Resolvent norm decay does not characterize norm continuity

We show that there exists a reflexive Banach space (chi, parallel to.parallel to) and a strongly continuous semigroup (T(t))(t >= 0) with generator (A, D(A)) on (chi, parallel to.parallel to) such that lim(mu is an element of R,) (vertical bar mu vertical bar -> infinity) parallel to R(i mu, A)parallel to = 0 but (T(t))(t >= 0) is not eventually norm continuous. This answers a question of Amnon Pazy in the negative.