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\begin{document}$$T_1,\dots , T_n$$\end{document} commuting power-bounded operators on a Banach space we study under which conditions the equality ker(T1-I)⋯(Tn-I)=ker(T1-I)+⋯+ker(Tn-I)\documentclass[12pt]{minimal}
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\begin{document}$$\ker (T_1-\mathrm {I})\cdots (T_n-\mathrm {I})=\ker (T_1-\mathrm {I})+\cdots +\ker (T_n-\mathrm {I})$$\end{document} 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 Tj=T(tj),tj>0,j=1,⋯,n\documentclass[12pt]{minimal}
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\begin{document}$$T_j=T(t_j), t_j>0, j=1,\dots , n$$\end{document} for some one-parameter semigroup (T(t))t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(T(t))_{t\ge 0}$$\end{document}. We also look at a generalization of the periodic decomposition problem when instead of the cyclic semigroups {Tjn:n∈N}\documentclass[12pt]{minimal}
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\begin{document}$$\{T_j^n:n \in \mathbb {N}\}$$\end{document} more general semigroups of bounded linear operators are considered.
