= Koopman operator
{c}
{title2=$K_F$}
{wiki}
For a measurable map $F:X\to X$, the <Koopman operator> acts on observables by composition:
$$
K_Fg=g\circ F.
$$
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 $\|K_Fg\|_2=\|g\|_2$, 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 $F(x)=2x\pmod1$, 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 $\alpha$, $K_Fe_k=e^{2\pi ik\alpha}e_k$. Thus observable <eigenfunctions> encode exact oscillatory correlations, a useful bridge between nonlinear <dynamical systems> and <spectral theorem>.
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