Compactness of composition operators acting on weighted Bergman–Orlicz spaces

Ajay K. Sharma¹ ; S. Ueki²

¹ School of Mathematics Shri Mata Vaishno Devi University Kakryal Katra-182320, J&K, India
² Faculty of Engineering Ibaraki University Hitachi 316-8511, Japan

Annales Polonici Mathematici, Tome 103 (2012) no. 1, pp. 1-13

Résumé

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 $\mathcal{A}^{2}_{\alpha}.$

Détail

DOI : 10.4064/ap103-1-1

Keywords: characterize compact composition operators acting weighted bergman orlicz spaces mathcal psi alpha mathbb int mathbb psi alpha infty right where alpha psi strictly increasing subadditive convex function defined infty satisfying psi growth condition lim rightarrow infty displaystyle psi infty delta condition prove varphi compact mathcal psi alpha only compact weighted bergman space mathcal alpha