On the Structure of a Semigroup of Operators with Finite-Dimensional Ranges

In the present paper, we describe the structure of a strongly continuous operator semigroup $T(t)$ (where $T\colon \mathbb{R}_+ \to \operatorname{End}X$ and $X$ is a complex Banach space) for which $\operatorname{Im}{T(t)}$ is a finite-dimensional space for all $t>0$. It is proved that such a semigroup is always the direct sum of a zero semigroup and a semigroup acting in a finite-dimensional space. As examples of applications, we discuss differential equations containing linear relations, orbits of a special form, and the possibility of embedding an operator in a $C_0$-semigroup.