Invariance and Approximation of Koopman Operators in Native Spaces

This paper derives a new condition to ensure that a selected reproducing kernel Hilbert space is invariant under the action of the Koopman operator associated with a generally nonlinear system in discrete time. If the function that determines the dynamics is continuous over an invariant set supporting the dynamics, and the reproducing kernel that defines the space of observables is uniformly bounded above and below by positive constants over the invariant set, then the native space that contains the observables is invariant under the Koopman operator. This condition is used to derive error bounds for approximations of the Koopman operator in the strong operator topology. These bounds are explicit in the number of snapshots and dimension of the subspace used to construct estimates. Numerical examples illustrate the qualitative behavior of the error bounds.