Boundedness from below of Volterra type integral operators on the weighted Bloch spaces, the weighted Dirichlet spaces and H ∞

We characterize the Volterra type integral operators Tg with closed range on H∞, Bα, Bα,0, Bω, Bω,0, Dα and Dω. And we completely solve the cases left open there and extend all the results. Let 0<α<1, α≤β. If Tg : Bα →Bβ is bounded, then Tg is not bounded below from Bα to Bβ. Let −1<α<1, α≤β. If Tg : Dα →Dβ is bounded, then Tg is not bounded below from Dα to Dβ. If Tg : Bω,0 →Bω,0 is bounded, then Tg is not bounded below from Bω,0 to Bω,0. If Tg : H∞ →Bω,0 is bounded, then Tg is not bounded below from H∞ to Bω,0.