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Koopman operators and transfer operators represent dynamical systems throughtheir induced linear action on vector spaces of observables, enabling the useof operator-theoretic techniques to analyze nonlinear dynamics in state space.The extraction of approximate Koopman or transfer operator eigenfunctions (andthe associated eigenvalues) from an unknown system is nontrivial, particularlyif the system has mixed or continuous spectrum. In this paper, we describe aspectrally accurate approach to approximate the Koopman operator on $L^2$ formeasure-preserving, continuous-time systems via a ``compactification'' of theresolvent of the generator. This approach employs kernel integral operators toapproximate the skew-adjoint Koopman generator by a family of skew-adjointoperators with compact resolvent, whose spectral measures converge in asuitable asymptotic limit, and whose eigenfunctions are approximately periodic.Moreover, we develop a data-driven formulation of our approach, utilizing datasampled on dynamical trajectories and associated dictionaries of kerneleigenfunctions for operator approximation. The data-driven scheme is shown toconverge in the limit of large training data under natural assumptions on thedynamical system and observation modality. We explore applications of thistechnique to dynamical systems on tori with pure point spectra and the Lorenz63 system as an example with mixing dynamics.

Consistent spectral approximation of Koopman operators using resolvent compactification