 Koopman operator

For a measurable map $F:X\to X$, the Koopman operator acts on observables by composition:It is linear in $g$ even if the state map $F$ is nonlinear. To act on measurable functions modulo equality almost everywhere, require $F$ to be a nonsingular transformation; otherwise changing $g$ on a null set could change $g\circ F$ on a positive-measure set. Boundedness on a chosen Lp space is an additional question, treated by the bounded Koopman operator criterion for $L^2$.

For a measure-preserving transformation, change of variables gives $\Vert K_{F}g{\Vert }_2=\Vert g{\Vert }_2$, so the Koopman operator is an isometry. If $F$ has a measurable measure-preserving inverse modulo null sets, it is a unitary operator. The converse need not follow from isometry alone: for $F(x)=2x(\mathrm{mod}1)$, the Fourier observable $e_k(x)=e^{2\pi ikx}$ maps to $e_{2k}$, so the range misses odd Fourier modes and the operator is not onto.

For an irrational rotation of the circle by $\alpha$, $K_F{e}_k=e^{2\pi ik\alpha }{e}_k$. Thus observable eigenfunctions encode exact oscillatory correlations, a useful bridge between nonlinear dynamical systems and spectral theorem.