Given $$T_1,\dots , T_n$$ T 1 , ⋯ , T n commuting power-bounded operators on a Banach space we study under which conditions the equality $$\ker (T_1-\mathrm {I})\cdots (T_n-\mathrm {I})=\ker (T_1-\mathrm {I})+\cdots +\ker (T_n-\mathrm {I})$$ ker ( T 1 - I ) ⋯ ( T n - I ) = ker ( T 1 - I ) + ⋯ + ker ( T n - 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 $$T_j=T(t_j), t_j>0, j=1,\dots , n$$ T j = T ( t j ) , t j > 0 , j = 1 , ⋯ , n for some one-parameter semigroup $$(T(t))_{t\ge 0}$$ ( T ( t ) ) t ≥ 0 . We also look at a generalization of the periodic decomposition problem when instead of the cyclic semigroups $$\{T_j^n:n \in \mathbb {N}\}$$ { T j n : n ∈ N } more general semigroups of bounded linear operators are considered.