The linear (often infinite-dimensional) operator U that advances observables of a dynamical system by composition with the flow or map: (U g)(x)=g(f(x)) for a discrete map x↦f(x) or (U^t g)=g∘Φ^t for continuous flow Φ^t; it encodes nonlinear state evolution as linear action on functions of state.

Principle

Pushforward of observables: instead of evolving points in state space, one considers how scalar-valued observables are transported; the Koopman operator is linear because composition with the deterministic map is linear on the vector space of functions.

Demonstration

For a circle rotation θ_{n+1}=θ_n+ω mod 2π, the Koopman operator acts on Fourier observables e^{ikθ} by multiplication e^{ikω}; its eigenfunctions are the Fourier modes and eigenvalues are phase factors e^{ikω}, revealing persistent frequencies.

Misapplication

Assuming a finite-dimensional truncated approximation (basis projection) of the Koopman operator will capture all dynamics without verifying closure or convergence, leading to model error and spurious modes.

Consequence

Provides a framework to apply linear spectral theory to nonlinear systems: eigenfunctions and eigenvalues of U identify coherent structures, invariant sets, and oscillatory components; motivates data-driven algorithms like Dynamic Mode Decomposition.

Reversal

The Perron–Frobenius (transfer) operator is the dual object acting on densities (measures) by pullback; it emphasizes ensemble evolution rather than observables and yields complementary spectral information.

Boundary

Defined on chosen spaces of observables (e.g., L^2, continuous functions); properties and spectral decomposition depend on that choice and invariant measures — finite-dimensional linear representations are typically approximations and may fail for generic nonlinear systems.

Semantic Tension

Compare with linearization in state space (Jacobian-based): Koopman is a global linear representation in function space acting on observables, whereas Jacobian linearization is a local linear approximation of trajectories in state coordinates.

Synthesis

The Koopman operator is the linear pushforward on observables induced by the dynamics, turning nonlinear state evolution into linear operator action whose spectral features expose dynamical patterns.