  Given $T_1,\dots, T_n$ commuting power-bounded operators on a Banach space we
study under which conditions the equality $\ker (T_1-\mathrm{I})\cdots
(T_n-\mathrm{I})=\ker(T_1-\mathrm{I})+\cdots +\ker (T_n-\mathrm{I})$ holds
true. This problem, known as the periodic decomposition problem, goes back to
I. Z. Ruzsa. In this short note we consider the case when $T_j=T(t_j)$,
$t_j>0$, $j=1,\dots, n$ for some one-parameter semigroup $(T(t))_{t\geq 0}$. We
also look at a generalization of the periodic decomposition problem when
instead of the cyclic semigroups $\{T_j^n:n \in \mathbb{N}\}$ more general
semigroups of bounded linear operators are considered.
