Abstract
Let T(t); T B (t) denote the C 0-semigroups generated by linear operators A, A + B respectively on a Banach space X. We show, in this article, that if (i) T(t 0)−T B (t 0) is compact for some t 0 > 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.
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Communicated by J. A. Goldstein
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Guo, BZ. On the exponential stability of C 0-semigroups on Banach spaces with compact perturbations. Semigroup Forum 59, 190–196 (1999). https://doi.org/10.1007/s002339900043
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DOI: https://doi.org/10.1007/s002339900043