Isometric and Invertible Composition Operators on Weighted Bergman Spaces of Dirichlet Series

We show that a composition operator on weighted Bergman spaces of Dirichlet series Aμp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}_{\mu }^p$$\end{document} is invertible if and only if it is Fredholm if and only if it is an isometry.