Koopman operators are infinite-dimensional operators that globally linearize
nonlinear dynamical systems, making their spectral information useful for
understanding dynamics. However, Koopman operators can have continuous spectra
and infinite-dimensional invariant subspaces, making computing their spectral
information a considerable challenge. This paper describes data-driven
algorithms with rigorous convergence guarantees for computing spectral
information of Koopman operators from trajectory data. We introduce residual
dynamic mode decomposition (ResDMD), which provides the first scheme for
computing the spectra and pseudospectra of general Koopman operators from
snapshot data without spectral pollution. Using the resolvent operator and
ResDMD, we also compute smoothed approximations of spectral measures associated
with measure-preserving dynamical systems. We prove explicit convergence
theorems for our algorithms, which can achieve high-order convergence even for
chaotic systems, when computing the density of the continuous spectrum and the
discrete spectrum. We demonstrate our algorithms on the tent map, Gauss
iterated map, nonlinear pendulum, double pendulum, Lorenz system, and an
11-dimensional extended Lorenz system. Finally, we provide kernelized
variants of our algorithms for dynamical systems with a high-dimensional
state-space. This allows us to compute the spectral measure associated with the
dynamics of a protein molecule that has a 20,046-dimensional state-space, and
compute nonlinear Koopman modes with error bounds for turbulent flow past
aerofoils with Reynolds number >105 that has a 295,122-dimensional
state-space
