A note on the periodic decomposition problem for semigroups

Abstract

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 $$(T(t))_{t\ge 0}$$ . 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.