n Hilbert space H and perturbation are studied in this paper. By resolvent of infinitesimal generator, the sufficient and necessary conditions for ventually norm continuous semigroups are given. Using the result obtained, it is proved that if A is infinitesimal generator of an eventually norm continuous semigroup T (t), then there is a subspace ΞA of L(H) such that, for any B ∈ ΞA, the semigroup S(t) generated by A + B preserves the property of T (t).