  The Koopman operator has become an essential tool for data-driven analysis,
prediction and control of complex systems. The main reason is the enormous
potential of identifying linear function space representations of nonlinear
dynamics from measurements. This equally applies to ordinary, stochastic, and
partial differential equations (PDEs). Until now, with a few exceptions only,
the PDE case is mostly treated rather superficially, and the specific structure
of the underlying dynamics is largely ignored. In this paper, we show that
symmetries in the system dynamics can be carried over to the Koopman operator,
which allows us to massively increase the model efficacy. Moreover, the
situation where we only have access to partial observations (i.e.,
measurements, as is very common for experimental data) has not been treated to
its full extent, either. Moreover, we address the highly-relevant case where we
cannot measure the full state, such that alternative approaches such as delay
coordinates have to be considered. We derive rigorous statements on the
required number of observables in this situation, based on embedding theory. We
present numerical evidence using various numerical examples including the wave
equation and the Kuramoto-Sivashinsky equation.
