  In this paper, we develop the Koopman operator theory for dynamical systems
with symmetry. In particular, we investigate how the Koopman operator and
eigenfunctions behave under the action of the symmetry group of the underlying
dynamical system. Further, exploring the underlying symmetry, we propose an
algorithm to construct a global Koopman operator from local Koopman operators.
In particular, we show, by exploiting the symmetry, data from all the invariant
sets are not required for constructing the global Koopman operator; that is,
local knowledge of the system is enough to infer the global dynamics.
