A unifying framework for estimation of the Koopman operator between Reproducing Kernel Hilbert Spaces

Dynamical systems, and estimations thereof, play an important role in various disciplines. The Koopman operator encapsulates properties of a dynamical system. This thesis investigates the estimation of the Koopman operator within the context of Reproducing Kernel Hilbert Spaces (RKHSs). We begin by reviewing the relevant background on RKHSs, including the vector-valued case, and Koopman operators in their natural setting of continuous functions. A general framework that links Koopman theory and discrete-time dynamical systems is provided. We then compare two methods for estimating Koopman operators in a unified framework, namely ridge regression in spaces of Hilbert-Schmidt operators on a RKHS and kernel Extended Dynamic Mode Decomposition. The boundedness of the actual Koopman operator between RKHSs is investigated and illustrated through examples. Dynamics for which the Koopman operator is bounded between Gaussian RKHSs on Rd are characterized.