Composition operators between growth spaces‎ ‎on circular and strictly convex domains in complex Banach spaces‎

Document Type : Research Articles


1 Aligudarz Branch, Islamic Azad University

2 Khoy Faculty of Engineering, Urmia University, Urmia, Iran


‎Let $\Omega_X$ be a bounded‎, ‎circular and strictly convex domain in a complex Banach space $X$‎,
‎and $\mathcal{H}(\Omega_X)$ be the space of all holomorphic functions from $\Omega_X$ to $\mathbb{C}$‎.
‎The growth space $\mathcal{A}^\nu(\Omega_X)$ consists of all $f\in\mathcal{H}(\Omega_X)$‎
‎such that $$|f(x)|\leqslant C \nu(r_{\Omega_X}(x)),\quad x\in \Omega_X,$$‎
‎for some constant $C>0$‎, ‎whenever $r_{\Omega_X}$ is the Minkowski‎
‎functional on $\Omega_X$ and $\nu‎ :‎[0,1)\rightarrow(0,\infty)$‎
‎is a nondecreasing‎, ‎continuous and unbounded function‎.
‎For complex Banach spaces $X$ and $Y$‎
‎and a holomorphic map $\varphi:\Omega_X\rightarrow\Omega_Y$‎, ‎put‎
‎$C_\varphi( f)=f\circ \varphi,f\in\mathcal{H}(\Omega_Y)$‎.
‎We characterize those $\varphi$ for which the composition operator‎
‎$ C_\varphi:\mathcal{A}^{\omega}(\Omega_Y)\rightarrow\mathcal{A}^{\nu}(\Omega_X)$ is a bounded or compact operator‎.