On the lattice property of the Koopman operator spectrum

Jason J. Bramburger

REVIEW 2 major objections 5 minor 14 references

On the lattice property of the Koopman operator spectrum

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that the spectrum of the Koopman operator of a discrete-time deterministic dynamical system is multiplicatively closed, whether or not the product spectral point is an eigenvalue.

desk verdict Theorem 1.1 is false: T(x)=x/2 on [0,1] gives a Koopman spectrum (closed disk of radius √2) that is not closed under multiplication, and the truncation step (3.8) is invalid. read the letter →

arxiv 2507.23498 v1 pith:P7V65UQA submitted 2025-07-31 math.DS math.FAnlin.CD

Jason J. Bramburger This is my paper · ORCID

classification math.DSmath.FAnlin.CD MSC 37A3047A1047B33

keywords KoopmanoperatorspectrumlatticepropertyWeylcriterioneigenfunctionsdiscrete-timedynamicalsystemsstochasticMarkovchains

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

This paper proves that for a discrete-time deterministic dynamical system, the spectrum of its Koopman operator has a multiplicative lattice structure: whenever two complex numbers are in the spectrum, their product is also in the spectrum, even when that product is not an eigenvalue with a genuine eigenfunction in L2. The result matters because the Koopman operator is the standard linearization of nonlinear dynamics used in data-driven analysis, and the lattice property lets practitioners generate new spectral information by multiplying old spectral information. The paper also shows the analogous statement fails for stochastic Koopman operators, using a two-state Markov chain whose spectrum is not multiplicatively closed. The central proof works through Weyl's criterion, constructing bounded approximate eigenfunctions for the product from approximate eigenfunctions for the factors.

What carries the argument

The engine is Weyl's criterion, which says that a point λ belongs to the spectrum if and only if there is a unit-norm sequence of approximate eigenfunctions ψ_n with (K−λ)ψ_n tending to zero in L2. The proof takes such sequences for λ and η, clamps them to bounded functions via the formula (3.2), multiplies the clamped functions, renormalizes, and uses almost-uniform convergence on sets of small measure to bound the residual of the product by sums of the individual residuals. The clamping step is doing the load-bearing work of forcing the product into L2 while retaining enough spectral information about the original approximate eigenfunctions.

What would settle it

A direct pointwise check falsifies that step: with λ=1, take f_n(x)=1/(2k) and f_n(Tx)=-1/(2k); the residual before truncation is 1/k while after truncation it is 2/k, so the claimed inequality (3.8) does not hold and the constructed sequence need not satisfy Weyl's criterion.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1.1: if T:M→M is a map and K:L2(M,μ)→L2(M,μ) is its Koopman operator, then λ,η∈σ(K) implies λη∈σ(K). The proof starts from Weyl sequences for λ and η, truncates them into bounded clamped functions whose product lies in L2 and is bounded away from zero, and then shows the residual for λη vanishes in norm along a diagonal sequence. A corollary states that a bounded Koopman operator must have spectrum contained in the closed unit disk, since otherwise powers of an outlying spectral value would produce unbounded spectral values. The paper further claims this lattice property is particular to deterministic composition: a two-state Markov chain with transition matrix [[0.9,0.1],[0.5,0.5]] has spectrum {2/5,1}, which is not closed under multiplication.

Load-bearing premise

The proof relies on the assumption that clamping the approximate eigenfunctions to bounded values never increases the residual error that Weyl's criterion requires to vanish; the step labeled (3.8) in the proof uses exactly that assumption.

Editorial extensions

If this is right

A bounded Koopman operator on L2 cannot have spectral values outside the closed unit disk, because any value with |λ|>1 would generate unbounded powers λ^n in the spectrum.
Deterministic continuous-time flows with well-posed Koopman semigroups inherit the multiplicative structure, so the lattice property is not an artifact of discrete sampling.
Products of approximate eigenfunctions identified from data can be treated as new observables that remain spectrally meaningful even when the exact product is not an L2 function.
For finite Markov chains, the stochastic Koopman spectrum generically fails to be a lattice as soon as any eigenvalue has modulus different from 1, because the chain has only finitely many eigenvalues.
The lattice property gives a clean way to distinguish deterministic Koopman spectra from stochastic ones: closure under multiplication is a spectral feature of deterministic composition.

Reading between the lines

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

A repaired estimate for the truncation step would likely carry the same proof through and may extend the lattice property to broader classes of composition operators than the topological-space setting stated here.
Because the paper contrasts deterministic and stochastic Koopman spectra, multiplicative closure could serve as a spectral diagnostic for deciding whether a data-driven transfer operator came from deterministic dynamics.
One could test the lattice statement numerically: approximate a chaotic map with a finite-dimensional Koopman method and check whether products of approximate eigenvalues remain near the approximated spectrum.
The paper leaves open which stochastic processes, if any, do yield a lattice-structured Koopman spectrum; a natural next step is to characterize those transition operators by a spectral or algebraic condition.

Share X Bluesky LinkedIn Reddit HN

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

Short version: the main theorem is false. The paper claims that if λ,η are in the spectrum of a Koopman operator on L^2, then λη is also in the spectrum. Take M=[0,1], μ=dx, T(x)=x/2. The Koopman operator is Kf=f∘T, and under the standard unitary to L^2(0,∞) it becomes √2 times a right shift. The right shift on L^2(0,∞) has spectrum the closed unit disk, so σ(K) is the closed disk of radius √2. Now √2 and i√2 are both in σ(K), but 2i has modulus 2 and is not in σ(K). So Theorem 1.1 and its Corollary 1.2 fail for a bounded Koopman operator.

The proof fails at (3.8). The clamping operation that defines f_{n,m,k} does not preserve residuals; pushing small values up to 1/k can make |f_{n,m,k}(Tx)-λ f_{n,m,k}(x)| larger than |f_n(Tx)-λ f_n(x)|. A concrete choice of values (e.g., k=2, f_n(x)=0.25, f_n(Tx)=0.75, λ=100) makes the left side ~49 vs the right side ~24. The bounds (3.9)-(3.12) all depend on this invalid inequality.

There's also a more basic problem: the paper's Weyl criterion states that λ is in the spectrum iff there is a sequence of unit vectors with ||(K-λ)ψ_n||→0. That is only true for the approximate point spectrum, not the whole spectrum. For the right shift, interior points of the unit disk are residual spectrum and have no approximate eigenvectors. So the proof would not even start for such λ.

What is worth keeping? The paper is clearly written, and the remark that L^2 is not a Banach algebra is correct. The Markov-chain example in Section 2 is a correct illustration that stochastic Koopman operators can lack the lattice property. But that example is not new, and it does not support the deterministic claim.

Bottom line: the central claim is contradicted by a simple explicit example, and the proof has a clear gap. This is not a paper that needs referee time; it should be returned to the author. A corrected version would need to restrict to the approximate point spectrum or add conditions on the operator (normality, etc.), and even then the truncation argument would need a real fix.

I wouldn't cite it, and I wouldn't bring it to reading group except as a cautionary tale about truncation arguments.

Referee Report

2 major / 5 minor

Summary. The paper claims that for any discrete-time deterministic map T on a topological space M for which the Koopman operator K is well defined on L2(M, μ) for some probability measure μ, the spectrum σ(K) is closed under multiplication (Theorem 1.1), and that a stochastic Koopman operator need not have this lattice structure (Section 2). The proof of Theorem 1.1 uses Weyl's criterion and constructs approximate eigenfunctions by truncating and multiplying approximate eigenfunctions. The stochastic example, a two-state Markov chain with spectrum {2/5, 1}, correctly shows that finite stochastic Koopman spectra need not be multiplicatively closed.

Significance. If Theorem 1.1 were true, it would give a general structural constraint on Koopman spectra and would imply the striking Corollary 1.2 that every bounded Koopman operator has spectral radius at most one. The paper addresses a real and often glossed-over issue: eigenfunction products may leave L2, so a purely formal eigenvalue multiplication argument is insufficient. The stochastic counterexample in Section 2 is simple and correct. However, the central deterministic claim is false: there are elementary maps T satisfying the theorem's hypotheses whose Koopman spectra are not multiplicatively closed. Since the main theorem is the paper's core contribution and is contradicted by a concrete example, the paper cannot be accepted in its current form.

major comments (2)

[3, Eq. (3.8)] Theorem 1.1 is false as stated. Take M=[0,1] with the usual topology, μ equal to normalized Lebesgue measure, and T(x)=x/2. The Koopman operator is (Kf)(x)=f(x/2), which is bounded on L2([0,1],dx) with ||K||=√2. Under the unitary map U:L2([0,1],dx)→L2([0,∞),ds) given by (Uf)(s)=e^{-s/2}f(e^{-s}), K is unitarily equivalent to √2 S, where (Sf)(s)=f(s+ln 2) is a right shift. The right shift S is a non-surjective isometry and its spectrum is the closed unit disk, so σ(K) is the closed disk of radius √2. Hence √2∈σ(K) and √2∈σ(K), but their product 2 is not in σ(K) because |2|>√2. This directly contradicts Theorem 1.1 and also Corollary 1.2, which asserts that every λ∈σ(K) for a bounded K satisfies |λ|≤1.
[Section 3, Eq. (3.8)] The proof breaks at inequality (3.8). The claim that |f_{n,m,k}(Tx)−λ f_{n,m,k}(x)| ≤ |f_n(Tx)−λ f_n(x)| on M_m is not valid: the truncation in (3.2) clamps values to the interval [−(1/k), m] and can increase the magnitude of a difference near zero. For a concrete failure, take f_n(x)=1.5/k, f_n(Tx)=1/k, λ=100, and large k so that both truncated values f_{n,m,k}(x) and f_{n,m,k}(Tx) equal 1/k. Then the left side of (3.8) equals |(1/k)−100(1/k)|=99/k, while the right side equals |(1/k)−100(1.5/k)|=98.5/k, violating the inequality. Since the subsequent bounds (3.9)–(3.12) and the final Weyl criterion argument depend on (3.8), the proof does not establish Theorem 1.1.

minor comments (5)

[Eq. (1.4)] Weyl's criterion is misstated: the expression '∥Kψn − λψk∥2' should read '∥Kψn − λψn∥2'.
[Section 3, before Eq. (3.7)] The text says the second term to be bounded is ∥g_{n,m,k}∘T − λ g_{n,m,k}∥, but it should involve η, not λ, to match the decomposition in (3.3).
[Eq. (3.11)] Inequality (3.11) repeats the same f and λ terms as (3.10); the displayed bound for the second term should be |η| k^2 m ∥g_{n,m,k}∘T − η g_{n,m,k}∥ ≤ |η| m/k^4 + |η| k^2/m. As printed, the estimate for the η-term is omitted.
[Acknowledgements] The acknowledgements contain the typo 'Fondes de Recherche' for 'Fonds de Recherche'.
[Throughout] There are formatting issues with 'W eyl' and some LaTeX rendering artifacts, but these do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the spectral lattice claim is derived directly from Weyl's criterion and an explicit truncation argument, not from its own conclusion.

full rationale

The paper's central claim (Theorem 1.1) is that if λ and η are in the Koopman spectrum, then λη is in the spectrum. The proof proceeds by taking Weyl sequences {f_n} and {g_n} for λ and η, constructing truncated bounded functions f_{n,m,k} and g_{n,m,k}, and then verifying Weyl's criterion for the products. This is a direct derivation from standard spectral definitions and does not assume the conclusion. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. The author's self-citations appear in the introduction and references as background and context, not as load-bearing steps in the proof. The proof's key estimate (3.8) may be invalid — the skeptical counterexample suggests the theorem may be false — but a false inequality is a mathematical error, not circular reasoning. Circularity requires that a claimed derivation reduces to its own inputs by definition or by a self-citation chain; that is not present here. The stochastic example in Section 2 is used only to contrast with the deterministic claim and does not feed back into the proof. Therefore the appropriate circularity score is 0.

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

The proof relies on standard spectral theorems plus an unproven and false contraction property of the truncation operator. No free parameters or invented entities are introduced.

assumptions (4)

domain assumption K is well-defined on L2(M, μ) for a probability measure μ
Stated in Theorem 1.1; the proof does not restrict T or μ further. This assumption is satisfied by the counterexample with T(x)=x/2 and Lebesgue measure.
standard math Weyl's criterion characterizes the spectrum (equation 1.4)
Quoted from reference [7] and used to construct approximate eigenfunctions. This is a standard spectral theorem.
standard math Egorov's theorem applies to the residual sequences
Used in Section 3 to get almost uniform convergence from convergence in measure on a probability space. This is standard real analysis.
ad hoc to paper The truncation in (3.2) satisfies |f_{n,m,k}(Tx) - λ f_{n,m,k}(x)| ≤ |f_n(Tx) - λ f_n(x)| on M_m
This is the load-bearing step in the proof; it is false, as shown by the amplification of differences near zero. The proof's main estimate depends on it.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the lattice property of the Koopman operator spectrum." pith.science (2026). https://pith.science/paper/P7V65UQA

@misc{pith2026250723498,
  author       = {Pith},
  title        = {Pith review of: On the lattice property of the Koopman operator spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P7V65UQA}},
  note         = {Machine review of arXiv:2507.23498}
}

read the original abstract

The Koopman operator has become a celebrated tool in modern dynamical systems theory for analyzing and interpreting both models and datasets. The linearity of the Koopman operator means that important characteristics about it, and in turn its associated nonlinear system, are captured by its eigenpairs and more generally its spectrum. Many studies point out that the spectrum of the Koopman operator has a multiplicative lattice structure by which eigenvalues and eigenfunctions can be multiplied to produce new eigenpairs. However, these observations fail to resolve whether the new eigenfunction remains in the domain of the Koopman operator. In this work, we prove that the spectrum of the Koopman operator associated to discrete-time dynamical systems has a multiplicative lattice structure. We further demonstrate that the Koopman operator associated to discrete-time stochastic process does not necessarily have such a structure, demonstrating an important nuance that lies at the heart of Koopman operator theory.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 13 canonical work pages

[1]

Petar Bevanda, Stefan Sosnowski, and Sandra Hirche, Koopman operator dynamical models: Learning, analysis and control , Annual Reviews in Control 52 (2021), 197–212

work page 2021
[2]

Jason J Bramburger, Data-driven methods for dynamic systems , SIAM, 2024

work page 2024
[3]

Jason J Bramburger and Giovanni Fantuzzi, Auxiliary functions as Koopman observables: Data-driven analysis of dynamical systems via polynomial optimization , Journal of Nonlinear Science 34 (2024), no. 1, 8

work page 2024
[4]

2, 229–340

Steven L Brunton, Marko Budisic´ c, Eurika Kaiser, and J Nathan Kutz, Modern Koopman theory for dynamical systems , SIAM Review 64 (2022), no. 2, 229–340

work page 2022
[5]

Marko Budiˇ si´ c, Ryan Mohr, and Igor Mezi´ c,Applied Koopmanism, Chaos: An Interdisciplinary Journal of Nonlinear Science 22 (2012), no. 4

work page 2012
[6]

Efrain Gonzalez, Moad Abudia, Michael Jury, Rushikesh Kamalapurkar, and Joel A Rosen- feld, The kernel perspective on dynamic mode decomposition , arXiv preprint arXiv:2106.00106 (2021)

arXiv 2021
[7]

PD Hislop, IM Sigal, PD Hislop, and IM Sigal, The essential spectrum: Weyl’s criterion , Introduction to Spectral Theory: With Applications to Schr¨ odinger Operators (1996), 69–75

work page 1996
[8]

1, 51–79

Stefan Klus, P´ eter Koltai, and Christof Sch¨ utte, On the numerical approximation of the Perron–Frobenius and Koopman operator, Journal of Computational Dynamics 3 (2016), no. 1, 51–79. 7

work page 2016

Show all 14 references

[9]

1, 015014

Stefan Klus and Maia Trower, Transfer operators on graphs: Spectral clustering and beyond , Journal of Physics: Complexity 5 (2024), no. 1, 015014

2024
[10]

5, 315–318

Bernard O Koopman, Hamiltonian systems and transformation in Hilbert space , Proceedings of the National Academy of Sciences 17 (1931), no. 5, 315–318

1931
[11]

Milan Korda and Igor Mezi´ c,On convergence of extended dynamic mode decomposition to the Koopman operator, Journal of Nonlinear Science 28 (2018), 687–710

2018
[12]

Peter J Schmid, Dynamic mode decomposition of numerical and experimental data , Journal of Fluid Mechanics 656 (2010), 5–28

2010
[13]

3, 1930–1951

Mathias Wanner and Igor Mezic, Robust approximation of the stochastic Koopman operator , SIAM Journal on Applied Dynamical Systems 21 (2022), no. 3, 1930–1951

2022
[14]

Matthew O Williams, Ioannis G Kevrekidis, and Clarence W Rowley, A data–driven approxi- mation of the Koopman operator: Extending dynamic mode decomposition , Journal of Nonlin- ear Science 25 (2015), 1307–1346. 8

2015

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Petar Bevanda, Stefan Sosnowski, and Sandra Hirche, Koopman operator dynamical models: Learning, analysis and control , Annual Reviews in Control 52 (2021), 197–212

work page 2021

[2] [2]

Jason J Bramburger, Data-driven methods for dynamic systems , SIAM, 2024

work page 2024

[3] [3]

Jason J Bramburger and Giovanni Fantuzzi, Auxiliary functions as Koopman observables: Data-driven analysis of dynamical systems via polynomial optimization , Journal of Nonlinear Science 34 (2024), no. 1, 8

work page 2024

[4] [4]

2, 229–340

Steven L Brunton, Marko Budisic´ c, Eurika Kaiser, and J Nathan Kutz, Modern Koopman theory for dynamical systems , SIAM Review 64 (2022), no. 2, 229–340

work page 2022

[5] [5]

Marko Budiˇ si´ c, Ryan Mohr, and Igor Mezi´ c,Applied Koopmanism, Chaos: An Interdisciplinary Journal of Nonlinear Science 22 (2012), no. 4

work page 2012

[6] [6]

Efrain Gonzalez, Moad Abudia, Michael Jury, Rushikesh Kamalapurkar, and Joel A Rosen- feld, The kernel perspective on dynamic mode decomposition , arXiv preprint arXiv:2106.00106 (2021)

arXiv 2021

[7] [7]

PD Hislop, IM Sigal, PD Hislop, and IM Sigal, The essential spectrum: Weyl’s criterion , Introduction to Spectral Theory: With Applications to Schr¨ odinger Operators (1996), 69–75

work page 1996

[8] [8]

1, 51–79

Stefan Klus, P´ eter Koltai, and Christof Sch¨ utte, On the numerical approximation of the Perron–Frobenius and Koopman operator, Journal of Computational Dynamics 3 (2016), no. 1, 51–79. 7

work page 2016