Abstract: Let S(t), t ≥ 0, be a strongly continuous semigroup on a Banach space, with infinitesimal generator A. The main results are as follows:Let either S(t) or S^* (t) be uniformly stable as t → + ∞, yet not uniformly exponentially stable as t →+ ∞. Then for any compact operator B on X, the semigroup S_B(t) generated by A+B cannot be uniformly exponentially stable as t→ + ∞. This result improves a result in[2] and[3], removing the assumption that S(t) is a contraction and X is a Hilbert space.