Koopman operator

Summary

The Koopman operator of a measure-preserving transformation $T$ acts on functions by $f \mapsto f \circ T$. It is the composition operator with symbol $T$ on the space $L^2$ of the invariant measure. There it is an isometry, and a unitary operator when $T$ is invertible. It turns a nonlinear dynamical system into a linear operator on a Hilbert space. Ergodicity, weak mixing and mixing all become properties of this operator, which is the starting point of the spectral theory of dynamical systems.