I apologize in advance as the question and title might be silly.
I started studying the Koopman operator framework and I, from what I understood, it describes a nonlinear system (I'll make the discrete example) $$x^+=F(x), \quad x\in\mathbb{R}^n,$$ in a linear manner, but in an infinite-dimensional space $\mathscr{H}$ of the observables $g:\mathbb{R}^n\rightarrow\mathbb{R}$ such as: $$[Ug](x)=g \circ F(x),$$ with the Koopman operator $U:\mathscr{H}\rightarrow\mathscr{H}$.
From what I got until now, in order to "lift" the nonlinear dynamics to the observable space we need the eigenfunctions $\psi(x)$ of $U$ such that: $$[U\psi](x)=\lambda\psi(x).$$ What I do not get is that in order to be able to describe the nonlinear system dynamics I need its states, so how do I return from the observable space to the state space?