Several authors have considered multiplicative perturbations of semigroups in the past and the present ([1, 2, 3, 5, 4] et al.). Some strong results are available where the perturbing operator B, which in general is bounded, commutes with the semigroup [2, 5]. They make use of functional calculi but need the extended notion of integrated semigroups. Other results deal with the case where B is of a special form. The (Z)-condition is particularly fruitful in that case, see [2]. However, about the intricate relationship between the perturbing operator and the semigroup to be perturbed far less is known than in the additive case.
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