Closed Range Weighted Composition Operators on H2 and Aα2

In this paper, first we investigate bounded below weighted composition operators C ψ , φ on a Hilbert space of analytic functions. Then for ψ ∈ H ∞ and a univalent map φ , we characterize all closed range weighted composition operators C ψ , φ on H 2 and A α 2 . Also we show that for ψ ∈ H ∞ which is bounded away from zero near the unit circle, the weighted composition operator C ψ , φ is bounded below on H 2 or A α 2 if and only if C φ has closed range. Moreover, we investigate invertible operators C ψ 1 , φ 1 C ψ 2 , φ 2 ∗ and C ψ 1 , φ 1 ∗ C ψ 2 , φ 2 on H 2 and A α 2 .