Koopman operator ($K_F$)

For a measurable dynamical map $F$, the Koopman operator acts on observables by $K_{F}g=g\circ F$. If $F$ is nonsingular, it defines an operator on equivalence classes of measurable functions.

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 map $F$ on a probability space, $K_F$ extends boundedly to $L^2$ exactly when $d(F_{\ast }\mu )/d\mu \in L^{\infty }$. In that case