  We consider Koopman operator theory in the context of nonlinear
infinitedimensional systems, where the operator is defined over a space of
nonlinear functionals. The properties of the Koopman semigroup are described
and a finite-dimensional projection of the semigroup is proposed, which
provides a linear finite-dimensional approximation of the underlying
infinite-dimensional dynamics. This approximation is used to obtain spectral
properties from data, a method which can be seen as a generalization of
Extended Dynamic Mode Decomposition for infinite-dimensional systems. Finally,
we exploit the proposed framework to identify (a finite-dimensional
approximation of) the Lie generator associated with the Koopman semigroup. This
approach yields a linear method for nonlinear PDE identification, which is
complemented with theoretical convergence results.
