  Matrix elements of observables in eigenstates of generic Hamiltonians are
described by the Srednicki ansatz within the eigenstate thermalization
hypothesis (ETH). We study a quantum chaotic spin-fermion model in a
one-dimensional lattice, which consists of a spin-1/2 XX chain coupled to a
single itinerant fermion. In our study, we focus on translationally invariant
observables including the charge and energy current, thereby also connecting
the ETH with transport properties. Considering observables with a
Hilbert-Schmidt norm of one, we first perform a comprehensive analysis of ETH
in the model taking into account latest developments. A particular emphasis is
on the analysis of the structure of the offdiagonal matrix elements $|\langle
\alpha | \hat O | \beta \rangle|^2$ in the limit of small eigenstate energy
differences $\omega = E_\beta - E_\alpha$. Removing the dominant exponential
suppression of $|\langle \alpha | \hat O | \beta \rangle|^2$, we find that: (i)
the current matrix elements exhibit a system-size dependence that is different
from other observables under investigation, (ii) matrix elements of several
other observables exhibit a Drude-like structure with a Lorentzian frequency
dependence. We then show how this information can be extracted from the
autocorrelation functions as well. Finally, our study is complemented by a
numerical analysis of the fluctuation-dissipation relation for eigenstates in
the bulk of the spectrum. We identify the regime of $\omega$ in which the
well-known fluctuation-dissipation relation is valid with high accuracy for
finite systems.
