On Decomposition of Generators

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