Let ϕ, (ϕ+ε) be closely analytic functions defined on D, such that ϕ(D)⊆D. The operator given by f⟼(ϕ+ε)(f∘ϕ) is called a weighted composition operator. We show the boundedness, compactness, weak compactness, and complete continuity of weighted composition operators on Hardy spaces H_(1+ε) (ε≥0). We show that such an operator is compact on H1 if and only if it is weakly compact on this space. This result depends on a technique proved by Contreras and Hernandez-Diaz which passes the weak compactness from an operator T to operators dominated in norm by T.

Keywords: weighted composition operators; Hardy spaces; compact operators; weakly compact operators; completely continuous operators.