ON THE CLOSED RANGE COMPOSITION AND WEIGHTED COMPOSITION OPERATORS

Let psi be an analytic function on D, the unit disc in the complex plane, and phi be an analytic self-map of D. Let B be a Banach space of functions analytic on D. The weighted composition operator W-phi,W-psi on B is defined as W(phi,psi)f = psi f circle phi, and the composition operator C-phi defined by C(phi)f= f circle phi for f is an element of B. Consider alpha > -1 and 1 <= p < infinity. In this paper, we prove that if phi is an element of H-infinity(D), then C-phi has closed range on any weighted Dirichlet space D-alpha if and only if phi(D) satisfies the reverse Carleson condition. Also, we investigate the closed rangeness of weighted composition operators on the weighted Bergman space A(alpha)(p).