On the multiplication operators and multipliers on weighted spaces of holomorphic functions on the upper halfplane

In this paper, we find necessary and sufficient conditions such that under these conditions a self map multiplication operator between weighted spaces of holomorphic functions is Fredholm or closed range operator. We also obtain a characterization of multipliers for certian type of weights operators between weighted spaces of holomorphic functions on the upper halfplane. Our results will remain valid for any simply connected domain in the complex plane instead of the upper halfplane .