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.