On the Exponential Stability of C0-Semigroups on Banach Spaces with Compact Perturbations

B (t) denote the C₀-semigroups generated by linear operators A,A+B respectively on a Banach space X. We show, in this article, that if (i) T(t₀)-T_B (t₀) is compact for some t₀>0, (ii) T(t) is asymptotically stable, (iii) T_B(t) is exponentially stable, then T(t) is also exponentially stable. This generalizes a result on compact perturbations proved by Triggiani Proc. AMS.,105(1989),375-383] on Hilbert spaces. Comunicated by