Koopman operator - Knowlpedia

The forward Koopman operator of a probability-preserving transformation $T$ is

U_T:L^2(X,\mu)\longrightarrow L^2(X,\mu),\qquad U_Tf=f\circ T.

It is a linear isometry. If $T$ is invertible modulo null sets, it is unitary, with $U_T^{-1}f=f\circ T^{-1}$ .

Verification

Measure preservation gives $\|U_Tf\|_2^2=\int|f|^2\circ T\,d\mu=\|f\|_2^2$ . It also sends equality almost everywhere to equality almost everywhere, so the formula is independent of representatives. Composition respects linear combinations of functions without requiring any linear structure on $X$ .

Direction of time

This convention gives $U_T^nf=f\circ T^n$ , as in time averages. For the left action $n\mapsto T^n$ , the Koopman representation instead has $\kappa(n)=U_T^{-n}$ . Under noncommuting transformations, forward pullback reverses composition order.

Noninvertible example

For the doubling map, $U_T e_k=e_{2k}$ . Odd Fourier frequencies are absent from its range. Thus it is an isometry that is not onto, and its nonnegative powers do not extend to a unitary representation of $\mathbb Z$ .

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §2.2.