We present a data-driven framework for spectral analysis of measure-preserving, ergodic dynamical systems utilizing techniques from reproducing kernel Hilbert space (RKHS) theory. A central element of this approach is to regularize the unbounded, skew-adjoint generator of the unitary Koopman group of the system by composing it with integral operators mapping into RKHSs of appropriate regularity. This results in a one-parameter family of skew-adjoint, compact operators on RKHS, with purely atomic spectral measures that converge to the spectral measure of the generator. In addition, the spectral measures of the regularized generators identify coherent observables under the dynamics through corresponding eigenfunctions, and have an associated Borel functional calculus, allowing one to approximate functions of the Koopman generator. In particular, exponentiating the regularized generators leads to approximations of the unitary Koopman group, which can be used to perform forecasting of observables. The RKHS structure also allows stable, data-driven formulations of this framework that converge under fairly mild assumptions on the system and observation modality. We illustrate this approach with applications to dynamical systems with both atomic and continuous spectra.
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