Spectral Conditions for Stability of One-Parameter Semigroups

Let {S(t) : t 0} be aC0-semigroup on a Banach spaceYwith generatorBand {T(t) : t 0} be a boundedC0-semigroup on a Banach spaceXwith generatorA. Suppose thatσ(B)∩iRis countable,Pσ(A*)∩iRis empty and that there is a bounded linear operatorC : Y→Xwith dense range which intertwines the two semigroups. Then T(t) x X→0 ast→∞, for eachxinX. This generalises results of W. Arendt and the author, Yu. I. Lyubich and V Quôc Phóng, and Falun Huang