On the boundedness of the differentiation operator between weighted spaces of holomorphic functions
We give necessary and sufficient conditions on the weights $v$ and $w$ such that the differentiation operator $D: Hv( \Omega) \rightarrow Hw( \Omega)$ between two weighted spaces of holomorphic functions is bounded and onto. Here $ \Omega = \mathbb C$ or $ \Omega = \mathbb D$. In particular we characterize all weights $v$ such that $ D:Hv( \Omega) \rightarrow Hw( \Omega)$ is bounded and onto where $w(r) = v(r)(1-r)$ if $ \Omega = \mathbb D$ and $w=v$ if $ \Omega = \mathbb C$. This leads to a new description of normal weights.