  The eigenstate thermalization hypothesis provides to date the most successful
description of thermalization in isolated quantum systems by conjecturing
statistical properties of matrix elements of typical operators in the
(quasi-)energy eigenbasis. Here we study the distribution of matrix elements
for a class of operators in dual-unitary quantum circuits in dependence of the
frequency associated with the corresponding eigenstates. We provide an exact
asymptotic expression for the spectral function, i.e., the second moment of
this frequency resolved distribution. The latter is obtained from the decay of
dynamical correlations between local operators which can be computed exactly
from the elementary building blocks of the dual-unitary circuits. Comparing the
asymptotic expression with results obtained by exact diagonalization we find
excellent agreement. Small fluctuations at finite system size are explicitly
related to dynamical correlations at intermediate times and the deviations from
their asymptotical dynamics. Moreover, we confirm the expected Gaussian
distribution of the matrix elements by computing higher moments numerically.
