Iterated Aluthge transforms of some composition operators on weighted Bergman spaces

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