## Some properties of weighted composition operators on algebras of analytic functions.(English)Zbl 1006.47027

Summary: The objects of study in this paper are the weighted composition operators on the disc algebra and the Hardy space $$H^\infty(\mathbb{D})$$. For those operators, we prove the equivalence of the compactness, the weak compactness and the complete continuity. Moreover, we give the necessary and sufficient condition for those operators to have closed range or to be Fredholm operators.

### MSC:

 47B33 Linear composition operators 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47A53 (Semi-) Fredholm operators; index theories 47B07 Linear operators defined by compactness properties