Generalized Koopman Operator (GeKo)

• Generalized Koopman Operator (GeKo) is a family of extensions that represents nonlinear dynamics linearly on enriched observable spaces, including controlled systems and structured liftings.
• It utilizes product Hilbert spaces to independently lift state and input observables, resulting in bilinear lifted dynamics that generalize traditional control models.
• Data-driven realizations using methods such as EDMD and Khatri–Rao regression, combined with model reduction techniques, enable efficient prediction and control of complex systems.

The Generalized Koopman Operator (GeKo) denotes a family of extensions of classical Koopman operator theory from autonomous state observables to richer observable spaces, controlled systems, structured liftings, and regularized operator realizations. In its most explicit recent formulation, GeKo is a linear operator on a product observable space of states and inputs, written as $K: H_x \otimes H_u \to H_x$, such that $\Psi_x(F(x,u)) = K(\Psi_x(x)\otimes \Psi_u(u))$, yielding lifted bilinear dynamics $z_{t+1}=K(z_t\otimes v_t)$ (Lazar, 10 Aug 2025). Earlier Koopman literature already described “generalized Koopman operators” as finite-dimensional approximations of the Koopman operator restricted to chosen dictionaries of observables, especially through EDMD and related methods [(Williams et al., 2014); (Snyder et al., 2021)]. This suggests that GeKo is best understood not as a single universally fixed object, but as a technically precise umbrella for operator-theoretic generalizations that preserve the central Koopman idea: nonlinear dynamics are represented through linear evolution on observables, even when the ambient construction, function space, or finite-dimensional realization is substantially enlarged.

1. Classical Koopman foundations and the emergence of GeKo

For a discrete-time autonomous system $x_{k+1}=F(x_k)$, the classical Koopman operator $\mathcal{K}$ acts on scalar observables $g$ by composition, $(\mathcal{K}g)(x)=g(F(x))$. In continuous time, the Koopman semigroup acts as $(\mathcal{K}^t g)(x)=g(\Phi^t(x))$, with infinitesimal generator $\mathcal{L}g=f(x)\cdot \nabla g(x)$. The crucial structural fact is that $\mathcal{K}$ is linear on observables even when the state dynamics are nonlinear, and its spectral objects—eigenvalues, eigenfunctions, and modes—organize nonlinear trajectories as linear superpositions of exponentially evolving features (Williams et al., 2014).

The first step toward a generalized Koopman viewpoint is the replacement of the full infinite-dimensional observable space by a chosen finite-dimensional subspace. In Dynamic Mode Decomposition (DMD), the observables are simply the state coordinates, and one fits a linear map $\Psi_x(F(x,u)) = K(\Psi_x(x)\otimes \Psi_u(u))$0 from snapshot pairs. In Extended Dynamic Mode Decomposition (EDMD), one instead selects a dictionary $\Psi_x(F(x,u)) = K(\Psi_x(x)\otimes \Psi_u(u))$1 or $\Psi_x(F(x,u)) = K(\Psi_x(x)\otimes \Psi_u(u))$2, lifts the state to a feature vector $\Psi_x(F(x,u)) = K(\Psi_x(x)\otimes \Psi_u(u))$3 or $\Psi_x(F(x,u)) = K(\Psi_x(x)\otimes \Psi_u(u))$4, and computes a matrix that approximates the Koopman operator on the span of that dictionary. This matrix is a finite-dimensional Koopman approximation restricted to a chosen observable subspace; different dictionaries therefore produce different finite-dimensional “generalized Koopman” representations of the same underlying dynamics (Snyder et al., 2021).

A central implication follows immediately. Classical Koopman theory is a statement about an operator on observables; GeKo begins when the observable space itself becomes a design variable. Polynomial, Fourier, radial basis, spectral-element, kernel, neural, or delay-coordinate dictionaries are not merely numerical conveniences. They define the finite-dimensional space on which the operator is represented, and therefore determine which Koopman spectral objects can be approximated and which nonlinear structures can be captured.

2. Product Hilbert spaces and the controlled GeKo formulation

The most rigorous controlled formulation of GeKo in the recent literature starts from the discrete-time nonlinear control system

$\Psi_x(F(x,u)) = K(\Psi_x(x)\otimes \Psi_u(u))$5

and constructs separate Hilbert spaces of state and input observables,

$\Psi_x(F(x,u)) = K(\Psi_x(x)\otimes \Psi_u(u))$6

together with their tensor-product space

$\Psi_x(F(x,u)) = K(\Psi_x(x)\otimes \Psi_u(u))$7

With state and input liftings $\Psi_x(F(x,u)) = K(\Psi_x(x)\otimes \Psi_u(u))$8 and $\Psi_x(F(x,u)) = K(\Psi_x(x)\otimes \Psi_u(u))$9, the generalized Koopman operator is defined as a single linear operator

$z_{t+1}=K(z_t\otimes v_t)$0

satisfying

$z_{t+1}=K(z_t\otimes v_t)$1

In trajectory form, this becomes the bilinear lifted system

$z_{t+1}=K(z_t\otimes v_t)$2

Under Riesz-basis assumptions on the state and input observables and the composition condition $z_{t+1}=K(z_t\otimes v_t)$3, the operator exists and admits the explicit representation $z_{t+1}=K(z_t\otimes v_t)$4, with $z_{t+1}=K(z_t\otimes v_t)$5 the Gram operator of the product basis and $z_{t+1}=K(z_t\otimes v_t)$6 built from inner products of $z_{t+1}=K(z_t\otimes v_t)$7 against that basis (Lazar, 10 Aug 2025).

This construction is more general than lifted linear models of the form $z_{t+1}=K(z_t\otimes v_t)$8, and also more general than control-affine lifted bilinear models $z_{t+1}=K(z_t\otimes v_t)$9. Its key distinction is that states and inputs are lifted independently and only combined through the tensor product. As a result, nonlinear dependence on the input is not forced into a fixed affine or channelwise-bilinear template; it is represented through the input observable space itself (Lazar, 1 Jun 2026).

The same product-space formulation also yields a nonlinear fundamental lemma. When outputs are included through an output Hilbert space $x_{k+1}=F(x_k)$0 and a linear output operator $x_{k+1}=F(x_k)$1, finite-horizon lifted trajectories of the nonlinear system can be characterized by a linear equation in a data matrix built from blocks of $x_{k+1}=F(x_k)$2 and output observables. In that sense, GeKo not only extends Koopman theory to control, but also provides a behavioral, data-driven representation of nonlinear trajectories in lifted bilinear coordinates (Lazar, 10 Aug 2025).

3. Data-driven realizations: EDMD, Khatri–Rao regression, and reduced models

In the EDMD formulation, one is given snapshot pairs $x_{k+1}=F(x_k)$3 with $x_{k+1}=F(x_k)$4, a dictionary $x_{k+1}=F(x_k)$5, and a feature map $x_{k+1}=F(x_k)$6. The least-squares Koopman matrix is

$x_{k+1}=F(x_k)$7

where

$x_{k+1}=F(x_k)$8

Its eigenvectors give approximations of Koopman eigenfunctions, and if the full-state observable lies in the span of the dictionary, corresponding Koopman modes can also be reconstructed. EDMD reduces to standard DMD when the dictionary is the set of coordinate projections; this makes DMD the minimal linear-dictionary case of the broader EDMD construction (Williams et al., 2014).

For the product-space GeKo, the finite-dimensional realization takes a Khatri–Rao form. With lifted state and input snapshots $x_{k+1}=F(x_k)$9, $\mathcal{K}$0, and $\mathcal{K}$1, define the feature matrix

$\mathcal{K}$2

A ridge-regularized one-step estimator is

$\mathcal{K}$3

but recent work replaces isolated one-step pairs by time-sequenced trajectory windows and solves a single multi-step ridge regression over horizontally stacked shifted blocks. This exposes the same operator $\mathcal{K}$4 to evolved states along trajectories and empirically reduces compounded prediction error in rollout. The same framework introduces a multi-step rollout diagnostic in feature space and decoded state space, $\mathcal{K}$5 and $\mathcal{K}$6, and uses the state-space error profile to choose the MPC horizon (Lazar, 1 Jun 2026).

Model reduction is essential because the lifted dimensions $\mathcal{K}$7, $\mathcal{K}$8, and especially $\mathcal{K}$9 can be large. One route is dictionary truncation and pseudoinverse regularization, as in EDMD. Another is a structured SVD reduction applied independently to state and input liftings, preserving the Khatri–Rao form. In the latter approach, one compresses $g$0 and $g$1 separately, refits the reduced operator in the compressed coordinates, and learns a linear decoder $g$2 to map reduced lifted states back to the physical state space. This yields a reduced-order GeKo model suitable for predictive control while retaining the tensor-product architecture (Lazar, 1 Jun 2026).

The earlier DMD/EDMD control literature already anticipated parts of this picture. DMD with control identifies $g$3 and $g$4 from

$g$5

while EDMD-based lifting replaces the raw state by a nonlinear observable vector and then fits a linear model in the lifted coordinates. These constructions were already described as finite-dimensional generalized Koopman operators acting on a chosen dictionary of observables, and they showed directly that the quality of the approximation depends on whether the dictionary approximates a Koopman-invariant subspace (Snyder et al., 2021).

4. Spectral, geometric, and representation-theoretic extensions

Generalization in Koopman theory is not limited to controlled tensor-product spaces. A second line of work extends the spectral and geometric side of the theory itself. For linear systems with nontrivial Jordan structure, the Koopman operator admits generalized eigenfunctions associated with Jordan chains, and for nonlinear systems near equilibria these generalized eigenfunctions can be transported through conjugacies. Invariant objects are then characterized as joint zero level sets of families of Koopman eigenfunctions: stable, unstable, and center subspaces in the linear case, and center, center-stable, and center-unstable manifolds in the nonlinear case. For globally attracting limit cycles and tori, the theory further introduces open eigenfunctions, isochrons, isostables, and function spaces such as Modulated Fock Spaces and Averaging Kernel Hilbert Spaces to obtain spectral expansions with lattice-type spectra (Mezic, 2017).

A broader representation-theoretic generalization replaces the scalar eigenproblem $g$6 by the vector-valued representation eigenproblem

$g$7

where $g$8 is a finite set of observables and $g$9 may be nonlinear. In that framework, a finite-dimensional representation of the dynamics is not necessarily linear, and the geometry of state space is encoded by the joint level sets of the components of $(\mathcal{K}g)(x)=g(F(x))$0. This formulation also extends to static maps between different spaces and connects naturally to learning with neural networks (Mezic, 2020). A related data-driven development pushes this idea further by learning nonlinear delay-lifted predictors of the form

$(\mathcal{K}g)(x)=g(F(x))$1

thereby relaxing the usual insistence that the finite-dimensional lifted evolution itself be linear. This suggests that, across the literature, “generalization” refers not only to enlarging observable spaces, but also to enlarging the admissible form of the finite-dimensional closure (Wilson, 2022).

At the spectral level, recent functional analysis has clarified an important deterministic–stochastic distinction. For deterministic discrete-time Koopman operators acting on $(\mathcal{K}g)(x)=g(F(x))$2, the full spectrum is closed under multiplication: if $(\mathcal{K}g)(x)=g(F(x))$3, then $(\mathcal{K}g)(x)=g(F(x))$4. But the proof must address the fact that $(\mathcal{K}g)(x)=g(F(x))$5 is not a Banach algebra and products of eigenfunctions need not remain in the domain. By contrast, the stochastic Koopman operator of a Markov process does not necessarily enjoy such a multiplicative lattice structure; finite-state Markov chains provide explicit counterexamples (Bramburger, 31 Jul 2025).

The fixed space of the Koopman operator also induces a canonical topological decomposition. For a compact Hausdorff topological dynamical system $(\mathcal{K}g)(x)=g(F(x))$6, the fixed space $(\mathcal{K}g)(x)=g(F(x))$7 defines a factor space whose fibers coincide with maximal level sets of invariant observables. A transfinite hierarchy of approximating orbits and superorbits recovers this decomposition and yields the finest partition into absolutely Lyapunov stable sets. In that setting, the system is topologically ergodic if and only if the fixed space is one-dimensional (Küster, 2019).

5. Structured, kernel, and physics-informed variants

Another major branch of GeKo research exploits prior structure rather than only data. For systems with time-scale separation and hierarchical control, the Koopman approximation is organized into block operators $(\mathcal{K}g)(x)=g(F(x))$8, and related blocks, reflecting slow–fast couplings and actuator-state hierarchies. In the singular limit $(\mathcal{K}g)(x)=g(F(x))$9, fast variables are replaced by equilibrium maps in lifted space, leading to reduced operators such as

$(\mathcal{K}^t g)(x)=g(\Phi^t(x))$0

This yields a structured ensemble of Koopman operators and derived reduced operators rather than a single monolithic matrix, and it enables stability analysis and optimal control at fast and slow time scales (Bakker, 18 Jun 2025).

For Euler–Lagrange systems, a different structural generalization is obtained by changing the state representation itself. Instead of the explicit state $(\mathcal{K}^t g)(x)=g(\Phi^t(x))$1, one uses the momentum-based state

$(\mathcal{K}^t g)(x)=g(\Phi^t(x))$2

which produces dynamics of the form

$(\mathcal{K}^t g)(x)=g(\Phi^t(x))$3

with a known, state-independent linear input channel. The resulting Koopman model is linear in lifted coordinates,

$(\mathcal{K}^t g)(x)=g(\Phi^t(x))$4

and only the passive dynamics need to be learned. The paper emphasizes that this reduces the number of learnable parameters by $(\mathcal{K}^t g)(x)=g(\Phi^t(x))$5 and $(\mathcal{K}^t g)(x)=g(\Phi^t(x))$6 compared to linear and bilinear Koopman formulations based on explicit state representations, respectively, and couples the model with a linear Generalized Extended State Observer for disturbance rejection (Singh et al., 21 Sep 2025).

A further generalization proceeds through the Koopman generator and reproducing-kernel methods. One recent method starts from the unbounded skew-adjoint Koopman generator $(\mathcal{K}^t g)(x)=g(\Phi^t(x))$7, applies a bounded functional transform $(\mathcal{K}^t g)(x)=g(\Phi^t(x))$8, smooths it by a Markov semigroup of kernel integral operators, and then reconstructs a family of skew-adjoint compact-resolvent operators $(\mathcal{K}^t g)(x)=g(\Phi^t(x))$9. These operators act on $\mathcal{L}g=f(x)\cdot \nabla g(x)$0 and, after transport, on an RKHS of observables; their eigendecomposition is obtained from a variational generalized eigenvalue problem, and the associated evolution operators preserve unitarity while converging spectrally to the original Koopman group in a suitable limit (Valva et al., 2024).

Kernel methods also appear in a more localized equilibrium-centered form. For a hyperbolic equilibrium, the principal Koopman eigenfunctions are decomposed into a linear part $\mathcal{L}g=f(x)\cdot \nabla g(x)$1, where $\mathcal{L}g=f(x)\cdot \nabla g(x)$2 is the left eigenvector of the Jacobian $\mathcal{L}g=f(x)\cdot \nabla g(x)$3, and a nonlinear remainder $\mathcal{L}g=f(x)\cdot \nabla g(x)$4 satisfying a first-order PDE,

$\mathcal{L}g=f(x)\cdot \nabla g(x)$5

with $\mathcal{L}g=f(x)\cdot \nabla g(x)$6 and $\mathcal{L}g=f(x)\cdot \nabla g(x)$7. The nonlinear component is then computed as the minimal-RKHS-norm solution under collocation constraints, yielding a kernel representation for $\mathcal{L}g=f(x)\cdot \nabla g(x)$8 and hence for the principal eigenfunction. Because the eigenvalues are anchored to the linearization, this construction avoids spectral pollution and spurious eigenvalues that can arise in operator-discretization methods (Lee et al., 2024).

6. Applications, limitations, and recurrent misconceptions

GeKo methods have been used in several control and prediction settings. In EDMD-based nonlinear control examples, an inverted pendulum and a cart-pole were lifted with polynomial and Fourier dictionaries, and linear control laws were designed on the lifted models. The pendulum case showed accurate reconstruction for both uncontrolled and controlled dynamics, whereas the cart-pole case exhibited large reconstruction errors in the uncontrolled regime and only partial improvement under control, underscoring the dependence on system complexity and dictionary choice (Snyder et al., 2021).

A more recent predictive-control realization was validated on the chaotic Lorenz system. With random Fourier feature liftings, structured SVD reduction, multi-step Khatri–Rao regression, and a learned decoder, the reduced GeKo model stabilized an unstable equilibrium from multiple initial conditions. The reported closed-loop result was convergence from 10 initial conditions in under $\mathcal{L}g=f(x)\cdot \nabla g(x)$9 s, with average final error $\mathcal{K}$0, under an MPC horizon chosen from rollout diagnostics (Lazar, 1 Jun 2026).

The product-Hilbert-space framework was also illustrated on the Van der Pol oscillator and compared with the benchmark Koopman with inputs and control formulation based on kernels over the stacked state–input vector. In that example, increasing the number of observables improved GeKo accuracy consistently, while the benchmark KIC method did not improve when the number of observables was increased from $\mathcal{K}$1 to $\mathcal{K}$2. The same work further tied GeKo to a nonlinear fundamental lemma, but also emphasized data richness requirements, injective liftings, and finite-dimensional rank conditions analogous to persistence of excitation (Lazar, 10 Aug 2025).

Several limitations recur across the literature. EDMD accuracy and usefulness hinge on the dictionary; if key eigenfunctions are not representable, one obtains missing eigenfunctions and spurious eigenfunctions of the projected operator rather than the true Koopman operator (Williams et al., 2014). In the multi-step Khatri–Rao framework, no formal convergence proofs are provided, online MPC remains nonconvex, and the reported solver was not real-time at $\mathcal{K}$3 ms sampling time (Lazar, 1 Jun 2026). The kernel PDE method for principal eigenfunctions currently treats real, simple eigenvalues near hyperbolic equilibria, leaving complex conjugate eigenpairs and broader attractor classes as extensions (Lee et al., 2024).

Two misconceptions are especially persistent. First, multiplicative spectral closure in deterministic Koopman theory does not imply that products of eigenfunctions are automatically valid observables in an $\mathcal{K}$4 setting, since domain issues remain central (Bramburger, 31 Jul 2025). Second, the deterministic lattice property of the Koopman spectrum does not generally extend to stochastic Koopman operators of Markov processes; finite-state counterexamples show that such spectra need not be closed under multiplication (Bramburger, 31 Jul 2025).

Taken together, these developments define GeKo as a technically diverse but conceptually coherent extension of Koopman theory. Its common core is the construction of linear operator descriptions of nonlinear dynamics on enlarged observable spaces—dictionary subspaces, product Hilbert spaces, RKHSs, structured block liftings, or regularized generator domains—together with finite-dimensional realizations that support prediction, control, and geometric analysis. The main unresolved questions concern the systematic design or learning of observable spaces, finite-data guarantees, stochastic extensions, and the treatment of continuous spectrum without forfeiting the operator-theoretic advantages that motivated Koopman methods in the first place.