Geodesic
On the Structure of a Semigroup of Operators with Finite-Dimensional Ranges
Matematičeskie zametki, Tome 91 (2012) no. 2, pp. 240-252

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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.
Keywords: operator semigroup, strong continuity, complex Banach space, Banach algebra, spectrum of an operator, bounded linear operator. 
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     author = {A. V. Pechkurov},
     title = {On the {Structure} of a {Semigroup} of {Operators} with {Finite-Dimensional} {Ranges}},
     journal = {Matemati\v{c}eskie zametki},
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     url = {http://geodesic.mathdoc.fr/item/MZM_2012_91_2_a6/}
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A. V. Pechkurov. On the Structure of a Semigroup of Operators with Finite-Dimensional Ranges. Matematičeskie zametki, Tome 91 (2012) no. 2, pp. 240-252. http://geodesic.mathdoc.fr/item/MZM_2012_91_2_a6/