Multiplicative perturbations in semigroup theory with the (Z)-condition

In 4] the (Z)-condition was introduced by G. W. Desch and W. Schappacher. This condition on a Banach spaceZ and a generatorA inX ensures thatA(I+B) and(I+B)A generateC ₀-semigroups, ifB has its range inZ. In this paper we will consider how certain properties of the semigroup generated byA are inherited by the semigroups generated byA(I+B) and(I+B)A. We shall furthermore investigate a related condition, that simplifies certain sufficiency assumptions given previously.