Equations of the first kind and the inversion of series of resolvents of a closed operator

A. R. Mirotin

REVIEW 2 major objections 4 minor 21 references

Equations of the first kind and the inversion of series of resolvents of a closed operator

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.02998 v2 pith:CWGFTVAH submitted 2019-08-08 math.FA

A. R. Mirotin This is my paper · ORCID

classification math.FA MSC 47A6047A2047A5245Q05

keywords closedoperatorleftinverseresolventseriesill-posedproblemintegralequationofthefirstkindregularizationRiesz-Dunfordfunctionalcalculussignalprocessing

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

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 $\alpha_j$.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

Colleague,

Read 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.

The 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.

The 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.

A 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.

I'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.

For reading group: maybe. I'd cite it if I worked in operator-theoretic inverse problems.

Best

Referee Report

2 major / 4 minor

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.

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 (2)

[Section 2, Definition 1 and Lemma 1] 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 3, Examples 1 and 2] 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.

minor comments (4)

[Abstract] '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.
[Introduction] 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 3, Example 3] 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 2, Corollary 2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the explicit left inverse is derived from Laurent expansion and the Riesz-Dunford functional calculus, not from the statement being proved.

full rationale

The central claim (Theorem 1) is not circular. The constants gamma and beta are computed directly from the Laurent expansion of 1/f at infinity, and h(z) is then defined as 1/f(z) - gamma - beta z. The proof then applies the identity 1 = (gamma + beta z)f(z) + h(z)f(z) to the operator A using the established multiplicativity properties of the Riesz-Dunford functional calculus for closed operators, which is cited to Dunford-Schwartz [6, VII.9]. Lemma 2, which locates the zeros of f inside conv({alpha_j}), is proved in the paper via a convex-separation argument and does not rely on the theorem or on prior work of the author. Lemma 1 identifies the operator series defining f(A) with the Riesz-Dunford integral; while the cited resolvent-norm estimate is mathematically questionable as a general fact, that is a correctness or rigor issue, not a circular dependency. Proposition 1 gives an independent counterexample when the spectral separation fails. The author's self-citations ([10] for the two-resolvent case and [11]-[13] for a functional calculus) are contextual and not load-bearing: the proof explicitly invokes [6] for the functional calculus properties, and Remark 2 openly credits Dunford-Schwartz for the existence of the left inverse, noting the novelty is the explicit formula. Nothing in the derivation reduces to fitting a parameter to the target quantity or to importing an unproved uniqueness theorem from the author's own prior work.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The main theorem is a pure operator-theoretic statement with no fitted parameters and no invented entities. It rests on standard functional calculus properties, on the standing domain assumptions about A and the coefficients, and on an erroneously stated resolvent norm estimate that is flagged as a red flag. The false estimate is not needed under the stronger assumptions of Theorem 1, where uniform resolvent boundedness holds.

assumptions (4)

standard math Riesz-Dunford functional calculus for unbounded closed operators has the expected product and composition properties, so that g(z)f(z)=1 implies g(A)f(A)=I.
Invoked in the proof of Theorem 1 via Dunford-Schwartz [6, VII.9] to pass from the scalar identity to the operator identity.
ad hoc to paper For a densely defined closed operator A, the resolvent norm satisfies ||(alpha-A)^{-1}|| <= 1/dist(alpha, sigma(A)).
Used in Lemma 1 and Definition 1 to justify convergence of the resolvent series. This inequality is false in general for nonnormal operators in Banach spaces; the correct relation is that the spectral radius of the resolvent equals 1/dist, which gives the reverse inequality. Under Theorem 1's spectral separation assumption, uniform boundedness of the resolvent on the compact convex hull repairs the convergence argument.
domain assumption A is a densely defined closed operator in a complex Banach space with spectrum avoiding the closed convex hull of the bounded set {alpha_j}.
This is the standing setup of Theorem 1 and the examples; it guarantees a positive distance between the poles and the spectrum, making the functional calculus applicable.
domain assumption The coefficients a_j are nonnegative and summable, and the set {alpha_j} is bounded.
Positivity is essential for Lemma 2, which says all zeros of f lie in the closed convex hull of the alpha_j. Summability and boundedness ensure the series and the inverse formula are well defined.

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Pith. "Pith review of Equations of the first kind and the inversion of series of resolvents of a closed operator." pith.science (2026). https://pith.science/paper/CWGFTVAH

@misc{pith2026190802998,
  author       = {Pith},
  title        = {Pith review of: Equations of the first kind and the inversion of series of resolvents of a closed operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CWGFTVAH}},
  note         = {Machine review of arXiv:1908.02998}
}

abstract

Let $A$ be a densely defined closed operator in a complex Banach space $X.$ Conditions for left invertibility of operators of the form $\sum_{j=1}^\infty a_j (\alpha_j -A)^{-1}$ are given. Several examples are considered.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

Данфорд, Н., Шварц, Дж. Т. Линейные операторы. Том 1. Общ ая теория. – М. : ИЛ, 1962. – 895 с

work page 1962

[2] [1]

Рассмотрим следующее интегральное уравнение первого рода в пространс тве Lp(R) (1 ≤ p < ∞): ∫ ∞ t k(s − t)x(s)ds = y(t)

Введение Как известно, обратные задачи, возникающие в различных раз делах науки, приводят к разнообразным уравнениям первого рода, решение которых представляет собой, как правило, некорректную задачу (см., например, [1 ], [2], [3], [4]). Рассмотрим следующее интегральное уравнение первого рода в пространс тве Lp(R) (1 ≤ p < ∞): ∫ ∞ t k(s − t)x(s)ds = y(t)...

work page

[3] [2]

Пусть {αj} /emdash.cyr не более чем счетное подмножествоC, A /emdash.cyr плотно определенный замкнутый оператор в комплексном бана ховом пространстве X

Основные результаты Определение 1. Пусть {αj} /emdash.cyr не более чем счетное подмножествоC, A /emdash.cyr плотно определенный замкнутый оператор в комплексном бана ховом пространстве X. Пусть функция f в некоторой окрестности спектра σ(A) допускает разложение в ряда типа Вольфа-Данжуа f (z) = ∞∑ j=1 aj αj − z , a j ∈ C (4) (в случае конечного множества ...

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Примеры Известные работы А. Ф. Леонтьева (см., например, [5]) посвя щены, в основ- ном, рядам экспонент с неограниченными показателями. В свя зи с примерами 1 и 2 отметим, что, как показано в [8, c. 199, следствие 4], напри мер, целые функции экспоненциального типа, меньшего единицы, разлагаются в а бсолютно сходящиеся ряды экспонент с ограниченными показ...

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S. I. Kabanikhin. Deﬁnitions and examples of inverse and ill-posed problems, J. Inv. Ill-Posed Problems 16 (2008), 317 – 357

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