Koopman operator ($K_F$)

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.

The finite positive measure representing an observable's autocorrelation sequence by the Bochner–Herglotz theorem. For a probability measure-preserving system, the Koopman operator is an isometry, so its forward autocorrelations, extended by complex conjugation to negative indices, form a positive-definite function on $\mathbb{Z}$. The measure has mass $\Vert f{\Vert }_2^2$, and its atoms are squared norms of projections onto Koopman operator eigenspaces, including for noninvertible transformations.

An $L^2$ observable of an invertible probability measure-preserving system is almost periodic when its Koopman operator orbit is totally bounded in the global $L^2$ norm. A noninvertible system uses its forward orbit. This is the Hilbert-space dynamical definition, distinguished from uniform almost periodicity for continuous functions of a real variable.

If the forward orbit of $f$ under an isometry $U$ is totally bounded, choose a finite net whose centers are $U^{n_j}f$ and let $M={\max }_j{n}_j$. For every sufficiently large $n$, some center is within $\epsilon$ of $U^{n}f$, so cancellation gives $n-n_j\in R_{\epsilon }$. Thus $R_{\epsilon }$ is a syndetic set. Telescoping also gives $\Vert U^{kn}f-f{\Vert }_2\le k\Vert U^{n}f-f{\Vert }_2$.

A probability measure-preserving system is compact if every $L^2$ observable has a relatively compact Koopman operator orbit in the global $L^2$ norm. For an invertible system one may use all integer times; for a general transformation use nonnegative times. This is a measure-theoretic definition, stronger than compactness of the underlying topological space.

For a probability measure-preserving system, if $⟨U_T^{n}f,f⟩\to |\int fd\mu {|}^2$, then $U_T^{n}f$ converges weakly in $L^2$ to $(\int fd\mu )1$. Centre $v=f-(\int f)1$; its autocorrelations tend to zero. The isometry identity gives the same decay against each $U_T^{k}v$, hence by approximation against their closed forward span. Against its orthogonal complement the correlations are exactly zero. Orthogonal decomposition by a closed subspace finishes the proof without invertibility.

For a nonsingular transformation $F$ of a probability space, let $h=d(F_{\ast }\mu )/d\mu$ be its Radon-Nikodym derivative. The Koopman operator extends to a bounded operator on $L^2(\mu )$ exactly when $h\in L^{\infty }(\mu )$, and thenIndeed, change of variables gives $\Vert K_{F}g{\Vert }_2^2=\int |g{|}^{2}hd\mu \le \Vert h{\Vert }_{\infty }\Vert g{\Vert }_2^2$. Conversely, if the squared operator norm is at most $C$, test $g=1_A$ to obtain ${\int }_{A}hd\mu \le C\mu (A)$ for every measurable $A$. The sets $\{h>C+ϵ\}$ must therefore be null, proving $h\le C$ almost everywhere and equality of the optimal constants. Nonsingularity supplies a well-defined composition on equivalence classes, while the essential bound supplies the norm estimate.