{"id":"f32fc632-ca31-48f5-818a-eaa4a1fe3130","arxiv_id":"2411.15636","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Norm limits of (M,N,lambda_n)-complementable operators remain complementable when lambda_n times the D-block error tends to zero, and complementable operators with infinite-dimensional range blocks are strong-operator boundary points of the non-complementable set.","lead":"This mathematics paper studies 'complementable operators', infinite-dimensional versions of the Schur complement from matrix theory, and asks when limits of such operators stay complementable. It gives sufficient conditions for limit preservation and shows which complementable operators sit on the boundary of the non-complementable set, though several proofs need repair.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.5's proof relies on a false implication about the running maximum β_n; as printed, the main closure argument is invalid, though a repair appears available.","rationale":"I read the paper as aiming to prove a stability threshold for complementability under norm convergence. The load-bearing statement is Theorem 3.5, and the proof of that theorem must transport the λ_n-inclusions to the limit. The printed proof does this through β_n, and the key implication λ_n‖D−D_n‖→0 ⇒ β_n‖D−D_n‖→0 is demonstrably false. This is more concrete than the other flagged risk, the unproved Theorem 2.5 from the authors' companion paper: the unit-ball characterization in Theorem 2.5 is a routine consequence of Douglas' lemma and can be independently checked, whereas the β_n step is a definite logical error in the central proof as printed. The error is repairable by the direct λ_n argument, so the theorem itself may be true, but the manuscript currently does not establish it. Lemma 3.10 is also false, for instance with T=I and α_n=n^{-2}; however, that lemma is peripheral to the main closure claim. The reader's CONDITIONAL verdict already captures the need for correction, and my preferred test would confirm both the invalidity of the printed step and the viability of the repair, so no verdict change is warranted.","tokens_in":15136,"tokens_out":21598,"duration_ms":202020,"concrete_test":"Re-derive the proof of Theorem 3.5 without any β_n estimate: for each x∈B_M, use C_nx=λ_nD_nz_n with ‖z_n‖≤1, verify λ_n(D_n−D)z_n→0, conclude D(λ_nz_n)→Cx, and apply closedness of R(D) to get Cx∈R(D); then repeat for B_n^*. Separately, plug the alternating sequences λ_{2k}=2k, λ_{2k+1}=1, ε_{2k}=(2k)^{-2}, ε_{2k+1}=(2k)^{-1} into the claimed implication to confirm that β_n ε_n fails to converge to 0. If the direct λ_n-based proof closes, the theorem survives but the manuscript must replace the faulty step; if not, the central claim is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.5 defines β_n = sup_{i≤n} λ_i and asserts that λ_n‖D−D_n‖→0 implies β_n‖D−D_n‖→0. This is false. For example, take λ_{2k}=2k, λ_{2k+1}=1, and let ε_{2k}=(2k)^{-2}, ε_{2k+1}=(2k)^{-1}. Then λ_n ε_n→0, but β_n ε_n=1 along odd indices, so β_n ε_n does not tend to 0. Since the printed argument uses this implication to compare β_nD y_n with β_nD_n y_n and to conclude that {β_nD y_n} is Cauchy, the central proof of Theorem 3.5 does not go through as written. The gap is likely repairable: for x∈B_M write C_nx=λ_nD_nz_n with ‖z_n‖≤1; then C_nx→Cx and λ_n(D_n−D)z_n→0 by the assumed product convergence, so D(λ_nz_n)→Cx. Because R(D) is closed, this gives Cx∈R(D), and the same argument handles B^*. But this repair is not present in the text, and Theorem 3.5 is the basis for Corollaries 3.6–3.8 and Theorem 4.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15256,"tokens_out":7823,"duration_ms":64521,"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":[{"comment":"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.","section":"Theorem 3.5 proof"},{"comment":"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).","section":"Lemma 3.10"},{"comment":"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.","section":"Theorem 2.5 / Section 2"},{"comment":"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.","section":"Corollary 3.8 proof"},{"comment":"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.","section":"Corollary 4.2 proof"}],"minor_comments":[{"comment":"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).","section":"Corollary 3.14 proof"},{"comment":"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)).","section":"Theorem 3.11 proof"},{"comment":"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.","section":"Abstract and Introduction"},{"comment":"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.","section":"Corollary 3.6 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claims are plausible and likely repairable, but the current version has a false implication in the proof of Theorem 3.5 and a false lemma (Lemma 3.10). These issues are substantive enough to require a major revision. The heavy reliance on the authors' own unpublished characterization (Theorem 2.5 from [19]) is also a concern for verification; I would advise the editor to request either a proof or a stable preprint citation. The constructive boundary results are a positive feature and deserve a careful second round of review after the repairs are made."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a genuinely new piece of work on how complementability behaves under limits. Theorems 3.12 and 3.13, identifying boundary points of the non-complementable set in the strong operator topology, are the strongest contributions and the constructions look sound on inspection. Theorem 3.5, giving a sufficient condition for complementability to survive norm limits, is the result people will cite, and it is true in substance but not as proved. The paper is honest—no fitting, no hidden parameters, and the examples (especially 3.4) are correct.\n\nNow the soft spots, in proportion. First, the proof of Theorem 3.5 uses the step \"λ_n‖D−D_n‖→0 implies β_n‖D−D_n‖→0\" for β_n = sup_{i≤n} λ_i. That implication is false; the stress-test example with alternating λ's and ε's is valid. The gap is repairable: for x ∈ B_M, write C_n x = λ_n D_n z_n with ‖z_n‖≤1, pass to the limit, and use closedness of R(D) to get Cx ∈ R(D). A direct argument avoids β_n entirely. But as printed, the central proof does not go through. Second, Lemma 3.10 is false: the \"if and only if\" fails on the only-if direction. Take T = I and α_n = 1/n². The series converges, but |α_n|^{1/n} → 1, so no β<1 works. The forward direction (which Theorem 3.11 uses) is fine, but the lemma must be corrected or weakened. Third, Section 4 is muddled. Theorem 4.1 as stated is trivially true because the notation omits a closure: the proof really establishes that the closure of ψ(M,N,λ) inside BCD(M,N) is contained in ψ(M,N,λ). Corollary 4.2 and 4.3 inherit that confusion, and the proof of 4.1 leans on Theorem 3.5. The section needs a clean rewrite rather than small patches. Minor: Corollary 3.14's proof has a typo (\"if R(D)\" should be \"if R(D) is infinite-dimensional\").\n\nA separate dependency worth flagging: every λ-indexed result rests on Theorem 2.5, quoted from the authors' own forthcoming paper [19]. That theorem may well be correct, but the referee will need to see its proof or the preprint; a \"to appear\" citation is not enough for a load-bearing characterization.\n\nWho this is for: functional analysts working on Schur complements, shorted operators, or range-inclusion methods. The paper is not revolutionary, but it fills a real gap in the literature and the topological boundary results are new. It deserves a serious referee. My recommendation: send it out, with instructions to check Theorem 3.5's repair, fix or weaken Lemma 3.10, and ask for a revised Section 4.","headline":"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.","tokens_in":15999,"tokens_out":4584,"would_cite":true,"duration_ms":40257,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A05","47A58"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["complementable operators","Schur complement","operator convergence","strong operator topology","range inclusion","Douglas factorization lemma","reduced minimum modulus","block operator matrices"],"falsifier":"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.","tokens_in":14754,"feed_emoji":"📐","tokens_out":19305,"duration_ms":155444,"temperature":0.7,"pith_summary":"Complementable operators generalize the classical Schur complement to bounded operators between Hilbert spaces: for closed subspaces $M$ and $N$, an operator $T$ written in block form with components $A,B,C,D$ is $(M,N)$-complementable when the off-diagonal blocks factor through $D$, which is equivalent to a pair of quantitative unit-ball inclusions. This paper asks when complementability survives taking limits. Its central theorem shows that if a sequence of $(M,N,\\lambda_n)$-complementable operators converges in operator norm to an operator whose $D$-block has closed range, and if $\\lambda_n\\|D-D_n\\|\\to 0$, then the limit is again $(M,N)$-complementable. The paper also shows which complementable operators lie on the boundary of the non-complementable set in the strong operator topology, and it analyzes powers, series, and closed subsets of these operators. These are stability statements for an infinite-dimensional structural decomposition used in operator equations and approximation arguments.","feed_headline":"Norm limits preserve complementability under a rate condition","feed_subtitle":"When the complementability constants stay small against the D-block error, the limit operator is again complementable.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $B(H,K)$."],"supporting_citations":[{"why":"supplies the quoted characterization of complementability by the pair of unit-ball inclusions (Theorem 2.5), on which all λ-indexed results rest.","marker":"[19]"},{"why":"introduces (M,N)-complementability and weak complementability for bounded operators and gives the range-inclusion and block-matrix framework used throughout.","marker":"[2]"},{"why":"Douglas's lemma provides the factorization and range-inclusion criterion used to connect the block inclusions to Douglas reduced solutions.","marker":"[11]"},{"why":"gives the reduced minimum modulus criterion γ(T)>0 iff R(T) is closed, which defines the class B_CD(M,N) and powers Corollary 3.8.","marker":"[14]"},{"why":"supplies the definitions of uniform and strong convergence of operator sequences that frame Theorem 3.2 and the convergence arguments.","marker":"[15]"}],"fun_headline_variants":["Norm limits keep complementability with matching D-blocks","Rate condition decides if a limit stays complementable","Strong limits can push complementable operators to boundary","Convergence of complementable operators: sharp boundary conditions","Norm limit preserves complementability, but strong limit may not"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Norm limits keep complementability with matching D-blocks","Rate condition decides if a limit stays complementable","Strong limits can push complementable operators to boundary","Convergence of complementable operators: sharp boundary conditions","Norm limit preserves complementability, but strong limit may not"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000768,"raw_usage":{"total_tokens":3405,"prompt_tokens":951,"completion_tokens":2454,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":2378}},"tokens_in":567,"tokens_out":2454,"duration_ms":16512,"temperature":1.0,"reasoning_tokens":2378,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:09:50.947057+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the quoted characterization of complementability by the pair of unit-ball inclusions (Theorem 2.5), on which all λ-indexed results rest."},{"cited_title":"Antezana, G","cited_arxiv_id":null,"evidence_quote":"introduces (M,N)-complementability and weak complementability for bounded operators and gives the range-inclusion and block-matrix framework used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Douglas's lemma provides the factorization and range-inclusion criterion used to connect the block inclusions to Douglas reduced solutions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the reduced minimum modulus criterion γ(T)>0 iff R(T) is closed, which defines the class B_CD(M,N) and powers Corollary 3.8."},{"cited_title":"Kreyszig","cited_arxiv_id":null,"evidence_quote":"supplies the definitions of uniform and strong convergence of operator sequences that frame Theorem 3.2 and the convergence arguments."}],"review_version":1}