{"id":"26b4e59b-1601-45a7-af5b-04d338aa0b48","arxiv_id":"2411.16426","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Second-order trace formulas are proved for the natural factorization class of divided differences, and modified trace formulas are proved for all n-times differentiable functions with bounded n-th derivative in the Schatten p setting.","lead":"This paper extends second-order trace formulas for self-adjoint, unitary and contractive operators to functions whose second divided difference admits a Hilbert-space factorization, and proves modified trace formulas for Schatten-class perturbations. It broadens the known admissible function classes and yields spectral shift functions in L1.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Contraction-case trace formula (Thm 4.20) hinges on an unverified application of the external convergence criterion [10, Thm 5.1], whose hypotheses are not checked for the Pn-compressed pairs.","rationale":"The reader's weakest-assumption analysis correctly points to Hypothesis 4.3 and to the external contraction-case dependencies (Lemma 4.17 and [10, Theorem 5.1]). I single out the convergence step in Theorem 4.20 because it is where the contractive extension under the full factorization class is actually completed: every earlier contraction result either covers a smaller function class (Theorem 4.14) or different operator classes (Theorems 4.21 and 4.22). The self-adjoint and unitary theorems are supported by detailed local arguments and standard MOI estimates, so they do not carry the same risk. The contraction case, by contrast, outsources its decisive limit to a reference whose exact hypotheses are never stated, and the paper does not show that the finite-dimensional approximations built from Lemma 4.17 satisfy those hypotheses. This is a concrete, checkable gap rather than a demonstrated error: if [10, Theorem 5.1] applies verbatim, the proof works. That is exactly the situation the reader's CONDITIONAL verdict describes, so I do not propose changing the verdict. The proposed test either verifies the reference applies or, failing that, independently establishes the needed convergence on a nontrivial example.","tokens_in":40678,"tokens_out":17606,"duration_ms":159190,"concrete_test":"Obtain [10, Theorem 5.1] and check its hypotheses against the sequence from Lemma 4.17; specifically verify the required convergence of the semi-spectral measures Es,n and the uniform bounds on Vn. As a self-contained alternative, test the case T0 = M_z on L2(T) with a rank-one V, taking Pn to be the span of indicators of a dyadic partition: compute eta_n from the formula in Theorem 4.16 and check directly that ||eta_n - eta_m||_{L1/H1} -> 0 and that lim \\int p'' eta_n equals Tr{RLin_2(p,T0,V)} for one or two test polynomials p. If the limit is not the expected trace, Theorem 4.20 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4.20, the second-order trace formula for a pair of contractions with T0 normal under Hypothesis 4.3, is the least secure link in the paper's central claim. After Theorem 4.19 establishes trace-norm approximation of RLin_2(f,T0,V) by finite-dimensional remainders, the remaining task is to show that the finite-dimensional spectral shift functions eta_n of Theorem 4.16 converge in L1(T)/H1(T). The paper says this 'readily follows from [10, Theorem 5.1]' and gives no further argument. The hypotheses of [10, Theorem 5.1] are not stated, and the paper does not verify them for the specific sequence (T0,n, Vn) = (PnT0Pn, PnVPn) produced by Lemma 4.17. Lemma 4.17 only guarantees convergence to zero in S2 of terms like Pn^\\perp T0 Pn and Pn^\\perp V; it is not shown that this is the right condition for the convergence criterion, nor that the semi-spectral measures of Ts,n converge in any topology needed by [10, Theorem 5.1]. Since Theorem 4.20 is the only route to the contraction version of the advertised extension, a failure of this step would leave the contraction claim unproved for the full Hypothesis 4.3 class.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":40920,"tokens_out":6508,"duration_ms":60829,"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":[{"comment":"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":"Section 5.3, proof of Theorem 5.4"},{"comment":"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":"Section 4.1, Proposition 4.4"},{"comment":"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.","section":"Section 3.2, Theorem 3.6"}],"minor_comments":[{"comment":"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":"Section 4.4.1, Lemma 4.17"},{"comment":"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":"Section 4.4.1, Theorem 4.16"},{"comment":"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":"Section 3.1"},{"comment":"There is a duplicated word in 'we always always refer to the Schäffer matrix unitary dilation'; this should be corrected.","section":"Section 4.4, Eq. (4.29)"},{"comment":"The notation 'cont(H)' appears in Eq. (4.29) whereas the paper otherwise uses 'Cont(H)' for contractions; please make the notation uniform.","section":"Section 5.3, Proposition 5.7"},{"comment":"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.","section":"Section 1, introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's contraction-case Theorem 4.20 depends on a convergence criterion from [10], a paper by the same research group; since the hypotheses of that criterion are not checked in the manuscript, the editor may wish to confirm that [10] is available and that the cited theorem indeed applies to the sequence of compressions constructed in Lemma 4.17. Similar reliance on earlier papers by the same group is otherwise standard in this area. The paper fits the scope of the journal, and the main self-adjoint and unitary results appear sound; the contraction results need the requested verification before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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<p<∞ are new and are proved by a coherent MOI-based approximation argument. The self-adjoint and unitary cases I read carefully; they hold up. The differentiation results for contractions in Section 3 are also a solid extension of Peller's work.\n\nSoft 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.\n\nThe 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.\n\nWho 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.","headline":"A substantive extension of Koplienko-Neidhardt trace formulas whose contraction case needs one missing verification before I would trust it fully.","tokens_in":41474,"tokens_out":2182,"would_cite":true,"duration_ms":20975,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A55","47A56","47B10","47B49"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Multiple operator integrals","operator derivatives","trace formulas","spectral shift functions","Schatten classes","divided differences","contractions","Hilbert-Schmidt perturbations"],"falsifier":"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.","tokens_in":40472,"feed_emoji":"∫","tokens_out":11911,"duration_ms":108385,"temperature":0.7,"pith_summary":"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)=<a(t1,t2),b(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<p<∞, any n-times differentiable function with bounded n-th derivative satisfies a weighted trace identity Tr(R_n · X)=∫ $f^{{(n)}}$ η, with the shift function depending continuously on the data. If true, this gives a single natural framework for second-order spectral shift functions and extends higher-order trace formulas to low-regularity functions.","feed_headline":"Trace formula extended to all factorized divided differences","feed_subtitle":"Under one inner-product condition, second-order Taylor remainders are trace class and their traces become spectral-shift integrals.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Characterizes when a triple operator integral takes values in the trace class via the inner-product factorization; this is the exact criterion behind Hypothesis 4.3.","marker":"[17]"},{"why":"Establishes higher-order S^p-differentiability and sharp remainder estimates for self-adjoint operators, used throughout Sections 3–5.","marker":"[12]"},{"why":"Provides the unitary analog of the S^p-differentiability and norm estimates used for the Neidhardt and modified unitary formulas.","marker":"[13]"},{"why":"Proves the second-order trace formula for a smaller admissible class; Theorem 4.8 extends it to the factorization class.","marker":"[16]"},{"why":"Establishes the earlier Koplienko trace formula for the Besov class B^2_{\\infty,1}, which is contained in the new class.","marker":"[38]"},{"why":"Supplies higher-order trace formulas for unitaries and contractions in the tensor-product class, including the shift functions used as the base in Theorem 4.12.","marker":"[11]"},{"why":"Gives the polynomial base case of the contraction trace formula that Theorem 4.14 extends.","marker":"[46]"},{"why":"Provides the higher-order contraction trace formula for polynomials that Remark 4.15 extends to the factorization class.","marker":"[43]"},{"why":"Supplies the finite-dimensional approximation and convergence criterion for contractions used to pass from finite rank to general pairs.","marker":"[10]"},{"why":"Introduces first-order modified trace formulas for pairs of unitaries and contractions, which Section 5 generalizes to all orders.","marker":"[3]"}],"fun_headline_variants":["Factorized differences extend trace formulas to all orders","Trace formulas for Schatten perturbations via factorization","One factorization condition makes remainders trace class","Koplienko-Neidhardt formulas for all factorized differences"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Factorized differences extend trace formulas to all orders","Trace formulas for Schatten perturbations via factorization","One factorization condition makes remainders trace class","Koplienko-Neidhardt formulas for all factorized differences"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00028,"raw_usage":{"total_tokens":1677,"prompt_tokens":980,"completion_tokens":697,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":635}},"tokens_in":596,"tokens_out":697,"duration_ms":7076,"temperature":1.0,"reasoning_tokens":635,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:12:13.867527+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Coine, C","cited_arxiv_id":null,"evidence_quote":"Characterizes when a triple operator integral takes values in the trace class via the inner-product factorization; this is the exact criterion behind Hypothesis 4.3."},{"cited_title":"Coine, Perturbation theory and higher order Sp-differentiability of operator functions , Rev","cited_arxiv_id":null,"evidence_quote":"Establishes higher-order S^p-differentiability and sharp remainder estimates for self-adjoint operators, used throughout Sections 3–5."},{"cited_title":"Functions of unitaries with $\\mathcal{S}^p$-perturbations for non continuously differentiable functions","cited_arxiv_id":"2406.13333","evidence_quote":"Provides the unitary analog of the S^p-differentiability and norm estimates used for the Neidhardt and modified unitary formulas."},{"cited_title":"Coine, C","cited_arxiv_id":null,"evidence_quote":"Proves the second-order trace formula for a smaller admissible class; Theorem 4.8 extends it to the factorization class."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the earlier Koplienko trace formula for the Besov class B^2_{\\infty,1}, which is contained in the new class."},{"cited_title":"Higher-Order Trace Formulas for Contractive and Dissipative Operators","cited_arxiv_id":"2407.02789","evidence_quote":"Supplies higher-order trace formulas for unitaries and contractions in the tensor-product class, including the shift functions used as the base in Theorem 4.12."},{"cited_title":"Potapov and F","cited_arxiv_id":null,"evidence_quote":"Gives the polynomial base case of the contraction trace formula that Theorem 4.14 extends."},{"cited_title":"Potapov, A","cited_arxiv_id":null,"evidence_quote":"Provides the higher-order contraction trace formula for polynomials that Remark 4.15 extends to the factorization class."},{"cited_title":"Chattopadhyay, S","cited_arxiv_id":null,"evidence_quote":"Supplies the finite-dimensional approximation and convergence criterion for contractions used to pass from finite rank to general pairs."},{"cited_title":"Lipschitz Estimates and an application to trace formulae","cited_arxiv_id":"2312.08706","evidence_quote":"Introduces first-order modified trace formulas for pairs of unitaries and contractions, which Section 5 generalizes to all orders."}],"review_version":1}