Closed-Range Composition Operators on Weighted Bergman Spaces

Authors
• 1 University of Arkansas, Department of Mathematics, Fayetteville, AR, 72701, USA , Fayetteville (United States)
Type
Published Article
Journal
Integral Equations and Operator Theory
Publisher
Birkhäuser-Verlag
Publication Date
Oct 13, 2011
Volume
72
Issue
1
Pages
103–114
Identifiers
DOI: 10.1007/s00020-011-1912-1
Source
Springer Nature
Keywords
For analytic self-maps φ of the unit disk, we show that recently given conditions that are both necessary and sufficient for the composition operator Cφ to be closed-range on the classical Bergman space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{A}^2}$$\end{document} are actually necessary and sufficient in the general setting of the weighted Bergman spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{A}^p_{\alpha}}$$\end{document} , for 1 ≤ p < ∞ and weights of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(1 - |z|^2)^{\alpha}}$$\end{document} ; where α > −1. Along the way, we show how work done in a paper of J. Akeroyd, P. Ghatage and M. Tjani can be used to improve upon the most definitive of these conditions, and we give applications.