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A note on the periodic decomposition problem for semigroups

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.