Resolvent norm decay does not characterize norm continuity (Q1001415)

The author studies the norm continuity of \(C_0\)-semigroups in general Banach spaces. He gives an example so that the semigroup is contractive and the resolvent satisfies condition \(\lim_{s\to\infty} \|R(is,A)\|=0\), but the semigroup is not norm continuous, even not eventually norm continuous. In addition, the author shows that the Spectral Mapping Theorem does not hold. This result gives a negative answer for the Pazy question in Banach space.