Relatively bounded extensions of generator perturbations

The Miyadera perturbation theorem provides as a by-product that operators defined on a core for the generator of a C_o-semigroup and satisfying the Miyadera condition have a relatively bounded extension to the domain of the generator. We show that a weakening of the Miyadera condition characterizes relative boundedness with respect to the generator. We also investigate extensions of these results to Hille-Yosida operators. The various conditions we use in the abstract part are illustrated by several examples.