ON THE CLOSED RANGE COMPOSITION AND WEIGHTED COMPOSITION OPERATORS

Let ψ be an analytic function on , the unit disc in the complex plane, and φ be an analytic self-map of . Let be a Banach space of functions analytic on . The weighted composition operator Wφ,ψ on is defined as Wφ,ψf = ψf ◦ φ, and the composition operator Cφ defined by Cφf = f ◦ φ for f ∈ . Consider α > -1 and 1 ≤ p < ∞. In this paper, we prove that if φ ∈ H∞( ), then Cφ has closed range on any weighted Dirichlet space α if and only if φ( ) satisfies the reverse Carleson condition. Also, we investigate the closed rangeness of weighted composition operators on the weighted Bergman space Apα.