Before revision:  This work investigates persistence of excitation conditions for discrete-time nonlinear systems that can be represented using Koopman operators. The Koopman operator presents a linear lifted representation of the evolution of a nonlinear system over a space of observable functions. We use this framework to derive spectral conditions for persistence of excitation, relating classical notions of persistence of excitation to the spectral properties of initial conditions, seen through the lens of Koopman observable functions .
After revision:  This work investigates persistence of excitation conditions exist. We use the input-Koopman operator method to model nonlinear systems and derive identifiability conditions for open-loop systems initialized from a single initial condition. We show that nonlinear identifiability is intrinsically tied to the rank of a given dataset's power spectral density, transformed through the lifted Koopman observable space .