A constructive proof of Gibson's stability theorem (Q1942789)

It was proved in [\textit{J. S. Gibson}, SIAM J. Control Optimization 18, 311--316 (1980; Zbl 0434.93044)] that, if the infinitesimal generator of an exponentially stable \(C_0\)-semigroup is additively perturbed by a compact operator, then the resulting semigroup is again exponentially stable, provided that it is strongly stable. The present paper gives another proof of this result. The authors' proof is constructive, provides a natural extension from a Hilbert to a Banach setting, and weakens slightly the compactness perturbation assumption. Applications are also discussed.