Iterated Aluthge transforms of some composition operators on weighted Bergman spaces

Abstract

In this paper, we compute the iterated Aluthge transforms ${\widetilde{C_{\ddot{I}}•}}^{(n)}$ of the composition operator $C_{\ddot{I}}•$ on the weighted Bergman spaces ${\mathcal{A}}_{\hat{I}}{\pm }^2(\mathbb{D})$, where $\ddot{I}•(z)=az+(1-a)$ for $0<a<1$. Also, we obtain the norm and numerical radius of ${\widetilde{C_{\ddot{I}}•}}^{(n)}$ on ${\mathcal{A}}_{\hat{I}}{\pm }^2(\mathbb{D})$. We establish that ${\widetilde{C_{\ddot{I}}•}}^{(n)}$ converges in the strong operator topology on ${\mathcal{A}}_{\hat{I}}{\pm }^2(\mathbb{D})$. The purpose of this paper is to examine the results of \cite{jung2015iterated} for the weighted Bergman spaces ${\mathcal{A}}_{\hat{I}}{\pm }^2(\mathbb{D})$. Additionally, by using the iterated Aluthge transforms of $C_{\ddot{I}}{•}^{\ast }$ on ${\mathcal{A}}_{\hat{I}}{\pm }^2(\mathbb{D})$, we derive the iterated Aluthge transforms of $C_{\ddot{I}}ƒ$, where $\ddot{I}ƒ(z)=\frac{az}{-(1-a)z+1}$ for $0<a<1$, on some weighted Hardy space $H^2({\hat{I}}_{\hat{I}}^2\pm )$ and study its convergence. Finally, we raise some questions that emerge from these findings.