The main purpose of this note is to prove Miyadera's theorem on perturbations of semigroup generators (Miyadera [4]; see Remark 2, c) below) under reduced assumptions. In our proof we obtain the perturbed semigroup by an iteration similar to the iteration used in the proof for bounded perturbations. By an example we show that the theorem is optimal with respect to a constant occurring in the theorem. In the followingX will be a Banach space. For a linear operator TinX we denote by D(T) its domain of definition and by R(T) its range. ~(X) denotes the bounded linear operators X ~X. A s.c. (strongly continuous) semigroup on X is a family (W(t);t>O) in ~(X) satisfying W(0) = I, W(t t + t2)= W(t,)W(t2) (tl, t2 =>0), [0, oo)~t~-~W(t)x is continuous (x6X). A s.c. semigroup satisfies an estimate If W(t)H < Le ~ (t ~ 0). The linear operator T defined by D(T)" = {xeX;Tx:= lim t-'(W(t)-I)x exists} t~O+