Weighted composition operators and‎ ‎dynamical systems on weighted Spaces of holomorphic functions on‎ ‎Banach spaces

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 Fréchet 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.