Compactness of composition operators acting on weighted Bergman–Orlicz spaces
Volume 103 / 2012
Annales Polonici Mathematici 103 (2012), 1-13
MSC: Primary 47B33, 46E10; Secondary 30D55.
DOI: 10.4064/ap103-1-1
Abstract
We characterize compact composition operators acting on weighted Bergman–Orlicz spaces \[ \mathcal{A}^{\psi}_\alpha = \left\{f \in H(\mathbb D) : \int_{\mathbb D} \psi(| f(z)|)\,d A_\alpha(z) < \infty\right \}, \] where $\alpha > -1$ and $\psi$ is a strictly increasing, subadditive convex function defined on $[0 , \infty)$ and satisfying $\psi(0) = 0,$ the growth condition $\lim_{t \rightarrow \infty}\displaystyle \psi(t)/t = \infty $ and the $\Delta_2$-condition. In fact, we prove that $C_{\varphi}$ is compact on $\mathcal{A}^{\psi}_\alpha$ if and only if it is compact on the weighted Bergman space $\mathcal{A}^{2}_{\alpha}.$
