Functional dimensionality of Koopman eigenfunction space

This work presents the general form solution of Koopman Partial Differential Equation for an autonomous system of N ordinary differential equations. We identify a domain in R^N for which any number in the complex plane is an eigenvalue of the Koopman operator, and all eigensolutions are obtained from N−1 functionally independent invariants of the system. Thus, we demonstrate that one may, in principle, diagonalize the system with only N functionally independent Koopman eigenfunctions.