Volterra type operators on Bergman spaces with exponential weights
type: Item
instance of: scholarly article
title: Volterra type operators on Bergman spaces with exponential weights
zbMATH Open document ID: 1262.30065
author: José Ángel Peláez
author: Jordi Pau
published in: Contemporary Mathematics
publication date: 2012-09-05T00:00:00Z
full work available at URL: http://hdl.handle.net/2445/96730
Mathematics Subject Classification ID: 30H20
Mathematics Subject Classification ID: 30H30
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 6077516
zbMATH Keywords: integration operators on Bergman spaces with exponential weights
MaRDI profile type: Publication
OpenAlex ID: W2282375482
DOI: 10.1090/CONM/561/11117
review text: For a holomorphic function \(g\) on the disc \(\mathbb{D}\) the authors consider the Volterra-type operator acting on holomorphic functions: NEWLINE\[NEWLINE J_g(f)(z)=\int_{0}^z f(\xi) g'(\xi) d\xi. NEWLINE\]NEWLINE A weight function \(w\) is a positive function on \(0\leq r<1\) that is integrable on \((0,1)\). For \(z\in\mathbb{D}\) set \(w(z)=w(|z|)\). For \(0<p<\infty\), the weighted Bergman space \(A^p(w)\) is the space of holomorphic functions such that NEWLINE\[NEWLINE \int_{\mathbb{D}} |f(z)|^p w(z) dA(z)<\infty. NEWLINE\]NEWLINE For the weights NEWLINE\[NEWLINE w_{\gamma,\alpha}(r)=(1-r)^{\gamma}\exp\left(-\frac{c}{(1-r)^{\alpha}}\right) NEWLINE\]NEWLINE with \(\gamma\geq 0\), \(\alpha>0\) and \(c>0\), the authors consider necessary and sufficient conditions on the symbol \(g\) so that \(J_g\) is bounded and compact. In particular, they show that \(J_g:A^{p}(w_{\gamma,\alpha})\to A^{p}(w_{\gamma,\alpha})\) is bounded if and only if NEWLINE\[NEWLINE \sup_{z\in\mathbb{D}}(1-|z|)^{1+\alpha}|g'(z)|<\infty. NEWLINE\]NEWLINE Similar results for compactness and Schatten ideal membership are also provided.NEWLINENEWLINEFor the entire collection see [Zbl 1232.30005].
reviewed by: Brett Wick
description: scientific article; zbMATH DE number 6077516