  We study the behaviour of sequences $U_2^n X U_1^{-n}$, where $U_1, U_2$ are
unitary operators, whose spectral measures are singular with respect to the
Lebesgue measure, and the commutator $XU_1-U_2X$ is small in a sense. The
conjecture about the weak averaged convergence of the difference $U_2^n X
U_1^{-n}-U_2^{-n} X U_1^n$ to the zero operator is discussed and its connection
with complex-symmetric operators is established in a general situation. For a
model case where $U_1=U_2$ is the unitary operator of multiplication by $z$ on
$L^2(\mu)$, sufficient conditions for the convergence as in the Conjecture are
given in terms of kernels of integral operators.
