Koopman Operator

The Koopman operator linearises a dynamical system by acting on observables instead of states. Fix a state space and a dynamics indexed by a totally ordered time monoid , acting by with . The two instances are the continuous case and the discrete case . On observables it acts by composition,

so is a representation of , a functor from time. For continuous it is a strongly continuous semigroup with generator (); for discrete it is the powers of a single operator. is linear even when is nonlinear, so it trades finite-dimensional nonlinearity for infinite-dimensional linearity.

Two structures make it useful:

• Spectrum. Eigenobservables satisfying , the Koopman modes, evolve by a scalar. Expanding a field in them turns the flow into independent decays and oscillations, the linear normal form of a nonlinear system.
• Duality. acts contravariantly on observables, the predicates, whereas its adjoint acts covariantly on densities, the states. That adjoint is the transfer operator, also called the Perron–Frobenius operator. This is exactly state-predicate-duality, with Koopman as the predicate-transformer and the transfer operator as the state-transformer. The state-side coalgebra picture is polynomial dynamics, and Koopman is its observable-side dual. Pairing this temporal operator with a spatial Laplacian gives the spacetime construction.

Remark. An algebra homomorphism

Observables form an algebra under pointwise product, and , so each is an algebra homomorphism, not merely a linear map. Koopman is a monoid map landing among algebra endomorphisms.

Example. Dynamic mode decomposition

DMD is the data-driven finite-rank approximation. From snapshot pairs it fits the best linear operator advancing a chosen dictionary of observables, a truncated Koopman. It recovers the operator when the dynamics is known only through data.

References

• Koopman, “Hamiltonian Systems and Transformation in Hilbert Space,” PNAS 17 (1931)
• Mezic, “Spectral Properties of Dynamical Systems, Model Reduction and Decompositions,” Nonlinear Dynamics (2005)
• Schmid, “Dynamic Mode Decomposition of Numerical and Experimental Data,” Journal of Fluid Mechanics (2010)