Let $\mathcal{A}$ be the infinitesimal generator of a strongly continuous semigroup on a Banach space $\mathcal{E}$. Two classes of bounded operators $\mathcal{P}$ on $\mathcal{E}$ are introduced for which the operators $\mathcal{A}\mathcal{P}$ and $\mathcal{P}\mathcal{A}$ also generate semigroups on $\mathcal{E}$. It is shown that many important operators can be decomposed into the form $\mathcal{A}\mathcal{P} + \mathcal{R}$ or $\mathcal{P}\mathcal{A} + \mathcal{R}$ where $\mathcal{A}$ is a generator of a simpler structure and $\mathcal{R}$ is a bounded operator. Applications of decompositions of this type to infinite dimensional system theory are discussed.