Decoherence entails exponential forgetting in systems complying with the eigenstate thermalization hypothesis

According to the eigenstate thermalization ansatz, matrices representing generic few-body observables take on a specific form when displayed in the eigenbasis of a chaotic Hamiltonian. We examine the effect of environment-induced decoherence on the dynamics of observables that conform with said eigenstate thermalization ansatz. The obtained result refers to a description of the dynamics in terms of an integro-differential equation of motion of the Nakajima-Zwanzig form. We find that environmental decoherence is equivalent to an exponential damping of the respective memory kernel. This statement is formulated as a rigorous theorem. Furthermore, the implications of the theorem on the stability of exponential dynamics against decoherence and the transition towards Zeno freezing are discussed.