Trace formulas for $\mathcal{S}^p$-perturbations and extension of Koplienko-Neidhardt trace formulas

Arup Chattopadhyay; Chandan Pradhan; Cl\'ement Coine; Saikat Giri

REVIEW 3 major objections 6 minor 53 references

Trace formulas for $\mathcal{S}^p$-perturbations and extension of Koplienko-Neidhardt trace formulas

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves second-order trace formulas for all functions whose divided difference factorizes, and modified trace formulas for all n-times differentiable functions under Schatten perturbations.

desk verdict A substantive extension of Koplienko-Neidhardt trace formulas whose contraction case needs one missing verification before I would trust it fully. read the letter →

arxiv 2411.16426 v3 pith:H5NDEPJ3 submitted 2024-11-25 math.FA

Arup Chattopadhyay , Clément Coine , Saikat Giri , Chandan Pradhan This is my paper · ORCID

classification math.FA MSC 47A5547A5647B1047B49

keywords MultipleoperatorintegralsderivativestraceformulasspectralshiftfunctionsSchattenclassesdivideddifferencescontractionsHilbert-Schmidtperturbations

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 asks which functions f are good enough that the second-order Taylor remainder f(H0+V)-f(H0)-f'(H0)[V] is a trace-class operator and its trace is given by integrating f'' against a spectral shift function. It argues that the right condition is a factorization of the second divided difference, f[2](t1,t2,t3)= with bounded Borel functions a and b. Under this condition the second-order remainder is trace class and the classical Koplienko–Neidhardt trace formula holds in the self-adjoint, unitary, and contraction settings, and the class contains all previously known admissible classes, including the Besov class $B^{2}$_{∞,1} and all polynomials. The paper also proves modified trace formulas: for a perturbation in S^p with 1

What carries the argument

The central tool is the theory of multiple operator integrals (MOIs), multilinear operators that feed operator arguments into a function symbol, generalizing f(A)=∫ f dE. The load-bearing characterization, recalled as Theorem 4.1(B), says that a triple MOI Γ_{A,B,C}(φ) maps $S^{2}$×$S^{2}$ into $S^{1}$ exactly when φ factorizes as ⟨a(t1,t2),b(t2,t3)⟩ with bounded Borel functions a and b; Hypothesis 4.3 asks precisely this of the second divided difference f[2]. Around this sit the S^p-boundedness estimates for MOIs with symbols f[n], a contraction-case extension of MOIs via unitary dilations and semi-spectral measures, and a finite-dimensional approximation lemma for perturbations of normal contractions that lets the trace formula pass from finite rank to the general setting.

What would settle it

Produce a single twice differentiable function satisfying the factorization (4.3) and a single pair (H0,V) with V Hilbert–Schmidt for which RLin_2(f,H0,V) is not trace class; the remainder is an explicit operator on a separable Hilbert space, so checking trace-class membership is a concrete calculation. Conversely, exhibit a twice differentiable f with bounded f'' whose second divided difference does not factor, yet whose second-order remainder is trace class for all such V; that would show the factorization hypothesis is stronger than necessary.

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Extended reading notes

Core claim

The paper's central claim is that the correct admissible class for the second-order trace formula is not a Besov space or a tensor-product space but the class of twice differentiable functions whose second divided difference factorizes as f[2](t1,t2,t3)=⟨a(t1,t2), b(t2,t3)⟩ for bounded Borel functions a and b. For exactly these f, the second-order Taylor remainder belongs to the trace class for every Hilbert–Schmidt perturbation, and Tr RLin_2(f,H0,V)=∫_R f'' η with η∈$L^{1}$ nonnegative; the same pattern holds for unitary pairs along the multiplicative path and for contraction pairs along both linear and multiplicative paths, with spectral shift functions in $L^{1}$. The proof reduces trace-class membership to the known characterization of when a triple operator integral lands in $S^{1}$, and then obtains the integral representation by finite-dimensional approximation or by induction from the first-order formula. The modified trace formulas extend the same idea to arbitrary order: for V∈S^p, 1<p<∞, and X in the conjugate Schatten class, Tr(R_n · X)=∫ $f^{{(n)}}$ η holds for every n-times differentiable f with bounded n-th derivative, and the shift function η obeys an explicit norm bound and varies continuously in V and X.

Load-bearing premise

The load-bearing premise is that admissible functions are required to have a second divided difference that factorizes as an inner product of bounded Borel functions; all second-order trace-class and spectral-shift conclusions depend on this factorization, and the paper does not prove it is necessary.

Editorial extensions

If this is right

Every function in the factorization class—including the Besov class B^2_{∞,1}, the integral-projective-tensor-product class, and all polynomials—becomes admissible for the second-order trace formula in the self-adjoint, unitary, and contraction settings.
Trace-class membership of the second-order remainder is guaranteed by the same Hilbert-space factorization that characterizes trace-class triple operator integrals, so the trace formula is pushed to the threshold where the remainder is known to be trace class for structural reasons.
For modified trace formulas, n-times differentiability with bounded n-th derivative is enough, and the shift function satisfies ‖η‖_1 ≤ c‖V‖_p^n‖X‖_q, giving quantitative control for V in S^p and X in the conjugate class.
The spectral shift functions vary continuously in the perturbation and weight, so finite-rank compressions of the remainder can be approximated uniformly and bounded in terms of Schatten norms.
In finite dimension the weight X=I recovers the classical unweighted trace formula, so the modified identities are genuine generalizations and not merely weighted curiosities.

Reading between the lines

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

The paper stops short of proving that the factorization class is optimal; if the converse trace-class characterization carries over to the remainder, Hypothesis 4.3 would be necessary as well as sufficient, answering the open question the authors point toward.
The modified trace formulas suggest a general regularization principle: multiplying the Taylor remainder by a Schatten weight lowers the differentiability requirement, so one could try the same trade-off for Besov or Sobolev symbols, for higher-order divided differences, or for non-self-adjoint perturbations.
The continuity and differentiability of the shift functions in the weight gives a bootstrapping tool: differentiating Tr(R_n · X(t)) with respect to t should yield identities linking shift functions of adjacent orders and linear response formulas.
Because the contraction results go through unitary dilations, the same factorization class is likely the right one for any operator class admitting a unitary-dilation model, such as dissipative operators.

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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

Fair and useful read. The paper's real contribution is the admissible class in Hypothesis 4.3: second-order divided differences that factor as inner products of bounded Borel functions. This genuinely subsumes the Besov class and the tensor-product class, and Example 4.6 gives a concrete function not covered before. The modified trace formulas for every n and every 1

Soft spots are concentrated in the contraction case, and they are real but not fatal. Theorem 4.20 (normal T0) ends with 'readily follows from [10, Theorem 5.1]' without stating that theorem or checking its hypotheses for the specific sequence of finite-dimensional compressions. The stress-test note is right: Lemma 4.17 only gives S2-convergence of the off-diagonal terms, and it is not shown that this is what [10, Thm 5.1] requires. As written, the full contraction claim is not self-contained. A referee should ask for the missing verification or a replacement argument. Several smaller delegations—Proposition 4.4 (proof omitted), Lemma 4.11 (left to the reader), Theorem 5.4 ('follows along the lines of Theorem 5.1')—are less serious, but they add work for the reader and should be tightened.

The citation pattern is fine. The heavy reliance on earlier papers by the same group is a genuine dependency, not padding, because the results are cited precisely where needed. I did not find circularity: the base polynomial cases are established elsewhere, and approximation transfers them.

Who should read this: anyone working on higher-order spectral shift functions, multiple operator integrals, or perturbation theory in Schatten classes. It deserves a serious referee. The right outcome is probably conditional acceptance with a request to fix the contraction-case gap and expand the delegated proofs. I would not desk-reject this.

Referee Report

3 major / 6 minor

Summary. The paper develops second-order Koplienko–Neidhardt trace formulas and higher-order modified trace formulas for Schatten-class perturbations. The central admissible class is given by Hypothesis 4.3: a twice differentiable function f is admissible when its second divided difference f[2] factorizes as an inner product of two bounded Borel functions on the relevant domain (R or T). Under this hypothesis, the paper proves trace-class membership of the second-order Taylor remainder and an integral representation with a spectral shift function, in the self-adjoint case (Theorem 4.8), the unitary/multiplicative case (Theorem 4.12), and the contraction case (Theorems 4.14, 4.20, 4.21, 4.22). Section 5 treats modified trace formulas for n-times differentiable functions with bounded n-th derivative, for S^p perturbations with p=n+epsilon, including self-adjoint, unitary, and contraction settings, and concludes with non-negativity, continuity, and differentiability properties of the resulting spectral shift functions.

Significance. If the results are correct, the paper gives a substantial and natural extension of the admissible function class for second-order trace formulas, unifying the Besov class B^2_{infty,1}, the tensor-product class of [11], and polynomials under a single factorization condition that mirrors Peller's operator-Lipschitz characterization for first-order formulas. The modified trace formulas in Section 5 are also genuinely broader than earlier results, covering merely n-times differentiable functions, and the paper states explicit norm bounds for the spectral shift functions. The authors provide detailed arguments for several difficult steps, including the trace-class estimate (4.29) and the finite-dimensional approximation scheme of Lemma 4.17, and the main self-adjoint and unitary arguments appear coherent. However, the contraction case contains a load-bearing step that is delegated to an external theorem without verification, and one of the Section 5 contraction results is proved only by a one-sentence reference to an analogous argument.

major comments (3)

[Section 5.3, proof of Theorem 5.4] The final paragraph asserts that convergence of the finite-dimensional spectral shift functions {η_n} in L1(T)/H1(T) 'readily follows from [10, Theorem 5.1]', but the hypotheses of [10, Theorem 5.1] are not stated and are not verified for the specific sequence (T0,n, Vn) produced by Lemma 4.17. Lemma 4.17 only shows convergence to zero in S2 of terms such as P_n^⊥ T0 P_n and P_n^⊥ V; it is not shown that this implies the conditions on the semi-spectral measures E_{s,n} required by [10, Theorem 5.1]. Since Theorem 4.20 is the only route in the paper to the contraction-case trace formula for the full Hypothesis 4.3 class, this is a load-bearing gap. Please either state [10, Theorem 5.1] and verify its assumptions in detail, or provide a direct proof of the L1(T)/H1(T) convergence of {η_n}.
[Section 4.1, Proposition 4.4] The proof of Theorem 5.4 consists of the single sentence 'The proof follows along the line of the proof of Theorem 5.1.' This is not adequate for the contraction case: the argument must handle functions in the disk algebra, the differentiability results of Section 3, approximation by polynomials, and passage to the limit of the shift functions in the quotient L1(T)/H1(T). None of these steps is written out, so the contraction version of the modified trace formula is not proved as it stands. Please expand this proof or explicitly identify which external result supplies each step, with the relevant hypotheses checked.
[Section 3.2, Theorem 3.6] Proposition 4.4 states that functions in the Besov class B^2_{infty,1}(T) and functions whose f[2] belongs to the integral projective tensor product satisfy Hypothesis 4.3, but the proof is omitted with the comment that 'a similar proof works'. These inclusions are used to substantiate the paper's claim that the new admissible class encompasses all previously known classes. Please provide a proof of both inclusions or give precise references to the exact statements, including the unitary/tensor-product cases, so that the scope claim is fully documented.

minor comments (6)

[Section 4.4.1, Lemma 4.17] The statement says the function ϕ is Gâteaux differentiable on [0,1]; at the endpoints this should be understood as one-sided differentiability, and the sentence should say so explicitly.
[Section 4.4.1, Theorem 4.16] In the statement of Lemma 4.17, the numbered items are operators, while the claim is that they 'converge to zero'. The intended statement is that their S2-norms converge to zero; please rephrase for clarity.
[Section 3.1] The phrase 'The approximation in the right hand side of (4.24) is trivial' is too terse; please specify the boundedness and pointwise convergence of the integrands that justify the limit.
[Section 4.4, Eq. (4.29)] There is a duplicated word in 'we always always refer to the Schäffer matrix unitary dilation'; this should be corrected.
[Section 5.3, Proposition 5.7] The notation 'cont(H)' appears in Eq. (4.29) whereas the paper otherwise uses 'Cont(H)' for contractions; please make the notation uniform.
[Section 1, introduction] In the differentiability statement, the proof uses η_{n,H0,V,Y(t)} but the statement does not explicitly identify Y(t) as the derivative or state the resulting derivative identity. Please add the derivative formula to the statement.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity: the factorization class is not defined in terms of the trace formula, and the main results extend known polynomial formulas by approximation; however, several base cases and the contraction-case convergence step are imported from same-group references [10], [11], [16], creating a modest self-citation burden.

full rationale

The central derivation is not circular by construction. Hypothesis 4.3 defines the admissible class by the Hilbert-space factorization f[2](t1,t2,t3)=<a(t1,t2),b(t2,t3)>; this is a condition on divided differences, not on the trace of RLin_2, so the trace formula Tr(RLin_2)=∫ f'' eta is not built into the hypothesis. The self-adjoint result (Theorem 4.8) is obtained by first treating the continuous second derivative case via the measure construction in [16, proof of Theorem 5.1], then extending to the factorization class by mollification and dominated convergence; the extension argument is supplied in the paper (Eq. (4.5)-(4.6)). The unitary result (Theorem 4.12) reduces the f[2]-factorization case to a polynomial base formula (4.10) whose shift functions xi_k are imported from [11, Theorem 3.2], and then passes to the limit using polynomial approximation with uniform and pointwise bounds; this is a standard extension, not an identity. The contraction results (Theorems 4.14, 4.19, 4.20) likewise derive the trace-class remainder from the factorization via Eq. (4.29) and then approximate by polynomials or finite-dimensional compressions; the final convergence of the finite-dimensional spectral shift functions in L1/H1 is asserted to follow from [10, Theorem 5.1] without verification of its hypotheses, which is a rigor gap in the proof of Theorem 4.20, but not a circularity because the cited theorem is a separate convergence criterion and the present paper does not define its own conclusion in terms of that citation. The modified trace formulas in Section 5 are derived from Riesz representation and L1-duality, not from any fitted parameter. No step in the paper reduces a predicted quantity to an input by construction, and no fitted value is renamed as a prediction. The paper is therefore not circular at the level of its central claims. The only reason the score is not 0 is the heavy reliance on prior work by the same research group ([9], [10], [11], [16]) for base polynomial formulas and the L1/H1 convergence criterion, which is a self-citation burden but not evidence that the derivation is circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted and no new entities are postulated. The central claims rest on standard operator theory, on the cited characterization and boundedness theorems for multiple operator integrals, and on the dilation and finite-dimensional approximation machinery used for contractions.

assumptions (5)

standard math Scalar-valued spectral measures and the w*-continuous extension of multiple operator integrals to L-infinity symbols, as in Section 2.1 following Conway [19].
This is the foundational construction used to define all multiple operator integrals in the paper.
standard math Theorem 4.1(B), the characterization that a triple operator integral maps S2 x S2 into S1 if and only if its symbol factorizes as phi(t1,t2,t3) = <a(t1,t2), b(t2,t3)> with bounded Borel functions a and b, proved in [17, Theorem 6.2].
This external theorem makes Hypothesis 4.3 the natural admissible class and is used throughout Section 4 to prove trace-class membership.
standard math Schatten class boundedness of multiple operator integrals for self-adjoint and unitary operators, Theorem 2.2, quoted from [12] and [13].
These estimates control Taylor remainders and traces and are used in every main theorem.
domain assumption Sz.-Nagy dilation and the explicit Schaeffer matrix unitary dilation for contractions, introduced in Section 3.1.
The contraction results are obtained by lifting operator integrals to unitary dilations and compressing back, so the dilation framework is a necessary modeling assumption.
domain assumption Voiculescu finite-dimensional approximation in Lemma 4.17 and the convergence criterion [10, Theorem 5.1] for the normal contraction case.
Theorem 4.20 depends on reducing to finite-dimensional compressions and on the L1/H1 convergence of the associated spectral shift functions; the paper does not reproduce these external arguments.

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Pith. "Pith review of Trace formulas for $\mathcal{S}^p$-perturbations and extension of Koplienko-Neidhardt trace formulas." pith.science (2026). https://pith.science/paper/H5NDEPJ3

@misc{pith2026241116426,
  author       = {Pith},
  title        = {Pith review of: Trace formulas for $\mathcalS^p$-perturbations and extension of Koplienko-Neidhardt trace formulas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H5NDEPJ3}},
  note         = {Machine review of arXiv:2411.16426}
}

abstract

In this paper, we extend the class of admissible functions for the trace formula of the second order in the self-adjoint, unitary, and contraction cases for a perturbation in the Hilbert-Schmidt class $\mathcal{S}^2(\mathcal{H})$ by assuming a certain factorization of the divided difference $f^{[2]}$. This class is the natural one to ensure that the second order Taylor remainder is a trace class operator. It encompasses all the classes of functions for which the trace formula was previously known. Secondly, for a Schatten $\mathcal{S}^p$-perturbation, $1<p<\infty$, we prove general modified trace formulas for every $n$-times differentiable functions with bounded $n$-th derivative in the self-adjoint and unitary cases and for every $f$ such that $f$ and its derivatives are in the disk algebra $\mathcal{A}(\mathbb{D})$ in the contraction case.

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