Koopman operators, representations, and spectrum

Core idea

Translate dynamics into linear operators. Follow fixed vectors to ergodicity, then eigenfunctions, spectral measures, mixing, and continuous-time generators.

Prerequisite doorway

• Hilbert space
• Unitary operator
• Strongly continuous unitary representation
• Stone's theorem for one-parameter unitary groups
• Matrix coefficient of a unitary representation
• Følner condition

Reading path

• Koopman operator
• Koopman representation
• Reduced Koopman representation
• Fixed-function criterion for ergodicity
• Correlation of dynamical observables
• Spectral theorem for a unitary operator
• Spectral measure of an observable
• Koopman eigenfunction
• Weak mixing
• Strong mixing
• Spectral characterization of weak mixing
• Discrete spectrum of a dynamical system
• Halmos–von Neumann theorem
• Kronecker factor
• Infinitesimal generator of a Koopman flow
• Transfer operator of a probability-preserving map
• Invariant vector of a unitary representation
• Almost-invariant vectors
• Spectral gap of a probability-preserving action
• Kazhdan property (T)
• Howe–Moore theorem
• Koopman representation of a nonsingular action

Return to Ergodic theory.

Differentiating representations

Continue to smooth and distribution vectors for common generator domains, the smooth-vector Fréchet topology, nuclearity, and the Heisenberg example.