{"id":"443a1d55-6b41-4c7b-a5c6-b57d10f06d6a","arxiv_id":"1908.02998","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a closed operator A, a positive-coefficient series of resolvents whose poles avoid the spectrum's convex hull has an explicit left inverse of the form constant plus beta A plus a bounded correction.","lead":"This mathematics paper gives conditions under which an infinite sum of resolvent operators has a left inverse, and writes that inverse explicitly. The result applies to first-kind integral equations and discrete-time filters, where it exposes the cause of ill-posedness and suggests a regularization scheme.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 1's resolvent norm bound is false, so Definition 1 and the identification of f(A) with the Riesz-Dunford integral are unproved; the main theorem needs the repair the reader requested, though the claim appears sound under its hypotheses.","rationale":"The paper's central theorem is very likely correct, but the written proof contains a false resolvent estimate in Lemma 1 that is used to define f(A) and to identify it with the Riesz-Dunford calculus. This is a genuine, concrete defect in the manuscript's main line of argument, not merely a stylistic issue. However, it is repairable: under the theorem's spectral-separation hypothesis, the alpha_j lie in a compact subset of the resolvent set, where the resolvent norm is bounded, so the operator series converges in norm and the termwise integration can be justified. Thus the concern does not overturn the claimed result; it means the paper needs a corrected Lemma 1 and a reworded Definition 1, exactly the kind of conditional acceptance the reader proposed. I mark agreement as partial because the reader's stated weakest assumption is the spectral-separation condition, whereas the most load-bearing defect I find is the false resolvent-norm estimate underlying Definition 1; the spectral-separation condition is indeed needed for the repair, but it is not by itself the source of the faulty step. I recommend keeping the reader's CONDITIONAL verdict, so verdict_should_be is UNCHANGED.","tokens_in":8824,"tokens_out":27663,"duration_ms":304392,"concrete_test":"Compute ||(I - N)^{-1}|| for the nilpotent shift N on ell^2 defined by N e_1 = e_2 and N e_k = 0 for k >= 2. Since N^2 = 0, (I - N)^{-1} = I + N, and ||(I + N)e_1|| = sqrt(2) > 1 = 1/dist(1, {0}), refuting Lemma 1's estimate. Then re-derive the convergence of sum a_j(alpha_j - A)^{-1} under Theorem 1's hypotheses by using boundedness of the resolvent on the compact set conv({alpha_j}) subset rho(A); if the re-derivation succeeds, the central statement survives and only Definition 1 and Lemma 1 need correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 1 and Lemma 1 rest on the estimate ||(alpha_j - A)^{-1}|| <= 1/dist(alpha_j, sigma(A)). This inequality is not valid for general closed operators: for a nilpotent shift N with N^2 = 0 and ||N|| = 1, at alpha = 1 we have dist(1, sigma(N)) = 1 but ||(1 - N)^{-1}|| = ||I + N|| > 1, because (I + N)e_1 = e_1 + e_2 has norm sqrt(2). More generally, for alpha_j = 1/j and a_j = 1/j^3, one has sum a_j/dist(alpha_j, sigma(N)) = sum 1/j^2 < infinity, yet ||(alpha_j - N)^{-1}|| >= j^2, so sum a_j ||(alpha_j - N)^{-1}|| diverges. Thus the convergence criterion in Definition 1 does not guarantee that the operator series defining f(A) converges in norm, and the termwise-integration argument identifying the series with the Riesz-Dunford integral is unjustified. Since Theorem 1 invokes Lemma 1 to treat f(A) as a Riesz-Dunford function, the proof of the central claim is incomplete as written. The gap is repairable: under Theorem 1's assumptions, the points alpha_j lie in the compact set conv({alpha_j}) subset rho(A), on which the resolvent is continuous and hence bounded, so the series converges in norm once sum |a_j| < infinity is used; but the paper does not supply this corrected argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies left invertibility of operators of the form f(A)=Σ a_j(α_j−A)^{-1}, where A is a densely defined closed operator in a Banach space and the series is interpreted in the norm topology. Definition 1 introduces such series under a convergence condition based on distances to the spectrum; Lemma 1 claims their agreement with the Riesz–Dunford functional calculus. The main result (Theorem 1) asserts that if all a_j are nonnegative, 0<Σa_j<∞, the α_j form a bounded set whose closed convex hull does not meet σ(A), then f(A) is left invertible on dom(A) with explicit inverse γ+βA+h(A), where β=−1/Σa_j, γ=Σa_jα_j/(Σa_j)^2, and h=1/f−γ−βz is a Riesz–Dunford function. Consequences include Hadamard ill-posedness for unbounded A, a regularization scheme (Corollary 3), and applications to integral equations and to a discrete-time filtering problem.","tokens_in":9146,"tokens_out":15576,"duration_ms":175144,"significance":"If the proof is repaired, the theorem is a natural and useful extension of the two-resolvent case: it gives a fully explicit left inverse and identifies the unbounded term βA as the source of ill-posedness, which supports the regularization result in Corollary 3. The paper is honest in Remark 2 that existence of the left inverse already follows from [6, Thm 9] and that the novelty is the explicit form. The convex-hull zero localization in Lemma 2 and the counterexample in Proposition 1 are also valuable. However, the central proof currently depends on a false resolvent norm estimate, so the claims in the submitted form are not fully established; the gap is localized and repairable.","major_comments":[{"comment":"The proof of Lemma 1 relies on the inequality ||(α_j−A)^{-1}|| ≤ 1/dist(α_j,σ(A)), which is false for general closed operators in Banach spaces. For example, a nilpotent shift N with N^2=0 and ||N||=1 satisfies dist(1,σ(N))=1 but ||(1−N)^{-1}||=||I+N||>1. More directly relevant to Definition 1, with α_j=1/j and a_j=1/j^3 one has Σ a_j/dist(α_j,σ(N))=Σ1/j^2<∞, yet ||(α_j−N)^{-1}||≥j and hence Σ a_j||(α_j−N)^{-1}|| diverges. Consequently, the convergence of the series in (5) is not established by the stated hypothesis, and the termwise integration identifying f(A) with the Riesz–Dunford integral is unjustified. Because Theorem 1 uses Lemma 1 to replace the series f(A) by the functional-calculus object in the identity (γ+βA)f(A)+h(A)f(A)=I, the proof of the central result is incomplete. The gap is repairable: under the hypothesis conv({α_j})⊂ρ(A), the resolvent is bounded on this compact set, so the series converges in norm whenever Σ|a_j|<∞; the proof should be rewritten accordingly.","section":"Section 2, Definition 1 and Lemma 1"},{"comment":"The hypotheses stated for the kernels in Examples 1 and 2 (Σ|a_j|<∞ and Re α_j>0, respectively Im β_j<0) do not imply the convergence condition Σ|a_j|/dist(α_j,σ(A))<∞ required by Definition 1 when the α_j accumulate on the spectrum. For the differentiation operator D with σ(D)=iR, taking α_j=2^{−j} and a_j=2^{−j} gives Σa_j<∞ but Σ a_j/dist(α_j,iR)=Σ1=∞. The examples should either impose a lower bound on dist(α_j,σ(A)) or verify the stronger summability condition explicitly.","section":"Section 3, Examples 1 and 2"}],"minor_comments":[{"comment":"'Key wards' should be 'Key words', and 'resol vents' contains a stray space; the source also contains visible '/emdash.cyr' artifacts that should be cleaned before publication.","section":"Abstract"},{"comment":"The statement that a two-term combination is left invertible if and only if the weighted average is not in the point spectrum needs a precise definition of left invertibility; for bounded operators with continuous spectrum, such as a multiplication operator whose symbol vanishes at a point, the bounded left inverse does not exist even though the point spectrum is empty.","section":"Introduction"},{"comment":"In ℓ2(Z), where σ(T) is the unit circle, the hypothesis σ(T)∩conv{z_j}=∅ means that all roots z_j of the characteristic polynomial lie strictly inside the unit disk; please state this stability condition explicitly rather than only deriving it from the general hypothesis.","section":"Section 3, Example 3"},{"comment":"The phrase 'корректна по Тихонову ... если и только если оператор A непрерывен в относительной топологии множества f(A)M' would be clearer if the intended topology on f(A)M were specified as the norm topology induced by X.","section":"Section 2, Corollary 2"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the repair to Lemma 1 is straightforward, but the current proof has a genuine gap in a load-bearing estimate. The examples also need to be checked against the corrected convergence condition. I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead Mirotin's paper on inverting series of resolvents. The core idea is nice: when the poles lie in a compact convex set disjoint from sigma(A), an infinite positive-coefficient sum of resolvents has an explicit left inverse of the form gamma + beta A + h(A), with h holomorphic at infinity. That genuinely extends the earlier two-resolvent paper [10] and provides a formula where Dunford–Schwartz only promised existence. The applications to first-kind integral equations and the signal-processing example are sensible, and Corollary 3 on regularization is a clean byproduct.\n\nThe soft spot is real, and the stress-test note lands. Lemma 1 uses ||(alpha - A)^{-1}|| <= 1/dist(alpha, sigma(A)), which is false for general closed operators; the nilpotent shift example is a valid counterexample. The convergence criterion in Definition 1 as written does not guarantee norm convergence of the series defining f(A), and the termwise integration with the Riesz–Dunford integral is not justified. That is a genuine gap in the proof as written.\n\nThe good news: Theorem 1's own hypotheses repair it. Since the alpha_j lie in a compact set disjoint from sigma(A), the resolvent is bounded there, so sum |a_j| < infinity gives norm convergence. The paper does not say this; it appeals to the false estimate instead. So Theorem 1's statement is very likely correct, but the proof is incomplete.\n\nA second, smaller issue: Example 3 says for l2(Z) the solution is unique because sigma(T) is the unit circle, then states the problem is well-posed by Hadamard. Well-posedness also needs boundedness of the inverse operator, which follows from boundedness of T and the formula, but the stability condition (poles strictly inside the unit disk) is only implicit; worth stating explicitly.\n\nI'd send this to a serious referee. The main result is useful, the gap is localized and repairable, and the paper is honest (Remark 2 acknowledges that existence of the left inverse is already known). A referee can verify the repair quickly.\n\nFor reading group: maybe. I'd cite it if I worked in operator-theoretic inverse problems.\n\nBest","headline":"Useful extension of the two-resolvent inversion to infinite Wolf–Denjoy series, but the convergence lemma is false as stated and Theorem 1's proof needs a local repair.","tokens_in":9673,"tokens_out":1477,"would_cite":true,"duration_ms":16992,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A60","47A20","47A52","45Q05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a positive-coefficient series of resolvents of a closed operator is left-invertible under a spectral separation condition, with an explicit inverse built from the operator itself plus a bounded analytic function.","keywords":["closed operator","left inverse","resolvent series","ill-posed problem","integral equation of the first kind","regularization","Riesz-Dunford functional calculus","signal processing"],"falsifier":"For the necessity direction, take a densely defined closed operator $A$ with an eigenvector $x$ at a spectral point $\\lambda$ and choose $\\alpha_1,\\alpha_2,\\alpha_3$ whose convex hull contains $\\lambda$; set $a_\\nu=k_\\nu|\\alpha_{j_\\nu}-\\lambda|^2$ as in Proposition 1 and verify directly that $f(A)x=0$, which rules out left-invertibility. For the sufficiency direction, take $A=D$ on $L^p(\\mathbb{R})$ with $\\operatorname{Re}\\alpha_j>0$ and check on a dense set of smooth compactly supported $y$ that $(\\gamma+\\beta D+h(D))f(D)y=y$ holds as the identity predicts.","tokens_in":8575,"feed_emoji":"🧮","tokens_out":9794,"duration_ms":98510,"temperature":0.7,"pith_summary":"The paper proves that operators of the form $f(A)=\\sum_j a_j(\\alpha_j-A)^{-1}$, with $a_j\\ge 0$ and $0<\\sum a_j<\\infty$, are left-invertible whenever the closed convex hull of the points $\\alpha_j$ does not meet the spectrum of $A$. It writes the left inverse explicitly as $\\gamma+\\beta A+h(A)$, where $\\gamma$ and $\\beta$ are constants computed from the series and $h$ is a function analytic near the spectrum and at infinity. This yields a unique solution for a class of integral equations of the first kind, including convolution equations with exponential kernels and discrete-time filter inversion problems. The explicit form shows that ill-posedness of the inverse problem is caused entirely by the unbounded term $\\beta A$; when $A$ is bounded the problem becomes Hadamard well-posed. The proof rests on the fact that the zeros of a positive-coefficient series of this type stay inside the convex hull of the $\\alpha_j$.","feed_headline":"Explicit left inverse found for resolver series","feed_subtitle":"Formula splits the inverse into a bounded analytic part and one unbounded term, pinpointing ill-posedness.","key_machinery":"The argument is carried by a zero-location lemma: any zero of $f(z)=\\sum_j a_j/(\\alpha_j-z)$ with $a_j\\ge 0$ and not all $a_j$ zero lies in the closed convex hull of the $\\alpha_j$. To prove it, an exterior zero is separated from the convex hull by a line, the plane is rotated so that line becomes $\\operatorname{Re} w=a$ with $a>0$, and the fractional linear transformation $\\zeta=1/w$ sends that half-plane to a disk containing all the points $\\zeta_j=1/(\\alpha_j-z_0)$; the transformed series $\\sum_j a_j\\zeta_j$ is a nontrivial sum of nonnegative terms and therefore cannot vanish. This lemma makes $1/f$ holomorphic outside the convex hull of the $\\alpha_j$, so its Laurent expansion at infinity exists. The paper reads off $\\beta=\\lim_{z\\to\\infty} (1/f(z))/z$ and $\\gamma=\\lim_{z\\to\\infty}(1/f(z)-\\beta z)$, and the remainder $h(z)=1/f(z)-\\gamma-\\beta z$ lies in the Riesz-Dunford class $F(A)$. Applying the holomorphic functional calculus to the identity $1=(\\gamma+\\beta z)f(z)+h(z)f(z)$ gives $I=(\\gamma+\\beta A)f(A)+h(A)f(A)$, which is precisely the claimed left inverse.","core_discovery":"The central discovery is Theorem 1: for a densely defined closed operator $A$ in a complex Banach space, with a bounded set $\\{\\alpha_j\\}$ whose closed convex hull avoids $\\sigma(A)$, and for $f(z)=\\sum_j a_j/(\\alpha_j-z)$ with all $a_j\\ge 0$ and $0<\\sum_j a_j<\\infty$, the operator $f(A)=\\sum_j a_j(\\alpha_j-A)^{-1}$ has a left inverse on $\\operatorname{dom}(A)$ given by $g(A)=\\gamma+\\beta A+h(A)$, where $\\gamma=(\\sum_j a_j\\alpha_j)/(\\sum_j a_j)^2$, $\\beta=-1/\\sum_j a_j$, and $h(z)=1/f(z)-\\gamma-\\beta z$ lies in the Riesz-Dunford class of functions holomorphic in a neighborhood of $\\sigma(A)$ and at infinity. Consequently $(\\gamma+\\beta A+h(A))f(A)x=x$ for every $x\\in X$, and $f(A)x$ belongs to $\\operatorname{dom}(A)$. The paper further shows that when a spectral point lies in the convex hull of the $\\alpha_j$ but not among the $\\alpha_j$ themselves, there is a positive-coefficient rational function of the same form for which $f(A)$ is not left-invertible, so the spectral separation condition is essential.","pith_inferences":["Beyond the paper, the explicit split into $\\beta A$ and $h(A)$ suggests a concrete numerical strategy for deconvolution: truncate the absolutely convergent series that defines $h(A)$ and apply standard regularization only to the derivative-type term $\\beta A$; a testable prediction is that this converges to the true inverse as the truncation length grows.","The zero-location mechanism may extend to signed coefficients whenever all zeros of $f$ still lie in a fixed compact set away from $\\sigma(A)$; if that holds, the same explicit inverse formula would survive, so the positivity assumption could be relaxed for specific rational-function families.","In the filter example, the spectral condition for $\\ell^2(\\mathbb{Z})$ is equivalent to the transfer-function poles lying strictly inside the unit disk, a standard stability condition; the explicit inverse then gives a direct algebraic recovery algorithm that avoids solving a Toeplitz system.","The necessity result suggests the spectral separation assumption is close to optimal: any eigenvalue inside the convex hull of the $\\alpha_j$ can be exploited to build a positive-coefficient resolvent series that annihilates the corresponding eigenvector, so left-invertibility should fail generically in that regime."],"forward_implications":["The inverse problem $f(A)x=y$ is Hadamard well-posed exactly when $A$ is bounded; for unbounded $A$, the failure of well-posedness is fully captured by the single term $\\beta A$.","If $A$ has a bounded inverse $K$ and a regularizing family is known for $Kx=y$, then $R_\\alpha=\\gamma+\\beta R^0_\\alpha+h(A)$ is a regularizing family for $f(A)x=y$.","For the convolution equation with kernel $k(t)=\\sum a_j e^{-\\alpha_j t}$, the unique solution exists precisely for $y\\in\\operatorname{dom}(D)$ and is $x(t)=-\\gamma y(t)-\\beta y'(t)-h(D)y(t)$.","For a discrete-time recursive filter on $\\ell^2(\\mathbb{Z})$, the input can be recovered from the output by an explicit formula involving the shift operator, provided the characteristic polynomial's roots avoid the unit circle so that $\\sigma(T)\\cap\\operatorname{conv}\\{z_j\\}=\\emptyset$.","Rational functions with simple poles and positive residues are covered directly, and the inverse decomposes into a sum of negative powers $(\\alpha_j-A)^{-k}$; this makes the formula computable in finite form."],"supporting_citations":[{"why":"Supplies the Riesz-Dunford holomorphic functional calculus for closed operators, the theorem guaranteeing existence of the left inverse under the spectral condition, and the resolvent norm bound used in Lemma 1.","marker":"[6]"},{"why":"Defines the Wolf-Danjoy series class of the form (4) and gives the uniqueness result that makes the functional calculus in Definition 1 well defined.","marker":"[8]"},{"why":"Establishes the finite two-pole case with positive coefficients that Theorem 1 generalizes to countable series.","marker":"[10]"},{"why":"Supplies the discrete-time recursive filter model used in Example 3 to illustrate the recovery formula.","marker":"[17]"}],"fun_headline_variants":["Explicit left inverse for positive resolvent sums","Spectral separation yields resolvent-series inverse","Inverting resolvent sums with an explicit formula","Left inverse found when spectrum avoids coefficient hull"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the spectrum of $A$ avoids the closed convex hull of the points $\\alpha_j$ and that all coefficients $a_j$ are nonnegative; if either fails, the zero-location lemma breaks and with it the explicit inverse formula.","fun_headline_variants_meta":{"raw":{"variants":["Explicit left inverse for positive resolvent sums","Spectral separation yields resolvent-series inverse","Inverting resolvent sums with an explicit formula","Left inverse found when spectrum avoids coefficient hull"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000272,"raw_usage":{"total_tokens":1588,"prompt_tokens":860,"completion_tokens":728,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":476,"completion_tokens_details":{"reasoning_tokens":671}},"tokens_in":476,"tokens_out":728,"duration_ms":8451,"temperature":1.0,"reasoning_tokens":671,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:28:03.478509+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the necessity direction, take a densely defined closed operator $A$ with an eigenvector $x$ at a spectral point $\\lambda$ and choose $\\alpha_1,\\alpha_2,\\alpha_3$ whose convex hull contains $\\lambda$; set $a_\\nu=k_\\nu|\\alpha_{j_\\nu}-\\lambda|^2$ as in Proposition 1 and verify directly that $f(A)x=0$, which rules out left-invertibility. For the sufficiency direction, take $A=D$ on $L^p(\\mathbb{R})$ with $\\operatorname{Re}\\alpha_j>0$ and check on a dense set of smooth compactly supported $y$ that $(\\gamma+\\beta D+h(D))f(D)y=y$ holds as the identity predicts.","supporting_citations":[{"cited_title":"М., Романов В","cited_arxiv_id":null,"evidence_quote":"Supplies the Riesz-Dunford holomorphic functional calculus for closed operators, the theorem guaranteeing existence of the left inverse under the spectral condition, and the resolvent norm bound used in Lemma 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Wolf-Danjoy series class of the form (4) and gives the uniqueness result that makes the functional calculus in Definition 1 well defined."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the finite two-pole case with positive coefficients that Theorem 1 generalizes to countable series."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the discrete-time recursive filter model used in Example 3 to illustrate the recovery formula."}],"review_version":1}