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The Miyadera perturbation theorem provides as a by-product that operators defined on a core for the generator of a C 0 -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 bydoi:10.1216/rmj-2009-39-3-947 fatcat:fhv3kbueczc55p7hedhyrm7xlm