Koopman/Perron–Frobenius Operators in RKHS

• Koopman/Perron–Frobenius in RKHS is a framework that embeds transfer operators in a reproducing kernel Hilbert space to enable spectral analysis and forecasting of complex dynamical systems.
• Finite-data approximations using Gram matrices and regularized inversions yield explicit operator estimates with convergence guarantees and rigorous error control.
• The approach integrates methods like kernel-EDMD, operator-valued kernels, and functional calculus, providing actionable insights for stability analysis and control in both deterministic and stochastic settings.

A reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions equipped with a positive-definite kernel, enabling pointwise evaluation as a bounded linear operation. Embedding transfer operators—particularly the Koopman and Perron–Frobenius (PF) operators—into RKHS structures has led to powerful techniques for spectral analysis, forecasting, and nonparametric modeling of complex dynamical systems. Within the RKHS setting, these operators admit explicit finite-data approximations, convergence guarantees, and facilitate rigorous error control for practical algorithms.

1. RKHS Foundations and Transfer Operator Construction

Let $X$ denote the state space of a stochastic or deterministic system, and $k : X \times X \to \mathbb{R}$ a strictly positive-definite kernel. The associated RKHS $\mathcal{H}_k$ comes with a canonical feature map $\phi(x) = k(x, \cdot)$, characterized by $$k(x,x') = \langle \phi(x), \phi(x') \rangle_{\mathcal{H}}$$ (Mollenhauer et al., 2018).

In the context of time-homogeneous Markov or deterministic processes, the following operators are defined:

• Covariance operator $C_{XX} : \mathcal{H} \to \mathcal{H}$: $$C_{XX}f = \mathbb{E}_{x}[\langle \phi(x), f \rangle_{\mathcal{H}} \phi(x)]$$,
• Cross-covariance operator $C_{YX} : \mathcal{H} \to \mathcal{H}$: $$C_{YX}f = \mathbb{E}_{(x_t, x_{t+1})}[\langle \phi(x_t), f \rangle_{\mathcal{H}} \phi(x_{t+1})]$$.

The Koopman operator on $\mathcal{H}$ is given by $K_\mathcal{H} = C_{XX}^{-1} C_{YX}$ (or regularized $(C_{XX} + \varepsilon I)^{-1} C_{YX}$), yielding optimal RKHS-valued approximations to conditional expectation mappings. The Perron–Frobenius operator on $\mathcal{H}$, often described as $P_\mathcal{H} = C_{XX}^{-1} C_{XY}$ with $C_{XY} = C_{YX}^*$, pushes embedded densities forward.

Empirical finite-rank estimators use Gram matrices $G_{XX} = \Phi^T \Phi$, $G_{YX} = \Psi^T \Phi$ for data pairs $(x_i, y_i)$, yielding matrix versions:

• $$\widehat{K}_\mathcal{H} = (G_{XX} + n\varepsilon I)^{-1} G_{YX}$$,
• $$\widehat{P}_\mathcal{H} = (G_{XX} + n\varepsilon I)^{-1} G_{XY}$$ (Mollenhauer et al., 2018, Zanini et al., 2021).

2. Spectral Approximation and Data-Driven Algorithms

Spectral analysis relies on solving the generalized eigenvalue problem

$$(G_{XX} / n + \varepsilon I)^{-1} (G_{YX} / n) w = \lambda w$$

and reconstructing eigenfunctions $v = \Phi w$ in $\mathcal{H}$. Singular value decomposition in the non-reversible case uses the block matrix

$$T = \begin{bmatrix} 0 & G_{YX}/n \ G_{XY}/n & 0 \end{bmatrix}.$$

Top eigenvector segments yield right and left singular vectors, facilitating SVD-based operator approximations (Mollenhauer et al., 2018).

Related kernel-EDMD methods operate as Galerkin projections, with analogous matrix formulations and regularization for invertibility, providing strong consistency as sample size grows and regularization decays (Zanini et al., 2021, Ikeda et al., 2022, Boullé et al., 18 Jun 2025). Asymptotic convergence and error bounds are now available for sparse online learning via conditional mean embeddings and stochastic approximation (Hou et al., 27 Jan 2025), as well as for Koopman Kernel Regression with operator-valued kernels and convex least-squares formulations (Bevanda et al., 2023, Withanachchi, 3 Jul 2025).

Algorithmic skeletons involve:

• Gram matrix assembly from feature maps,
• Operator estimation via least-squares or stochastic approximation,
• Spectral decomposition (EVP/SVD),
• Extraction and normalization of RKHS eigen/singular functions for forecasting and system identification.

3. Rigorous Functional Analytic Theory

On a theoretical level, Koopman and PF operators in RKHS are adjoint under the Hilbert structure. If $K$ is Koopman ($Kf = f \circ F$) and $P$ is PF ($P^* = K$), then for all $g,h \in \mathcal{H}$, $\langle K g, h \rangle = \langle g, P h \rangle$ (Ikeda et al., 2022). Adjointness supports identification of eigenfunctions (Koopman: $K\varphi = \lambda \varphi$; PF: $P\psi = \mu \psi$) and Rayleigh quotient-based spectral principles (e.g., $\lambda$ maximizes $|\langle K\varphi, \varphi \rangle|$).

Compactness of covariance operators (from stationarity and kernel regularity) guarantees discrete spectra modulo accumulation at zero, ensuring well-behaved and numerically accessible eigen-decompositions (Mollenhauer et al., 2018, Giannakis et al., 2018). Invariant kernels are constructed to enforce operator invariance (e.g., Koopman-invariant direct sum kernels) (Bevanda et al., 2023).

4. Extensions: Operator-Valued, Banach, and C*-module RKHSs

The RKHS formalism generalizes further:

• Operator-valued RKHSs permit prediction of vector fields and spatio-temporal systems, using block-structured kernels with Sobolev regularity, extending representer theorems and ensuring spectral accuracy for kernel-approximated Koopman operators (Withanachchi, 3 Jul 2025).
• Reproducing kernel Hilbert C*-modules (RKHM) encode matrix- or algebra-valued features and inner products, facilitating joint spectral analysis and orthonormalization, with PF operator and Gram–Schmidt routines inherited from module theory (Hashimoto et al., 2020, Hashimoto et al., 2023). RKHM approaches are critical when covariance or dynamic coupling is algebraically nontrivial (e.g., multivariate or networked dynamics).
• Banach space generalization extends the above to non-Hilbert settings, preserving adjoint and symmetry properties relevant in RKBS (Ikeda et al., 2022).

5. Advanced Spectral Theory and Computational Optimality

Recent works have formalized the convergence rates, complexity-theoretic optimality, and solvability indices for RKHS-based Koopman/PF approximation (Boullé et al., 18 Jun 2025). The Solvability Complexity Index (SCI) hierarchy precisely quantifies how many nested data or numerical limits are necessary for reliable spectrum/pseudospectrum computation, with explicit impossibility results for randomized oracles and adversarial dynamical systems.

For finite-data algorithms:

• Koopman and PF operators are approximated via Galerkin compression using kernel-dictionaries,
• Pseudospectrum approximations utilize residual minimization over spectral grids,
• Spectral measures—needed for forecasting and dynamic mode extraction—are computed via Stone's formula or rational approximations, with explicit error control.

6. Functional Calculus, Invariant Subspaces, and Rigged Hilbert Spaces

Compactification of unitary evolution groups on RKHS is used to define compact, skew-adjoint generators (e.g., $W_\tau$) whose spectra converge in the strong resolvent sense to the original generator as regularization vanishes (Giannakis et al., 2018). These generators yield discrete projections and enable Borel functional calculus, out-of-sample eigenfunction evaluation, and pointwise data-driven forecasting.

Rigged Hilbert spaces (Gelfand triples) allow analytic continuation of Koopman eigenfunctions and precise characterization of weak or generalized eigenmodes, supporting JetDMD and spectral decompositions beyond the original RKHS (Ishikawa et al., 2024).

7. Connections to Stability, Invariance, and Control

When the underlying RKHS is designed with invariant structures (e.g., product of linear and radial Wendland kernels), close connections emerge between spectral properties of Koopman operators and system stability:

• Asymptotic stability of equilibrium points corresponds to the confinement of Koopman spectrum within the unit disk,
• Bifurcations induce spectrum escape,
• Probabilistic certificates are available for data-driven stability analysis (Tang et al., 8 Nov 2025).
• For linear SDEs, Gaussian RKHSs remain invariant under Koopman evolution provided a Lyapunov-like matrix inequality holds, with explicit norm bounds and inclusion characterizations (Philipp et al., 2024).

8. Summary Table: RKHS Koopman/Perron-Frobenius Operator Frameworks

Approach	Operator Definition	Algorithmic/Spectral Features
Classical RKHS	$K f(x) = f(F(x))$, $P = K^*$	Gram-based EVP/SVD, regularization
Conditional Mean Emb.	$K = C_{YX} C_{XX}^{-1}$	Sparse online, stochastic approx.
Operator-valued RKHS	$K: f \mapsto f(\Phi_t(x), t)$	Block kernels, Sobolev norm, spectral conv.
RKHM (C*-module)	$K: \phi(x) \mapsto \phi(f(x))$, $\langle \cdot, \cdot \rangle$ in $A$	Orthonormal Gram-Schmidt, block QR, matrix eigenproblems
Compactified RKHS	$W_\tau = P_\tau V P_\tau^*$ Eq. (above)	Discrete spectrum, Borel functional calculus, out-of-sample extension

All frameworks exploit the kernel structure to construct feature maps, compute covariance/cross-covariance, extract spectral information, and generate forecasts or control certificates directly from sampled trajectories, with convergence and error control governed by functional analysis and RKHS theory.

9. Active Research Directions and Open Problems

Challenges and extensions in this domain include:

• Rigorous characterization of necessary and sufficient operator invariance conditions (e.g., Lyapunov-inequality tightness for Gaussian RKHS (Philipp et al., 2024)),
• Data-driven learning of invariant and high-regularity operator-valued kernels for complex field dynamics,
• Integration with deep learning and hybrid architectures for automated feature discovery (Hashimoto et al., 2023, Withanachchi, 3 Jul 2025),
• Extension to stochastic and infinite-dimensional systems with diffusive generators,
• Physical constraint enforcement via kernel adaptation (e.g., incompressibility in fluid mechanics).

This body of work furnishes a robust, general, and rigorously founded toolkit for the spectral analysis, learning, and control of nonlinear dynamical systems in reproducing kernel Hilbert space settings.