Explore Koopman operator theory for dynamical systems in this 28-minute video lecture. Delve into the history and recent resurgence of interest in Koopman operators due to increased data availability and advanced regression algorithms. Learn about open problems, key challenges, and emerging techniques in the field. Examine dynamical systems through the lens of Koopman operators, including examples of Koopman linear embedding and cases without easy closure. Understand how Koopman eigenfunctions define invariant subspaces and discover the Dynamic Mode Decomposition (DMD) technique. Gain valuable insights from Steve Brunton's expertise in this comprehensive overview of Koopman spectral analysis.