-free and its stated assumptions do not include any target result of this paper, so it does not by itself make the derivation circular — it is flagged here only as an omitted-proof caveat. The derived results (Corollaries 3.6–3.8, Theorem 3.11, Theorem 4.1) build on Theorem 3.5's argument rather than restating the definition of ψ(M,N,λ); Theorem 4.1 uses the same hypothesis with constant λn = λ, so no step assumes its own conclusion. Separately, the printed proof of Theorem 3.5 contains a correctness gap unrelated to circularity: the step 'Since λn‖D−Dn‖ → 0, it follows that βn‖D−Dn‖ → 0' for βn = sup{λi : 1 ≤ i ≤ n} is false in general (e.g., λ2k = 2k, λ2k+1 = 1 with ‖D−Dn‖ = (2k)^−2 on even and (2k)^−1 on odd indices gives βn‖D−Dn‖ = 1 along odd indices). A direct repair exists (work with Cnx = λnDnyn and λn(Dn−D)yn → 0), and the circularity verdict is unaffected.
Assumptions & free parameters
assumptions (3)
- standard math Douglas factorization theorem: R(A) subset of R(B) iff A = BC for some bounded C, with norm controlled by the smallest lambda satisfying AA* <= lambda BB*.
- standard math An operator has closed range if and only if its reduced minimum modulus gamma(T) is positive.
- domain assumption Theorem 2.5: T is (M,N)-complementable iff, for some lambda > 0, C(B_M) subset of lambda D(B_M^perp) and B*(B_N) subset of lambda D*(B_N^perp).
Cite this review
Pith. "Pith review of Convergence of Complementable Operators." pith.science (2026). https://pith.science/paper/EXZI646B
@misc{pith2026241115636,
author = {Pith},
title = {Pith review of: Convergence of Complementable Operators},
year = {2026},
howpublished = {\url{https://pith.science/paper/EXZI646B}},
note = {Machine review of arXiv:2411.15636}
}
read the original abstract
Complementable operators extend classical matrix decompositions, such as the Schur complement, to the setting of infinite-dimensional Hilbert spaces, thereby broadening their applicability in various mathematical and physical contexts. This paper focuses on the convergence properties of complementable operators, investigating when the limit of sequence of complementable operators remains complementable. We also explore the convergence of sequences and series of powers of complementable operators, providing new insights into their convergence behavior. Additionally, we examine the conditions under which the set of complementable operators is the subset of set of boundary points of the set of non-complementable operators with respect to the strong operator topology. The paper further explores the topological structure of the subset of complementable operators, offering a characterization of its closed subsets.
Reference graph
Works this paper leans on
-
[19]
S. M. Naik and P. S. Johnson. Complementable operators a nd their Schur complements. Indian J Pure Appl Math , To appear
-
[1]
T. Ando. Generalized Schur complements. Linear Algebra Appl. , 27:173–186, 1979
work page 1979
-
[2]
J. Antezana, G. Corach, and D. Stojanoff. Bilateral shorte d operators and parallel sums. Linear Algebra Appl. , 414(2-3):570–588, 2006
work page 2006
-
[3]
M. L. Arias, G. Corach, and A. Maestripieri. On complemen table operators in the sense of T. Ando. Linear Algebra Appl. , 594:158–176, 2020
work page 2020
-
[4]
M. L. Arias and M. C. Gonzalez. Reduced solutions of Dougl as equations and angles between subspaces. J. Math. Anal. Appl. , 355(1):426–433, 2009
work page 2009
-
[5]
C. A. Butler and T. D. Morley. A note on the shorted operato r. SIAM J. Matrix Anal. Appl., 9(2):147–155, 1988. 18
work page 1988
-
[6]
D. Carlson. What are Schur complements, anyway? Linear Algebra Appl. , 74:257–275, 1986
work page 1986
-
[7]
D. Carlson and E. V. Haynsworth. Complementable and almo st definite matrices. Linear Algebra Appl., 52/53:157–176, 1983
work page 1983
Show all 21 references
-
[8]
Contino, A
M. Contino, A. Maestripieri, and S. Marcantognini. Schu r complements of selfadjoint Krein space operators. Linear Algebra Appl. , 581:214–246, 2019
2019
-
[9]
Corach, A
G. Corach, A. Maestripieri, and D. Stojanoff. Generalized schur complements and oblique projections. Linear Algebra and its Applications , 341(1-3):259–272, 2002
2002
-
[10]
R. W. Cottle. On manifestations of the Schur complement . Rend. Sem. Mat. Fis. Milano , 45:31–40, 1975
1975
-
[11]
R. G. Douglas. On majorization, factorization, and ran ge inclusion of operators on Hilbert space. Proc. Amer. Math. Soc., 17:413–417, 1966
1966
-
[12]
J. I. Giribet, A. L. Maestripieri, and F. D. Mart ´ ınez Pe r ´ ıa. Shorting selfadjoint operators in Hilbert spaces. Linear Algebra Appl. , 428(8-9):1899–1911, 2008
1911
-
[13]
H. Goller. Shorted operators and rank decomposition ma trices. Linear Algebra Appl. , 81:207–236, 1986
1986
-
[14]
I. S. Hwang and W. Y. Lee. The reduced minimum modulus of o perators. J. Math. Anal. Appl., 267(2):679–694, 2002
2002
-
[15]
Kreyszig
E. Kreyszig. Introductory functional analysis with applications . Wiley Classics Library. John Wiley & Sons, Inc., New York, 1989
1989
-
[16]
Maestripieri and F
A. Maestripieri and F. Mart ´ ınez Per ´ ıa. Schur complem ents in Krein spaces. Integral Equations Operator Theory , 59(2):207–221, 2007
2007
-
[17]
Massey and D
P. Massey and D. Stojanoff. Generalized Schur complement s and P -complementable oper- ators. Linear Algebra Appl. , 393:299–318, 2004
2004
-
[18]
S. K. Mitra and M. L. Puri. Shorted matrices—an extended concept and some applications. Linear Algebra Appl. , 42:57–79, 1982
1982
-
[20]
Ouellette
D. Ouellette. Schur complements and statistics. Linear Algebra Appl. , 36:187–295, 1981
1981
-
[21]
Zhang, editor
F. Zhang, editor. The Schur complement and its applications , volume 4 of Numerical Methods and Algorithms . Springer-Verlag, New York, 2005. 19
2005
Reviewed August 12, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.