Generalized Multiple Operator Integrals for Operators with Finite Dimensions

Shih-Yu Chang

REVIEW 5 major objections 5 minor 1 cited by

Generalized Multiple Operator Integrals for Operators with Finite Dimensions

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A generalized multiple operator integral extends matrix calculus to non-Hermitian matrices.

desk verdict The GMOI construction is a real algebraic extension, but the derivative theorem fails for generic non-normal matrices because it silently requires differentiating Jordan data that need not exist. read the letter →

arxiv 2506.19971 v1 pith:ILTXH2GT submitted 2025-06-24 math.FA math.OAmath.SP

Shih-Yu Chang This is my paper · ORCID

classification math.FAmath.OAmath.SP MSC 47A6047A5515A60

keywords multipleoperatorintegralsnon-HermitianmatricesJordandecompositiondivideddifferencesperturbationformulamatrixfunctionderivativesFrobeniusnormestimatesfunctionalcalculus

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 proposes Generalized Multiple Operator Integrals (GMOIs), a finite-dimensional replacement for Multiple Operator Integrals that works for arbitrary square matrices, not only Hermitian ones. It defines the GMOI through the Jordan decomposition of each parameter matrix, adding divided-difference terms that carry the nilpotent parts, so the integral encodes the full Jordan structure rather than just eigenvalues. The paper claims this recovers conventional MOIs as a special case, supplies Frobenius-norm upper and lower bounds, and proves a perturbation formula with explicit correction terms. From that it derives Lipschitz estimates and a continuity theorem for the GMOI under matrix perturbations, and it represents the n-th derivative of a matrix function as a combination of GMOIs and correction terms. A sympathetic reader would care because non-Hermitian matrices are the rule in numerical linear algebra and non-Hermitian physics, and this gives them a functional-calculus calculus analogous to the Hermitian one.

What carries the argument

The load-bearing object is the GMOI of Eq. (17): a finite sum over eigenvalue indices and geometric components in which each parameter matrix contributes either its spectral projector $P_{k,i}$ or a nilpotent power $N^q_{k,i}$, weighted by normalized partial derivatives (divided differences) of the kernel $\beta$. The binary-expansion notation $\Psi$ in Section 4 organizes the $2^{\zeta+1}$ choices of projector versus nilpotent at each parameter slot; this expansion is what makes the norm estimates tractable. The perturbation formula in Theorem 3 rests on Lemma 2, a divided-difference identity that converts differences $\beta^{[k]}(\ldots,\lambda,\ldots)-\beta^{[k]}(\ldots,\mu,\ldots)$ into $(\lambda-\mu)\beta^{[k+1]}(\ldots,\lambda,\mu,\ldots)$, so eigenvalue differences are traded for one more GMOI slot. The correction terms $X(\ldots)$ collect all residual terms involving differences of nilpotents $N_C-N_D$.

What would settle it

Take $D$ to be a $2\times 2$ Jordan block and $V$ a rank-one perturbation that makes $D+tV$ diagonalizable for every $t>0$; compute the GMOI difference and the correction term $\overline{X}(\ldots)$ as $t\to 0$. If the correction term is not $o(t)$, or if the second-derivative formula of Example 3 fails to match a direct finite-difference evaluation of $\frac{d^2}{dt^2}f(D+tV)|_{t=0}$, then Lemma 3 and Theorem 7 would be contradicted.

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Extended reading notes

Core claim

The central claim is that the family defined in Eq. (17), built from spectral projectors $P_{k,i}$ and nilpotents $N_{k,i}$ of each matrix $X_p$ via Jordan decomposition, is the correct generalization of multiple operator integrals. For any analytic kernel $\beta$, the GMOI $T^{X_1,\ldots,X_{\zeta+1}}_{\beta}(Y_1,\ldots,Y_\zeta)$ expands into one term using only projectors plus partial sums over nilpotent powers with divided-difference weights $\beta^{(q)}(\lambda)/q!$, so it reproduces the Taylor expansion of $f(X_1,\ldots,X_r)$ from Theorem 1. Theorem 3 gives a perturbation formula expressing the difference between GMOIs evaluated at $C$ and $D$ as GMOIs of $C-D$ plus correction terms, and Theorem 7 uses iterated perturbation to write $d^n f(X+tY)/dt^n$ at $t=0$ as $n!$ times a GMOI with divided-difference kernel $f^{[n]}$, minus sums of GMOIs with nilpotent parameter pairs and correction terms $X^{(i)}$.

Load-bearing premise

The proof assumes that, as one matrix is smoothly deformed into another, its Jordan blocks (the projectors and nilpotents) can also be chosen to deform smoothly as long as eigenvalue multiplicities do not change; without that assumption, the continuity and derivative conclusions do not follow.

Editorial extensions

If this is right

Conventional Hermitian MOIs become the projector-only summand of the GMOI, so the Hermitian theory is a special case rather than a separate construction.
The Frobenius-norm bounds give explicit control on how large a GMOI can be in terms of kernel maxima and nilpotent norms, which is what makes perturbation series manageable.
The Lipschitz estimate bounds the change in a GMOI when the argument matrices $Y_i$ change, with scalar constants computed from the upper bounds.
Continuity of the GMOI under $X_{i,\ell}\to X_i$ holds when the Jordan data vary continuously; under that condition perturbative expansions converge.
Every $n$-th derivative of a matrix function at a non-Hermitian matrix $X$ is expressed as a finite combination of GMOIs with divided-difference kernels plus computable correction terms.

Reading between the lines

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

If the continuity theorem is right, the GMOI gives a route to Fréchet derivative formulas for non-analytic matrix functions defined through divided differences on non-normal matrices, not just analytic functions.
A natural test is to compute the second-derivative formula of Example 3 for a $2\times 2$ Jordan block with $Y$ coupling the chain; direct finite differences should match the GMOI expression only if the correction terms are retained.
The paper treats finite dimensions throughout; the same projector-nilpotent expansion suggests a path toward infinite-dimensional operators with discrete non-normal spectra, but continuous-spectrum cases need a mechanism other than Jordan decomposition.
Where geometric multiplicities change (eigenvalue coalescence), the paper's smooth-Jordan assumption fails, so one would expect the perturbation formula to require additional spectral projectors or to break down entirely.

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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

Shih-Yu Chang's paper develops a Jordan-decomposition-based multiple operator integral for arbitrary finite matrices, with norm bounds, perturbation formulas, and a claimed formula for derivatives of matrix functions. The algebraic core is the genuinely new part: it rewrites the standard non-commutative divided-difference expansion of f(X1,...,Xr) in terms of projectors and nilpotents, and the composition rule in Proposition 2 shows the formalism has internal coherence. The upper norm bound is a routine triangle inequality, and the Lipschitz estimate in Theorem 4 follows from telescoping plus that bound; those parts are basically fine.

The soft spots are where the reader's report put them. The lower bound in Eq. (41) is not justified for non-orthogonal projectors: the step bounding the norm of A0 below by min|beta| times the product of the Y-norms uses a spectral-norm-type property that Frobenius sums of non-orthogonal projections do not have. That is a real error, though it is not load-bearing for everything else.

The load-bearing problem is Theorem 7. The proof, including the X^(i) correction terms, requires differentiating the Jordan projectors and nilpotents of X+tY at t=0. For non-normal X this fails in general. The stress-test example X=E12, Y=E21 is clean: X+tY has eigenvalues ±√t, so the spectral projectors have 1/√t singularities as t→0, and the t-derivatives of the Jordan data do not exist, while the second derivative of f(X+tY) is perfectly finite for analytic f. So Theorem 7 as stated, for arbitrary matrices X and Y, is false. The manuscript itself concedes after Lemma 5 that X^(1) has "no simple formula," which is a claim-without-derivation at the hinge. A hypothesis of stable geometric multiplicities along the ray might repair it, but that is not what is stated, and the example shows the missing hypothesis is not cosmetic.

The continuity theorem also rests on Lemma 3, whose proof explicitly assumes stable geometric multiplicities; the theorem statement omits that assumption, and norm convergence alone does not imply it. So the continuity claim is not established in the stated generality.

Who is this for? A functional analyst working on non-Hermitian perturbation theory will find the GMOI definition useful as a formal calculus, and the algebraic properties are worth knowing. But the analytic claims as stated are not proved. I would not cite the main theorems in current form, though I might cite the GMOI definition as a notation if needed. The paper deserves a serious referee—it is not incoherent algebra—but it should return only after the derivative theorem is either proved under explicit stable-multiplicity assumptions or downgraded to a formal identity. Send it to peer review with the smooth-Jordan hypothesis and the projector lower bound flagged as mandatory revisions.

Referee Report

5 major / 5 minor

Summary. The manuscript introduces Generalized Multiple Operator Integrals (GMOIs) for arbitrary square matrices via Jordan decompositions, claiming to unify classical multiple operator integrals and to provide norm estimates, perturbation formulas, Lipschitz bounds, continuity theorems, and a representation of n-th derivatives of matrix functions in terms of GMOIs. The central advertised result is Theorem 7 (Eq. (110)), which purports to express d^n f(X+tY)/dt^n at t=0 for any matrices X and Y.

Significance. If correct, the framework would offer a useful extension of MOI calculus beyond Hermitian matrices, potentially impacting non-Hermitian perturbation theory and matrix functional calculus. The paper has a systematic algebraic setup, and the upper-norm bound in Theorem 2 (Eq. (38)) is a routine triangle inequality that appears sound. However, the main derivative and continuity claims rely on smoothness properties of Jordan decompositions that are neither stated nor generally true, and the principal theorem fails on a simple 2x2 non-normal example. The significance of the paper is therefore not established in its current form.

major comments (5)

[Section 7, Theorem 7 (Eq. (110)) and Lemma 5 (Eqs. (106)-(107))] Theorem 7 is stated for 'any matrices X and Y', but its proof requires the derivatives at t=0 of the Jordan projectors P_{kd,id}(t) and nilpotents N_{kd,id}(t) of X+tY, as made explicit in Example 5 (Eq. (132)). For X = [[0,1],[0,0]] and Y = [[0,0],[1,0]], the matrix X+tY has eigenvalues ±√t for t>0, and the associated spectral projectors contain factors of 1/√t, so P(t) and N(t) are not differentiable at t=0 even though d²/dt² f(X+tY)|_{t=0} exists for analytic f. Consequently, the correction terms X^(i) appearing in Eq. (110) are undefined in this basic non-normal case. The theorem requires an additional hypothesis such as stable geometric multiplicities along the ray, which is neither stated nor satisfied in the example.
[Section 6, Lemma 3 (proof, Eqs. (91)-(93))] The proof of Lemma 3 assumes 'under smooth variation of the Jordan decomposition (i.e., assuming stable geometric multiplicities)' and then invokes Fréchet derivatives of the Jordan projectors and nilpotents. This hypothesis is not stated in Lemma 3 or in Theorem 5, and it fails for non-normal matrices where eigenvalues can coalesce. The cancellation argument for the difference T3−T4 also relies on unproved convergence of the index pairs (kc,ic) to (kd,id). As a result, the continuity theorem (Theorem 5, Eq. (96)) is not established for the claimed generality.
[Section 5.1, Theorem 3] The perturbation formula in Theorem 3 is load-bearing for both Theorem 5 and Theorem 7, but its proof only verifies identities (65) and (66), with the remaining identities (67)-(72) asserted to follow similarly without proof. Moreover, the derivation of Eq. (73) contains the assertion 'P_{kc,ic}P_{kc,ic}=P_{kd,id}P_{kd,id}=I', which is false for idempotent projectors unless a summation over all indices is intended; the displayed algebra does not rigorously justify the claimed correction terms. This gap undermines the foundation for the subsequent continuity and differentiation results.
[Section 7, Lemma 4 (Eq. (104))] The proof of Lemma 4 applies Theorem 3 with C=(X+tY)_N and D=X_N, but the two GMOI terms displayed in Eq. (104) are identical (both have argument tY) and cancel trivially. No argument is given to show that the difference of nilpotent parts equals tY, nor that the correction term X(...) vanishes. Since Lemma 4 is used in the proof of Theorem 7 to discard certain derivative terms, this is a substantive gap in the main theorem.
[Section 4, Theorem 2 (Eq. (41) and Eq. (45))] The lower-bound estimate uses the step ∥Σ β(...) P...P ∥ ≥ min|β| ∥Σ P...P ∥, which does not follow from the triangle inequality and is not generally valid for non-orthogonal projectors. The equality ∥Σ P Y P ∥ = ∥Y_1...Y_ζ∥ requires the projectors to form a resolution of the identity, which is not stated or proved in the non-Hermitian setting. A rigorous justification of Eq. (45) is needed before the lower bound in Eq. (41) can be accepted.

minor comments (5)

[Throughout] There are numerous typos and unclear notations, including 'Fr ´echet', 'Propotion', 'geometry multiplicities' for 'geometric multiplicities', and malformed expressions such as 'B4,3))' in Eq. (34). A careful editorial pass is needed.
[Sections 2 and 7] The foundational Theorem 1 (from the author's preprint [10]) and Theorem 6 (from the author's preprint [9]) are cited rather than proved. Since these results are load-bearing for the GMOI definition and the derivative application, the paper is not self-contained, and the dependence on unpublished preprints should be addressed.
[Eq. (17)] The definition of the GMOI sets Y_{ζ+1}=I but the notation and index ranges for the parameter matrices and argument matrices should be checked for consistency; the current presentation is confusing.
[Section 6, Theorem 5] The assumption that all partial derivatives of β[ζ+1] up to any order are bounded is odd because β[ζ+1] is evaluated on finite spectra; the intended smoothness hypothesis should be stated precisely.
[Section 7, Example 5] The formulas (132)-(138) for high-order derivatives of the correction terms are stated without proof and assume the differentiability of spectral data. Given the counterexample above, these formulas are not valid for arbitrary non-normal matrices.

Circularity Check

2 steps flagged · score 4.0 of 10

Central derivative formula rests on the author's own unproved Theorem 6 and Theorem 1; no fitted-input circularity, but the framework is not self-contained.

self citation load bearing [Section 7, Theorem 6 and proof of Theorem 7 (Eqs. 100, 110-116)]
"Recall Theorem 11 [9], we have Theorem 6 Let f be a complex-valued and differentiable function, we have df(X+tY)/dt = T^{X+tY,X+tY}_{f[1]}(Y), where T^{X+tY,X+tY}_{f[1]}(Y) is the GDOI with matrices X and Y."

The proof of Theorem 7 starts from Eq. (112), which is exactly this cited identity, and obtains all higher derivatives by repeatedly differentiating GMOIs built from it. Theorem 6 is not proved in this paper; the cited [9] is the author's own preprint and is not machine-checked or independently verified. Hence the central n-th derivative representation has no independent base: its first-order input is a self-citation whose own content is the same GDOI framework being advertised.
ansatz smuggled in via citation [Section 2, Eqs. (9)-(16); Section 3.1, Eq. (17)]
"Let us present Theorem 3 in [10] first. ... According to Theorem 1, GMOIs can be defined as follows."

The GMOI definition in Eq. (17) is a direct transcription of Theorem 1's expansion (Eq. (6)) with Y matrices inserted between the projector/nilpotent factors, and Theorem 1 is 'Theorem 3 in [10]', the author's own preprint. The Section 2 'unification' likewise obtains Eq. (8) 'by applying Theorem 1.' Thus the foundational ansatz—derivative terms driven by nilpotents—is imported from a self-citation and is not derived or independently validated here.

full rationale

The paper contains no parameter fitting and no prediction that is equivalent to its inputs by construction: the GMOI is a definition, the norm estimates are derived from that definition via triangle inequalities, and the perturbation formula is an identity with correction terms that are later given explicit expressions. However, the derivation chain is not self-contained. The foundational expansion of matrix functions (Theorem 1) is quoted from the author's own preprint [10] and is used both to define GMOIs and to 'unify' conventional MOIs, while the base first-derivative identity (Theorem 6) is quoted from the author's preprint [9] and is the starting point for every higher derivative in Theorem 7. These are load-bearing self-citations rather than independent, machine-checked, or externally verified results, so the central claims rest on the author's prior unproved framework. In addition, Lemma 3's proof explicitly inserts a smooth-variation assumption ('assuming stable geometric multiplicities') and Theorem 7 leaves correction terms as derivatives of Jordan data ('tedious and there is no simple formula'); these are correctness risks rather than circular reductions, and they are not scored as circularity here. Overall, the self-citation chain is substantial but the paper does contain independent algebraic content, so a moderate score of 4 is appropriate.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No data fitting, no empirically fitted constants, and no new physical entities are postulated. The GMOI is a mathematical construct, not an entity with independent empirical evidence. The main unmotivated inputs are the self-cited Theorem 1 from [10] and the smoothness assumption on Jordan decompositions.

assumptions (6)

ad hoc to paper Theorem 1 of [10] (same author) provides the expansion of an analytic multivariable function of matrices in terms of spectral projectors and nilpotents from Jordan decompositions.
The GMOI definition and the Section 2 unification are built on this theorem; it is cited, not proved, and no independent verification is supplied.
ad hoc to paper The Jordan decomposition of a matrix varies smoothly under one-parameter perturbations when geometric multiplicities are stable.
Assumed in the proof of Lemma 3 (Section 6) and used for continuity and perturbation formulas; not stated as a hypothesis in Theorem 5 and false in general for non-diagonalizable matrices.
standard math Spectral mapping theorem for matrices and the standard expansion of f(X) via Jordan form.
Used in Section 2 for spectral decompositions.
standard math Divided difference identities and the Faà di Bruno formula for derivatives of compositions.
Used in Lemma 2 and Example 5.
standard math Frobenius norm satisfies the triangle inequality and reverse triangle inequality; projectors of a matrix sum to the identity.
Used throughout Section 4; the paper's application of the reverse triangle inequality to A_0 is where the lower bound error appears.
domain assumption The GMOI defined by Jordan basis projectors is independent of the choice of basis within each eigenspace and across geometric components.
The paper never proves well-definedness; the expressions involve individual Jordan-block projectors and nilpotents that are basis-dependent. This is an unstated assumption in the definition (Eq. 17).

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Cite this review

Pith. "Pith review of Generalized Multiple Operator Integrals for Operators with Finite Dimensions." pith.science (2026). https://pith.science/paper/ILTXH2GT

@misc{pith2026250619971,
  author       = {Pith},
  title        = {Pith review of: Generalized Multiple Operator Integrals for Operators with Finite Dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ILTXH2GT}},
  note         = {Machine review of arXiv:2506.19971}
}

read the original abstract

Multiple Operator Integrals (MOIs) have played a foundational role in operator theory and functional calculus, particularly for analyzing Hermitian matrices via spectral decomposition. Conventional MOIs rely on the assumption of self-adjointness, making them analytically tractable for computing Frechet derivatives, establishing trace formulas, and deriving commutator estimates. However, many problems in mathematics, science, and engineering involve matrices that are fundamentally non-Hermitian, arising in contexts such as numerical discretization of differential operators, signal processing, control systems, and non-Hermitian physics. These cases necessitate a more general framework for operator integration. In this paper, we propose and develop the theory of Generalized Multiple Operator Integrals (GMOIs), which extends MOI techniques to arbitrary matrices, including non-Hermitian and non-normal cases. We unify conventional MOIs under the perspective of the Spectral Mapping Theorem and present a rigorous construction of Generalized Double Operator Integrals (GDOIs) in finite-dimensional setting via Jordan decomposition of any matrix. We establish key algebraic properties, derive norm estimates, and prove continuity and Lipschitz-type perturbation formulas. Finally, we demonstrate how GMOIs can be used to compute matrix function derivatives in non-Hermitian environments, thus significantly broadening the applicability of MOI-based methods in both theoretical and practical domains.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Generalized Multiple Operator Integrals and Perturbation Theory for Operators with Continuous Spectra
math.FA 2025-07 reject novelty 4.0 of 10

The paper defines generalized multiple operator integrals for continuous-spectrum operators and derives perturbation and spectral shift formulas, but the assumed spectral decomposition is not valid for the intended op...

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Works this paper leans on

11 extracted references · 5 canonical work pages · cited by 1 Pith paper

[1]

Generalized choi–davis–jensen’s operator inequalities and their applications,

S. Y . Chang and Y . Wei, “Generalized choi–davis–jensen’s operator inequalities and their applications,” Symmetry, vol. 16, no. 9, p. 1176, 2024

work page 2024
[2]

Random multiple operator integrals,

S.-Y . Chang, “Random multiple operator integrals,”arXiv preprint arXiv:2210.09392, 2022

arXiv 2022
[3]

Random double tensors integrals,

S. Y . Chang, “Random double tensors integrals,”arXiv preprint arXiv:2204.01927, 2022

arXiv 2022
[4]

Random parametrization double tensors integrals and their applications,

——, “Random parametrization double tensors integrals and their applications,”arXiv preprint arXiv:2205.03523, 2022

arXiv 2022
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Non-hermitian photonics based on parity–time symmetry,

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-hermitian photonics based on parity–time symmetry,”Nature Physics, vol. 14, no. 1, pp. 11–19, 2018. 51

work page 2018
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Non-Hermitian physics,

Y . Ashida, Z. Gong, and M. Ueda, “Non-Hermitian physics,”Advances in Physics, vol. 69, no. 3, pp. 249–435, 2020

work page 2020
[8]

Generalized double operator integrals: Finite dimensions,

S.-Y . Chang, “Generalized double operator integrals: Finite dimensions,”arXiv preprint arXiv:2503.16239, 2025

arXiv 2025

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[9]

Generalized double operator integrals for continuous spectrum operators,

——, “Generalized double operator integrals for continuous spectrum operators,”arXiv preprint arXiv:2505.02188, 2025

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[10]

Operator characterization via projectors and nilpotents,

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Skripka and A

A. Skripka and A. Tomskova,Multilinear operator integrals theory and applications. Springer, 2019. 52

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Generalized choi–davis–jensen’s operator inequalities and their applications,

S. Y . Chang and Y . Wei, “Generalized choi–davis–jensen’s operator inequalities and their applications,” Symmetry, vol. 16, no. 9, p. 1176, 2024

work page 2024

[2] [2]

Random multiple operator integrals,

S.-Y . Chang, “Random multiple operator integrals,”arXiv preprint arXiv:2210.09392, 2022

arXiv 2022

[3] [3]

Random double tensors integrals,

S. Y . Chang, “Random double tensors integrals,”arXiv preprint arXiv:2204.01927, 2022

arXiv 2022

[4] [4]

Random parametrization double tensors integrals and their applications,

——, “Random parametrization double tensors integrals and their applications,”arXiv preprint arXiv:2205.03523, 2022

arXiv 2022

[5] [5]

L. N. Trefethen and D. Bau,Numerical Linear Algebra. SIAM, 1997

work page 1997

[6] [6]

Non-hermitian photonics based on parity–time symmetry,

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-hermitian photonics based on parity–time symmetry,”Nature Physics, vol. 14, no. 1, pp. 11–19, 2018. 51

work page 2018

[7] [7]

Non-Hermitian physics,

Y . Ashida, Z. Gong, and M. Ueda, “Non-Hermitian physics,”Advances in Physics, vol. 69, no. 3, pp. 249–435, 2020

work page 2020

[8] [8]

Generalized double operator integrals: Finite dimensions,

S.-Y . Chang, “Generalized double operator integrals: Finite dimensions,”arXiv preprint arXiv:2503.16239, 2025

arXiv 2025