On the closed range composition and weighted composition operators
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Let $\psi$ be an analytic function on $\D$, the unit disc in the complex plane, and $\varphi$ be an analytic self-map of $\D $. Let $\mathcal{B}$ be a Banach space of functions analytic on $\D$. The weighted composition operator $\W $ on $\mathcal{B}$ is defined as $\W f=\psi f\circ \varphi$, and the composition operator $\C$ defined by $\C f=f\circ \varphi$ for $f\in \mathcal{B}$. Consider $\alpha >-1$ and $1\leq p<\infty$. In this paper, we prove that if $\varphi\in H^\infty(\D)$, then $\C$ has closed range on any weighted Dirichlet space $\d$ if and only if $\varphi(\D)$ satisfies the reverse Carleson condition. Also, we investigate the closed rangeness of weighted composition operators on the weighted Bergman space $\A$.
Abstract
Let $\psi$ be an analytic function on $\D$, the unit disc in the complex plane, and $\varphi$ be an analytic self-map of $\D $. Let $\mathcal{B}$ be a Banach space of functions analytic on $\D$. The weighted composition operator $\W $ on $\mathcal{B}$ is defined as $\W f=\psi f\circ \varphi$, and the composition operator $\C$ defined by $\C f=f\circ \varphi$ for $f\in \mathcal{B}$. Consider $\alpha >-1$ and $1\leq p<\infty$. In this paper, we prove that if $\varphi\in H^\infty(\D)$, then $\C$ has closed range on any weighted Dirichlet space $\d$ if and only if $\varphi(\D)$ satisfies the reverse Carleson condition. Also, we investigate the closed rangeness of weighted composition operators on the weighted Bergman space $\A$.
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