The Koopman Decade: From Silent Foundations to Frontier Geometry and the VORT Framework

This work provides the first unified historical and structural synthesis of Koopman operator theory, tracing its evolution across four periods—Silent, Realization, Challenge, and Frontier—while showing how ideas originating with Koopman (1931) grew into a multi-disciplinary framework spanning dynamical systems, numerical simulation, machine learning, robotics, and autonomous decision-making. By combining a broad literature review with a conceptual analysis of what the community has achieved and where it has struggled, the monograph clarifies the core bottlenecks that persisted for decades: lifting consistency, spectral drift, reconstruction instability, loss of admissibility, and the absence of mechanisms for structural identity and correction. The central contribution is the Vector–Operator Reconstruction Theory (VORT), a six-stage mathematical architecture that unifies all Koopman and Koopman-type operator-learning approaches and explains both their successes and predictable failure modes. Through this lens, the Frontier period becomes coherent: the field now possesses not only powerful computational tools but a complete operator-theoretic structure capable of supporting modern high-dimensional, multi-physics, and safety-critical systems.