  The eigenstate thermalization hypothesis (ETH) and the theory of linear
response (LRT) are celebrated cornerstones of our understanding of the physics
of many-body quantum systems out of equilibrium. While the ETH provides a
generic mechanism of thermalization for states arbitrarily far from
equilibrium, LRT extends the successful concepts of statistical mechanics to
situations close to equilibrium. In our work, we connect these cornerstones to
shed light on the route to equilibrium for a class of properly prepared states.
We unveil that, if the off-diagonal part of the ETH applies, then the
relaxation process can become independent of whether or not a state is close to
equilibrium. Moreover, in this case, the dynamics is generated by a single
correlation function, i.e., the relaxation function in the context of LRT. Our
analytical arguments are illustrated by numerical results for idealized models
of random-matrix type and more realistic models of interacting spins on a
lattice. Remarkably, our arguments also apply to integrable quantum systems
where the diagonal part of the ETH may break down.
