Koopman Spectral Theory Overview

• Koopman spectral theory is an operator-theoretic framework that linearizes nonlinear dynamics by lifting state evolution into an infinite-dimensional space of observables.
• It uses spectral decomposition through eigenvalues and eigenfunctions to provide modal insights into oscillatory and dissipative behavior across complex systems.
• Data-driven techniques like DMD and EDMD extract Koopman modes, offering practical tools for control applications, forecasting, and model reduction.

Koopman spectral theory is the operator-theoretic framework in which nonlinear dynamical systems are globally linearized by lifting the state evolution to the infinite-dimensional space of observables. The Koopman operator encodes the full dynamics via its spectrum—eigenvalues and eigenfunctions—which offer modal decompositions analogous to those in finite-dimensional linear systems. In recent years, Koopman spectral analysis has become central to dynamical systems, machine learning, and control, with applications spanning physical, engineered, and data-driven contexts. This article surveys the mathematical construction of the Koopman operator, principal spectral properties, parameterizations for learning and control, data-driven computation, and recent empirical advances.

1. Mathematical Definition and Operator-Theoretic Structure

The Koopman operator $\mathcal{K}$ is defined for a (possibly nonlinear) discrete- or continuous-time dynamical system. For discrete time, if $x_{n+1}=T(x_n)$ with $T: \mathbb{R}^d \to \mathbb{R}^d$, then for any scalar observable $g: \mathbb{R}^d \to \mathbb{R}$, the operator acts as

$(\mathcal{K}g)(x) = g(T(x)).$

The mapping is linear on the Banach space of observables (e.g., $C(X)$ or $L^2(X,\mu)$), despite the nonlinearity of $T$. The same construction applies to continuous-time flows $\dot{x} = f(x)$ by defining the operator semigroup $U^t g(x) = g(\varphi^t(x))$ and infinitesimal generator $\mathcal{L}g = \nabla g \cdot f$ (Susuki et al., 2017, Forootani et al., 1 Feb 2026).

As applied to stochastic, controlled, or hybrid systems, the Koopman operator extends its domain accordingly, evolving observables on state-control pairs, composing with expectation over stochastic forcing, or gluing across hybrid mode transitions (Proctor et al., 2016, Komeno et al., 2022, Katayama et al., 2024).

2. Spectral Decomposition: Eigenvalues, Eigenfunctions, and Modal Expansions

Spectral theory for $\mathcal{K}$ centers on its eigenvalues $\lambda_j$ and eigenfunctions $\phi_j$, defined by $\mathcal{K}\phi_j = \lambda_j \phi_j$. These generate modal decompositions of the dynamics: time-evolution of observables admits expansions

$$g(T^n(x)) = (\mathcal{K}^n g)(x) = \sum_j \lambda_j^n \phi_j(x)\,m_j + \text{(continuous spectrum)},$$

where $m_j$ are Koopman modes (projection coefficients) specific to $g$ (Susuki et al., 2017, Mezic, 2017). For $|\lambda_j|=1$ (unitary case), the evolution is oscillatory (ergodic/measure-preserving); for $|\lambda_j|<1$, the modes are exponentially decaying (dissipative systems); and for continuous spectrum, non-modal aperiodic dynamics appear (Susuki et al., 2020).

Spectra of the Koopman operator in systems with multiscale or hybrid dynamics can inherit complicated geometry—such as lattices in the complex plane, spectral bands stratified by subsystem regimes, and pointwise or singular continuous eigenfunctions (Katayama et al., 2024, Katayama et al., 2024, Viennot, 15 May 2025).

Table: Koopman Operator Spectra in Dynamical Regimes

Dynamical Context	Koopman Spectrum Structure	Eigenfunction Interpretation
Linear/Equilibrium	Discrete, finite lattice	Linear modes; classical modal decomposition
Ergodic (Unitary)	Pure point + continuous	Oscillatory/frequency modes + aperiodic part
Limit Cycle/Quasiperiodic/Torus	Discrete lattice	Floquet or Fourier-like eigenfunctions
Chaotic/Liouville-mixed	Continuous	No global eigenfunctions, resonance bands
Singular-perturbed/Hybrid	Piecewise/concatenated	Glued or partitioned modal structure

3. Lattice Structure and Nonlinear Proliferation

Koopman spectral theory exhibits a lattice structure: products of eigenfunctions are eigenfunctions, with multiplicative eigenvalues (Bramburger, 31 Jul 2025, Zhang et al., 2020). If $\mathcal{K}f_1 = \lambda_1 f_1$ and $\mathcal{K}f_2 = \lambda_2 f_2$, then $$\mathcal{K}(f_1 f_2) = (\lambda_1 \lambda_2) (f_1 f_2)$$. This property generates infinite hierarchies (“proliferation”) of higher-order eigenfunctions under nonlinear interactions and enables a full modal expansion from a finite set of principal modes (Zhang et al., 2020, Mezic, 2017).

However, the lattice property is rigorously justified only for deterministic (measure-preserving or dissipative) systems with Banach algebra observables. In stochastic Markovian settings, or when working in $L^2$ with minimal regularity, the lattice structure may break down (Bramburger, 31 Jul 2025).

4. Data-Driven Algorithms and Spectral Approximation

Extraction of Koopman spectral objects from measurement data is enabled by Dynamic Mode Decomposition (DMD), Extended DMD (EDMD), Krylov/Prony methods, and from recent rigorous frameworks such as residual DMD (ResDMD) (Susuki et al., 2017, Colbrook et al., 2021, Li et al., 2022). DMD forms a best-fit linear operator from snapshot pairs and obtains its eigenvalues/modes, which converge to approximations of the Koopman spectrum under ergodic and richness assumptions (Susuki et al., 2017, Brunton et al., 2021).

EDMD generalizes to arbitrary dictionaries of observables, projecting the Koopman operator onto a learned low-dimensional subspace. Recent methods address noise, finite-sample convergence, and ill-conditioning by adopting residual-based validation, Fourier/Cauchy reparametrization, and rigorous convergence theorems to control spectral pollution and provide error bounds (Drmač et al., 2018, Colbrook et al., 2021, Zeng et al., 2024).

Spectral densities and pseudospectra for systems with continuous spectrum are estimated via convolution with Poisson kernels or by optimization over the residual pseudonorm, with theoretical guarantees in the large-data and large-dictionary limits (Colbrook et al., 2021).

5. Learnable Parameterizations and Spectral Control

Recent advances integrate Koopman parameterizations with deep learning and modern forecasting architectures. Koopman-enhanced models define a latent linear propagator $K_\phi$ acting in the latent representation of input windows encoded by a neural network (e.g., Transformer backbone) (Forootani et al., 1 Feb 2026). Explicit spectral control is achieved by parameterizing $K_\phi$ in an orthogonal–diagonal–orthogonal (ODO) form, factorizing the operator as

$K_\phi = U\,\operatorname{diag}(\Sigma)\,V^T,$

where $U,V$ are learned orthonormal matrices, and the spectrum is shaped by learnable nonlinear maps (scalar gate, per-mode gate, MLP, low-rank truncation). Stability constraints are imposed via spectral radius bound $|\sigma_i|<\rho_{max}<1$ (exponential contraction) and Lyapunov penalties to bias toward contractivity in a prescribed norm (Forootani et al., 1 Feb 2026).

Empirical results show that explicit spectral constraints yield superior bias–variance trade-offs and improved conditioning compared to unconstrained or non-spectrally-regularized models. The spectral distribution of learned propagators, analyzed by eigenvalue trajectory and violin plot diagnostics across tasks, confirm theoretical predictions: constrained models stabilize error and latent dynamics, while unconstrained models can lead to brittle, overdamped, or unstable representations.

6. Koopman Spectral Theory with Control and Hybrid Systems

Koopman operator theory is systematically extended to systems with control inputs by lifting the state–input evolution to observables depending jointly on state and actuators (Proctor et al., 2016). The input–output extension provides a natural operator-theoretic foundation for Dynamic Mode Decomposition with control (DMDc) and its data-driven variants. In this framework, eigenvalues and modes characterize dynamic responses and input-driven patterns, with rigorous algorithms for finite-dimensional approximation and model identification (Proctor et al., 2016, Komeno et al., 2022).

Spectral theory has further been expanded to hybrid dynamical systems: the construction of invariant observable spaces under flows with discrete jumps (“guards” and “resets”) allows for global Koopman analysis, preserving spectral completeness and enabling modal embedding even across piecewise-smooth or non-smooth transitions (Katayama et al., 2024).

7. Empirical Applications and Advanced Algorithms

Koopman spectral frameworks underpin a wide range of methods in time series prediction, dynamical model reduction, control, and uncertainty quantification. Learnable Koopman operators, physics-informed kernel spectral methods, and stochastic Koopman variants expand the reach of spectral analysis to high-dimensional, noisy, or partially observed systems (Forootani et al., 1 Feb 2026, Valva et al., 2024).

Adaptive spectral-collocation methods (ASK) achieve mesh-free, spectrally convergent solutions for deterministic ODEs by expanding evolution in Koopman eigenpairs and dynamically recentering the local basis (Li et al., 2022). Rigorous kernel- and residual-based spectral estimation enables spectral analysis of chaotic molecular systems (e.g., 20,000+ dimensions), turbulent flows, and complex cellular automata (Colbrook et al., 2021, Taga et al., 2021, Valva et al., 2024).

Koopman-based generative models exploit operator spectra for accelerated optimal transport, while Laplace-domain analysis using Koopman resolvents connects classical system synthesis with nonlinear dynamics, extending control-theoretic tools to nonlinear settings (Susuki et al., 2020, Xu et al., 21 Dec 2025).

• Learnable Koopman-Enhanced Transformer-Based Time Series Forecasting with Spectral Control (Forootani et al., 1 Feb 2026)
• Generalizing Koopman Theory to allow for inputs and control (Proctor et al., 2016)
• Koopman Analysis of the Singularly-Perturbed van der Pol Oscillator (Katayama et al., 2024)
• Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems (Susuki et al., 2020)
• Deep Koopman with Control: Spectral Analysis of Soft Robot Dynamics (Komeno et al., 2022)
• Applied Koopman Operator Theory for Power Systems Technology (Susuki et al., 2017)
• The Adaptive Spectral Koopman Method for Dynamical Systems (Li et al., 2022)
• On the lattice property of the Koopman operator spectrum (Bramburger, 31 Jul 2025)
• Data driven Koopman spectral analysis in Vandermonde-Cauchy form via the DFT (Drmač et al., 2018)
• Koopman Operators for Global Analysis of Hybrid Limit-Cycling Systems (Katayama et al., 2024)
• Spectrum of the Koopman Operator, Spectral Expansions in Functional Spaces, and State Space Geometry (Mezic, 2017)
• Generative Modeling through Spectral Analysis of Koopman Operator (Xu et al., 21 Dec 2025)
• Koopman operator framework for spectral analysis and identification of infinite-dimensional systems (Mauroy, 2021)
• Koopman spectral analysis of elementary cellular automata (Taga et al., 2021)
• Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems (Colbrook et al., 2021)
• Modern Koopman Theory for Dynamical Systems (Brunton et al., 2021)
• Physics-informed spectral approximation of Koopman operators (Valva et al., 2024)
• Koopman analysis of CAT maps onto classical and quantum 2-tori (Viennot, 15 May 2025)
• A theoretical framework for Koopman analyses of fluid flows, part 1: local Koopman spectrum and properties (Zhang et al., 2020)
• Koopman Spectral Analysis from Noisy Measurements based on Bayesian Learning and Kalman Smoothing (Zeng et al., 2024)