Generalized Koopman Operator Overview

• Generalized Koopman operator is a linear operator framework extended to controlled, stochastic, and infinite-dimensional dynamical systems.
• It employs observable lifting, bilinear formulations, and spectral techniques to capture nonlinear dynamics and enhance prediction accuracy.
• Numerical strategies like EDMD are used alongside advanced spectral methods, though challenges remain in approximating continuous spectra.

Searching arXiv for recent and foundational papers on generalized Koopman operator, including controlled, stochastic, nonlinear, and numerical formulations. The generalized Koopman operator denotes a family of extensions of the classical Koopman composition operator beyond autonomous, finite-dimensional, deterministic systems. In the classical discrete-time setting, for $x_{t+1}=T(x_t)$, the Koopman operator acts on an observable $h$ as $[\mathbb{U}h](x)=h(T(x))$; it is linear on observables even when $T$ is nonlinear, but it is generally infinite-dimensional (Sinha et al., 2020). In recent literature, the same operator-theoretic principle is extended to systems with control inputs, stochastic and random dynamics, infinite-dimensional evolution equations, and even observables of probability distributions, while parallel developments enlarge the associated spectral theory through generalized eigenfunctions, continuous spectrum, and function-space constructions tailored to dissipative dynamics (Lazar, 10 Aug 2025, Oprea et al., 15 Apr 2025, Mauroy, 2021, Mezic, 2017).

1. Classical framework and the source of the generalizations

Koopman analysis provides a framework for analyzing a nonlinear dynamical system in terms of a linear operator acting on an infinite-dimensional observable space. This perspective underpins dynamic mode decomposition algorithms and motivates finite-dimensional approximations such as DMD and EDMD, where the operator is approximated on a span of selected observables or lifted coordinates (Wilson, 2022, Williams et al., 2014). In the finite-dimensional EDMD setting introduced by Williams, Kevrekidis, and Rowley, snapshot pairs $\{(\mathbf{x}_m,\mathbf{y}_m)\}$ and a dictionary $\mathcal{D}=\{\psi_1,\dots,\psi_K\}$ yield the approximation

$\mathbf{K}=\mathbf{G}^{+}\mathbf{A},$

with $\mathbf{G}$ and $\mathbf{A}$ formed from empirical inner products of lifted observables; the eigenvalues, eigenfunctions, and modes of $\mathbf{K}$ approximate those of the Koopman operator (Williams et al., 2014).

This classical formulation already contains two pressures toward generalization. First, autonomous linear models in lifted coordinates can require very large observable spaces to approximate fundamentally nonlinear behavior, which creates overfitting risk and weak long-term predictive capability (Wilson, 2022). Second, many systems of interest are not autonomous maps on finite-dimensional state spaces: they may have control inputs, evolve on Hilbert spaces, contain stochastic forcing, or be observed only through aggregate distributions rather than trajectories (Lazar, 10 Aug 2025, Mauroy, 2021, Oprea et al., 15 Apr 2025). The phrase “generalized Koopman operator” therefore refers less to a single canonical object than to a program of extending Koopman linearization to these broader settings.

2. Controlled systems and the product-Hilbert-space formulation

A central modern usage of the term concerns nonlinear systems with inputs,

$h$0

Mircea Lazar’s construction places state observables $h$1 in a Hilbert space $h$2, input observables $h$3 in a Hilbert space $h$4, and forms the product Hilbert space

$h$5

The generalized Koopman operator is then an infinite-dimensional linear operator $h$6 satisfying

$h$7

where $h$8 and $h$9 (Lazar, 10 Aug 2025). Relative to the autonomous Koopman law $[\mathbb{U}h](x)=h(T(x))$0, the lifted controlled dynamics are bilinear in the lifted state and lifted input.

This construction is significant because the generalization of the Koopman operator to systems with control input and the derivation of a nonlinear fundamental lemma were identified as open problems that hinge on the construction of observable functions and their Hilbert space (Lazar, 10 Aug 2025). The product-space formulation gives an explicit basis construction through products $[\mathbb{U}h](x)=h(T(x))$1, proves existence of the generalized operator by orthonormal expansion, and yields a finite-dimensional data-driven estimator

$[\mathbb{U}h](x)=h(T(x))$2

where $[\mathbb{U}h](x)=h(T(x))$3 is the Khatri-Rao product of lifted state and input data (Lazar, 10 Aug 2025). Observable families reported in this framework include universal kernels, deep neural networks, and Takens delay embeddings (“HAVOK” observables) (Lazar, 10 Aug 2025).

The same bilinear structure is also the basis for the reported nonlinear extension of Willems’ fundamental lemma. Under suitable rank conditions, lifted trajectories can be represented by data-driven mixing of previously recorded lifted trajectories, linking operator-theoretic lifting to the behavioral formulation of nonlinear control (Lazar, 10 Aug 2025). On the controlled Van der Pol oscillator, the reported error for KIC does not decrease with increased observable dimension and can worsen, while the generalized Koopman embedding consistently improves as the observable space increases (Lazar, 10 Aug 2025).

3. Nonlinear, stochastic, and distributional generalizations

A second line of generalization does not preserve a globally linear finite-dimensional predictor, but instead retains nonlinear terms after lifting. In the nonlinear data-driven approximation of the Koopman operator, delay-embedded lifted coordinates $[\mathbb{U}h](x)=h(T(x))$4 are augmented by nonlinear functions $[\mathbb{U}h](x)=h(T(x))$5, and the predictor is written as

$[\mathbb{U}h](x)=h(T(x))$6

A reduced-order version is obtained by projection onto a POD basis, and an analogous controlled model

$[\mathbb{U}h](x)=h(T(x))$7

is identified by least squares (Wilson, 2022). The reported motivation is that finite-dimensional linear predictors are generally only able to provide short-term predictions for observable dynamics, while comparable nonlinear estimators provide accurate predictions on substantially longer timescales and replicate infinite-time behaviors that linear predictors cannot (Wilson, 2022).

In stochastic settings, Williams, Kevrekidis, and Rowley showed that EDMD applied to data generated by a Markov process approximates the eigenfunctions of the Kolmogorov backward equation, which they describe as the stochastic Koopman operator (Williams et al., 2014). A more recent extension moves the observables from states to probability distributions. The Distributional Koopman Operator (DKO) is defined by

$[\mathbb{U}h](x)=h(T(x))$8

where $[\mathbb{U}h](x)=h(T(x))$9 is the transfer operator propagating a distribution $T$0 and $T$1 is an observable of distributions (Oprea et al., 15 Apr 2025). The DKO is linear, has the semigroup property, generalizes the stochastic Koopman operator, and is especially useful when only aggregate distribution data is available and particle tracking is unavailable (Oprea et al., 15 Apr 2025).

This distributional viewpoint changes the class of admissible observables. Linear observables recover expectations, but nonlinear observables can encode variance and higher-order statistics, so the operator acts on evolving distributions rather than on individual sample paths (Oprea et al., 15 Apr 2025). In that sense, the generalization is not merely a larger dictionary for the same operator; it changes the base space on which Koopman analysis is performed.

4. Spectral generalization, infinite-dimensional systems, and state-space geometry

The generalized Koopman operator is also a spectral concept. Mezić develops generalized eigenfunctions of the Koopman operator for linear systems using Kato Decomposition, and then extends this structure to nonlinear systems through conjugacy properties and the notion of open eigenfunctions (Mezic, 2017). In this framework, stable, unstable, and center subspaces are interpreted in terms of joint zero level sets of generalized eigenfunctions, while global center manifolds, center-stable manifolds, and center-unstable manifolds are characterized in terms of families of Koopman eigenfunctions associated with nonlinear systems (Mezic, 2017). The same work introduces Modulated Fock Spaces and related Hilbert-space constructions to capture on- and off-attractor properties of dissipative dynamics, especially for systems with globally stable limit cycles and limit tori (Mezic, 2017).

A related generalization concerns infinite-dimensional nonlinear systems such as PDEs. For

$T$2

the Koopman semigroup is defined on bounded continuous functionals $T$3 by

$T$4

where $T$5 is the semiflow on the Hilbert space $T$6 (Mauroy, 2021). Its Lie generator is

$T$7

with $T$8 the Gâteaux derivative in the direction $T$9 (Mauroy, 2021). Finite-dimensional projections of the semigroup and its generator then provide linear approximations of nonlinear infinite-dimensional dynamics and a route to PDE identification from weak data (Mauroy, 2021).

These spectral generalizations connect operator theory to geometry. Zero level sets of eigenfunctions define invariant manifolds and isostables; lattice-type principal spectra provide coherent coordinates around equilibria, cycles, and tori; and finite-dimensional lifted models can be understood as truncations of richer spectral structures rather than as arbitrary regressions (Mezic, 2017, Mauroy, 2021). For systems with symmetries, equivariance implies that the Koopman operator commutes with the group action, and representation theory yields an isotypic decomposition in which finite-dimensional approximations become block diagonal (Salova et al., 2019).

5. Numerical approximation, scalable computation, and known failure modes

The generalized Koopman program is computational as well as conceptual. EDMD reduces Koopman approximation to a least-squares problem over a chosen dictionary, but the resulting matrix formulas often require a Moore-Penrose pseudoinverse, which becomes a bottleneck for high-dimensional systems or large datasets (Sinha et al., 2020, Williams et al., 2014). For the EDMD problem

$\{(\mathbf{x}_m,\mathbf{y}_m)\}$0

a Cholesky-based pseudoinverse formula,

$\{(\mathbf{x}_m,\mathbf{y}_m)\}$1

was proposed to avoid direct SVD or QR on large matrices (Sinha et al., 2020). The method was demonstrated on a network of coupled oscillators with state-space dimension up to $\{(\mathbf{x}_m,\mathbf{y}_m)\}$2 and on the IEEE 68 bus system with state-space dimension $\{(\mathbf{x}_m,\mathbf{y}_m)\}$3 and up to $\{(\mathbf{x}_m,\mathbf{y}_m)\}$4 time-points; the reported computation time scales linearly with system size or data size while eigenvalue spectra closely match those from standard methods (Sinha et al., 2020).

At the same time, the numerical analysis literature places explicit limits on what finite-dimensional approximations can recover. DMD- and EDMD-type methods can be interpreted as finite-section approximations of infinite-dimensional composition operators, and these approximations are not universally reliable (Mezic, 2020). For a mixing map such as $\{(\mathbf{x}_m,\mathbf{y}_m)\}$5 on the unit circle, the Koopman operator has no eigenvalues in the relevant $\{(\mathbf{x}_m,\mathbf{y}_m)\}$6 setting, and the finite-section method yields spurious eigenvalues collapsing to zero, so the approximation does not reflect the true spectrum (Mezic, 2020). Under smoothness and pure point spectrum assumptions, convergence rates of order $\{(\mathbf{x}_m,\mathbf{y}_m)\}$7 for empirical finite sections and pseudospectral convergence for Krylov-subspace approximations are reported, but the same paper emphasizes that continuous-spectrum settings remain problematic (Mezic, 2020).

This numerical caveat is conceptually important. Generalization does not eliminate the distinction between point spectrum and continuous spectrum, nor does a larger dictionary guarantee faithful spectral recovery. In systems with mixing dynamics, finite-dimensional modal outputs can be artifacts of the approximation scheme rather than genuine Koopman eigenstructure (Mezic, 2020).

6. Applications, misconceptions, and open directions

Generalized Koopman constructions have been applied well beyond state prediction. Numerical algorithms themselves can be treated as dynamical systems, leading to Koopman analysis of gradient descent, Nesterov acceleration, and Newton-Raphson methods, with data-adapted basis functions used to construct reduced operator representations on low-dimensional manifolds (Dietrich et al., 2019). In power systems, Koopman-based modal decompositions have been used for coherency identification, precursor diagnostics of instabilities, and stability assessment without mathematical models (Susuki et al., 2017). In signal processing, a generalized Nyquist-Shannon sampling theorem based on the Koopman operator extends exact reconstruction to certain non-band-limited signals in generator-bounded spaces (Zeng et al., 2023). In symbolic dynamics, Generalized Koopman Analysis (GKA) uses left eigenfunctions of finite-dimensional Koopman approximations to identify symbolic boundaries in multivariate chaotic series (Li et al., 8 Jun 2025). In generative modeling, Koopman spectral approximations have been used as surrogates for the inverse Langevin generator in Wasserstein gradient descent (Xu et al., 21 Dec 2025).

Several recurring misconceptions are corrected by this body of work. The first is that the generalized Koopman operator is a unique object; the literature instead presents controlled bilinear lifts, nonlinear lifted predictors, stochastic and distributional operators, generalized spectral theories, and semigroups on functionals as different but related extensions (Lazar, 10 Aug 2025, Wilson, 2022, Oprea et al., 15 Apr 2025, Mauroy, 2021). The second is that finite-dimensional linear lifts are universally sufficient. The reported evidence is the opposite: linear estimators may require high-dimensional observable spaces, may overfit, and generally provide only short-term predictions for strongly nonlinear behaviors, whereas some nonlinear lifted models recover basins of attraction, unstable fixed points, and periodic orbits that finite-dimensional linear predictors cannot reproduce (Wilson, 2022). The third is that data-driven approximation guarantees faithful Koopman spectra in all regimes; continuous-spectrum examples show otherwise (Mezic, 2020).

Survey work therefore emphasizes that finite approximations of the Koopman operator still lack an overarching framework and that questions of invariant subspaces, spectral convergence, truncation error, robustness, and controlled or stochastic extensions remain active (Bevanda et al., 2021). This suggests that “generalized Koopman operator” is best understood as an evolving operator-theoretic umbrella for nonlinear dynamics, rather than as a finalized definition.