  We consider smoothing equations of the form $$X ~\stackrel{\mathrm{law}}{=}~
\sum_{j \geq 1} T_j X_j + C$$ where $(C,T_1,T_2,\ldots)$ is a given sequence of
random variables and $X_1,X_2,\ldots$ are independent copies of $X$ and
independent of the sequence $(C,T_1,T_2,\ldots)$. The focus is on complex
smoothing equations, i.e., the case where the random variables $X,
C,T_1,T_2,\ldots$ are complex-valued, but also more general multivariate
smoothing equations are considered, in which the $T_j$ are similarity matrices.
Under mild assumptions on $(C,T_1,T_2,\ldots)$, we describe the laws of all
random variables $X$ solving the above smoothing equation. These are the
distributions of randomly shifted and stopped L\'evy processes satisfying a
certain invariance property called $(U,\alpha)$-stability, which is related to
operator (semi)stability. The results are applied to various examples from
applied probability and statistical physics.
