Koopman Resolvent Analysis

• Koopman resolvent is an operator-theoretic tool that provides a Laplace-domain representation linking the evolution of observables to spectral properties of nonlinear dynamics.
• It facilitates modal decomposition and frequency response analysis by identifying Koopman eigenvalues and associated modes, even in complex systems with chaotic behavior.
• Data-driven methods such as DMD and kernel-based approximations enable practical computation of the resolvent, ensuring robust spectral identification and convergence.

The Koopman resolvent is a central construct in modern operator-theoretic analysis of nonlinear deterministic and stochastic dynamical systems, providing a rigorous framework for Laplace-domain, spectral, and frequency-response representations well beyond the reach of classical linear system theory. It connects the dynamical evolution of observables under nonlinear flows to their integral, spectral, and computational properties, and underpins a range of rigorous and data-driven techniques for system identification, spectral analysis, and control.

1. Definition and Semigroup Foundations

Let $x \in \mathbb{R}^n$ evolve under an autonomous ODE $\dot x = F(x)$ with flow map $\varphi^t(x_0) = x(t)$. For a Banach (or Hilbert) space $F$ of observables $f: \mathbb{R}^n \to \mathbb{C}$, define the Koopman semigroup $(U^t)_{t\ge0}$ by $U^t f = f \circ \varphi^t$. Assuming strong continuity, the infinitesimal generator $L: D(L) \subset F \to F$ is

$L f := \lim_{t \to 0^+} \frac{U^t f - f}{t}$

and for smooth $f$, $(L f)(x) = \nabla f(x) \cdot F(x)$. The spectrum $\sigma(L) \subset \mathbb{C}$, with its complement $\rho(L)$ called the resolvent set, partitions the complex plane by invertibility of $sI - L$.

The Koopman resolvent is defined by

$R(s;L) := (sI - L)^{-1},\quad s \in \rho(L)$

and, whenever $\|U^t\| \leq M e^{\omega t}$ and $\mathrm{Re}\,s > \omega$, admits the Laplace representation (Susuki et al., 2020)

$R(s;L)f = \int_0^\infty e^{-s t} U^t f\, dt$

This formalism applies to both continuous and discrete time, deterministic and stochastic dynamics, including measure-preserving systems and Markov processes (Zhou et al., 10 Apr 2025).

2. Spectral Structure and Modal Decomposition

The spectral analysis of $R(s;L)$ reveals the operator's role as a Laplace-domain analog of the classical transfer function, generalizing modal and transient decompositions in nonlinear settings.

• Point Spectrum (Eigenvalues): Poles of $R(s;L)$ coincide with Koopman eigenvalues. The corresponding residues yield projections onto eigenspaces, allowing a representation such as (Susuki et al., 2020)

$$R(s;L) = \sum_{j} \frac{P_j}{s - \lambda_j} + \int_\mathrm{cont.}\frac{dE(\omega)}{s - i\omega}$$

for generators with both point and continuous spectrum.

• Modal Expansion: For an observable $f$ and initial state $x_0$, the Laplace transform of $y(t) = f(\varphi^t(x_0))$ is $Y(s;x_0) = [R(s;L) f](x_0)$. This decomposes as a sum of Koopman eigenfunctions times corresponding modes, with continuous spectral integrals in the presence of mixing or chaos.
• Examples:
  • On compact attractors with unitary Koopman group, the generator is skew-adjoint and the spectrum is decomposed into countable points plus continuous branch cuts (Susuki et al., 2020).
  • Near hyperbolic equilibria and limit cycles, the modal structure of $R(s;L)$ encodes nonlinear transients through repeated poles and Floquet-type expansions.
  • In stochastic systems, the generator includes diffusion; its resolvent structure is associated with the Feynman–Kac formula and elliptic operator theory (Hamzi et al., 1 Mar 2026, Zhou et al., 10 Apr 2025).

In all cases, the region of convergence aligns with classical causal Laplace transforms, e.g. $\mathrm{Re}\,s > 0$ for stable flows (Susuki et al., 2020).

3. Computation and Data-Driven Approximations

Direct computation of the Koopman resolvent is infeasible due to its infinite-dimensional nature. Recent advances provide several complementary strategies:

• Pseudo-Resolvent and DMD Frameworks: Finite approximations, such as Extended Dynamic Mode Decomposition (EDMD), may suffer spectral pollution. Data-driven "pseudo-resolvent" matrices $R_N(z) = (zI - \Psi_X^\dagger \Psi_Y)^{-1}$ constructed via Sherman–Morrison–Woodbury formulas (Xu et al., 31 Dec 2025) enable robust spectral analysis, with the resolvent norm used as a spectral indicator. Analytical guarantees, including Hausdorff convergence and algebraic multiplicity preservation for isolated eigenvalues, are established.
• Residual DMD and Rigged DMD: Residual DMD (ResDMD) and Rigged DMD algorithms use resolvent-based residuals, Galerkin projections, and kernel-smoothing to extract both discrete and continuous spectral components from snapshot data, with explicit high-order convergence and error bounds (Colbrook et al., 2021, Colbrook et al., 2024).
• Resolvent-Type Generator Learning: The Yosida approximation uses resolvent identities to approximate generators without numerical differentiation, enabling high-accuracy, low-sampling-rate estimation in both deterministic (Meng et al., 2024) and stochastic (Zhou et al., 10 Apr 2025) systems:

$$L_\lambda = \lambda^2 R(\lambda;L) - \lambda I \to L\quad \text{as}~\lambda \to \infty$$

• Kernel and Path-Integral Methods: For stochastic SDEs, kernel methods relate Koopman resolvent Green's functions, Feynman–Kac path integrals, and variational RKHS kernels, revealing the effects of diffusion and providing collocation-based computational frameworks (Hamzi et al., 1 Mar 2026).
• Operator-Theoretic Algorithms: Graph-based or analytic approaches for directly computing powers of resolvent-like operators for SDE generators achieve efficient, accurate evaluation of Koopman matrix entries (Ohkubo, 2021).

These computational advances allow extraction of modal decompositions, spectral densities, and system parameters robustly from data, often with rigorous convergence guarantees.

4. Frequency Response and Transfer Function Generalizations

The resolvent framework generalizes frequency response analysis to nonlinear and forced systems. For an input $u(t) = u_0 e^{i\omega t}$, augmenting the flow to a skew-product, the Laplace transform of the output is (Susuki et al., 6 Mar 2026):

$$\widehat{y}(s;x_0,u_0) = [R(s;L_{\text{forced}}) g](x_0,u_0)$$

Periodic or resonant steady-state responses correspond to poles at $s = i n\omega$; the associated residues yield generalized Bode plots for nonlinear plants. Sufficient conditions for existence of simple poles and explicit formulae are provided for LTI, globally stable analytic nonlinear, and ergodic systems (Susuki et al., 6 Mar 2026). The transfer function expression $c^T (i\omega I - A)^{-1}b$ for LTI models is a special case.

5. Theoretical and Algorithmic Impact Across Disciplines

The resolvent organizes spectral and dynamical information for ergodic, equilibrating, periodic, quasiperiodic, and mixing regimes, applicable to a wide range of models:

• Ergodic Theory and Attractor Physics: By providing explicit spectral decompositions combining unitary theory on attractors with analytic expansions off-attractor, the resolvent yields global Laplace-domain representations for autonomous flows, capturing both stationary and transient dynamics (Susuki et al., 2020).
• Fluid Mechanics and Navier–Stokes Analysis: In mean-linearized flows (e.g., wall turbulence), the resolvent bridges Koopman modes (of state) and resolvent modes (of harmonic forcing), providing the mapping between nonlinear forcing and state fluctuations (Sharma et al., 2016). Singular vectors of the resolvent are optimal approximations to physical mode structures.
• Symbolic and Analog Computation: The resolvent of the Koopman operator abstracts reachability and halting for computational systems, connecting computational complexity with spectral properties. For symbolic systems, resolvent poles correspond to cycles, invariant sets, or halting basins (Caravelli et al., 7 Oct 2025).
• Stochastic and High-Dimensional Systems: In SDEs, the resolvent connects via the Feynman–Kac formula to stochastic transition densities and Green's functions, offering robust kernel-based and Monte Carlo estimation strategies (Hamzi et al., 1 Mar 2026).

6. Open Problems and Connections

While the resolvent theory is fundamentally operator-theoretic, practical estimation in high dimension, continuous spectrum, or under noise remains a subject of ongoing research. Questions include the design of function/dictionary spaces for numerical stability, robustness to measurement noise, and extensions to non-autonomous, control, or networked systems. Theoretical questions persist regarding non-self-adjoint spectral theory in the presence of dissipation, optimal basis selection for dynamics with discrete and continuous spectrum, and connections to computational universality (Caravelli et al., 7 Oct 2025).

7. Summary Table: Key Properties of the Koopman Resolvent

Domain	Resolvent Expression	Modal Structure
Deterministic ODE	$\int_0^\infty e^{-s t} U^t f\,dt$	Poles: Koopman eigenvalues; residues: modes
Ergodic/Unitary	Series $\sum_j P_j/(s-i\omega_j)$, integral over continuous	Branch cuts: mixing/chaos
Stochastic SDE	$$\mathbb{E}\left[\int_0^\infty e^{-\lambda t}f(X_t)dt\right]$$	Corresponds to Green's function/kernel
Forced Nonlinear (Freq. Resp.)	$\left[R(s;L_{\text{forced}}) g\right](x_0,u_0)$	Residue at $s = i n\omega$ = frequency response

In all these settings, the Koopman resolvent $(sI - L)^{-1}$ or its data-driven approximation organizes the spectral, transient, and steady-state structure of nonlinear dynamical systems, providing a unified framework for analysis, computation, and system identification across deterministic, stochastic, and computational domains (Susuki et al., 2020, Xu et al., 31 Dec 2025, Colbrook et al., 2021, Meng et al., 2024, Susuki et al., 6 Mar 2026, Caravelli et al., 7 Oct 2025, Zhou et al., 10 Apr 2025, Hamzi et al., 1 Mar 2026).