arXiv:2405.07432v4 Announce Type: replace Abstract: The Koopman operator provides a powerful framework for representing the dynamics of general nonlinear dynamical systems. However, existing data-driven approaches to learning the Koopman operator rely on batch data. In this work, we present a sparse online learning algorithm that learns the Koopman operator iteratively via stochastic approximation, with explicit control over model complexity and provable convergence guarantees. Specifically, we study the Koopman operator via its action on the reproducing kernel Hilbert space (RKHS), and address the mis-specified scenario where the dynamics may escape the chosen RKHS. In this mis-specified setting, we relate the Koopman operator to the conditional mean embeddings (CME) operator. We further establish both asymptotic and finite-time convergence guarantees for our learning algorithm in mis-specified setting, with trajectory-based sampling where the data arrive sequentially over time. Numerical experiments demonstrate the algorithm's capability to learn unknown nonlinear dynamics.