Iterated Aluthge transforms of some composition operators on weighted Bergman spaces

TL;DR

This paper analyzes the iterated Aluthge transforms of specific composition operators on weighted Bergman spaces, computes their norms and radii, and studies their convergence properties.

Contribution

It extends the study of iterated Aluthge transforms to weighted Bergman spaces and derives convergence results for these transforms on such operators.

Findings

Iterated Aluthge transforms converge in the strong operator topology.

Explicit formulas for norms and numerical radii of the transforms.

Derived transforms for related composition operators on weighted Hardy spaces.

Abstract

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.