Definition:Koopman Operator on Complex L-2 Space

Definition

Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.

Let $\map {L^2_\C} \mu$ be the complex-valued $L^2$ space of $\mu$.

The Koopman operator $U_T : \map {L^2_\C} \mu \to \map {L^2_\C} \mu$ of $T$ is defined by:

$\forall f \in \map {L^2_\C} \mu  : \map {U_T} f = f \circ T$

where $f \circ T$ denotes the composition of $T$ and $f$.

Source of Name

This entry was named for Bernard Osgood Koopman.

Sources

• 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $2.4$: Associated Unitary Operators