Abstract
Let $B_{X}$ and $B_{Y}$ be the open unit balls of the Banach Spaces $X$ and $% Y$, respectively. Let $V$ and $W$ be two countable families of weights on $% B_{X}$ and $B_{Y}$, respectively. Let $HV\left( B_{X}\right) \left( \text{or }HV_{0}\left( B_{X}\right) \right) $ and $HW\left( B_{Y}\right) $ $\left( \text{or }HW_{0}\left( B_{Y}\right) \right) $ be the weighted Frechet spaces of holomorphic functions. In this paper, we investigate the holomorphic mappings $\phi :B_{X}\rightarrow B_{Y}$ and $\psi :B_{X}\rightarrow \mathbb{C}$ which characterize continuous weighted composition operators between the spaces $HV\left( B_{X}\right) \left( \text{or }HV_{0}\left( B_{X}\right) \right) $ and $HW\left( B_{Y}\right) $ $\left( \text{or }HW_{0}\left( B_{Y}\right) \right) .$ Also, we obtained a (linear) dynamical system induced by multiplication operators on these weighted spaces.
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