Volterra composition operators on logarithmic Bloch spaces
type: Item
instance of: scholarly article
title: Volterra composition operators on logarithmic Bloch spaces
zbMATH Open document ID: 1163.47022
author: Xiangling Zhu
published in: Banach Journal of Mathematical Analysis
publication date: 2008-12-09T00:00:00Z
full work available at URL: https://eudml.org/doc/55543
full work available at URL: v3n1.htm
review text: Let \(\mathbb B\) be the unit ball in \(\mathbb C^n\) and \(H(\mathbb B)\) the set of all holomorphic functions on \(\mathbb B\). Let \({\mathfrak R} f(z)=\sum_{j=1}^n z_j \frac{\partial f}{\partial z_j} (z)\) be the radial derivative of \(f\in H(\mathbb B)\) and \[ A_f(z):=(1-| z| ^2)\left( \log\frac{e}{1-| z| ^2} \right) | {\mathfrak R} f(z)| . \] The logarithmic Bloch space \(LB\) is the set of all functions \(f\in H(\mathbb B)\) for which \( \beta(f):=\sup_{z\in \mathbb B}A_f(z)<\infty,\) and its little sibling is given by \[ \text{LB}_0= \Big\{f\in \text{LB}: \lim_{| z| \to1} A_f(z)=0\Big\}. \] Let \(T_{g,\varphi}\) be defined as follows: \[ T_{g,\varphi}f(z)=\int_0^1 f(\varphi(tz)){\mathfrak R}g(tz)\,\frac{dt}{t}. \] Here, \(g\in H(\mathbb B)\) and \(\varphi\) is a holomorphic selfmap of \(\mathbb B\). The author characterizes boundedness and compactness of these operators between the logarithmic Bloch spaces LB and \(\text{LB}_0\).
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 32A37
zbMATH DE Number: 5379955
zbMATH Keywords: composition operators
zbMATH Keywords: logarithmic Bloch spaces on the ball
zbMATH Keywords: boundedness
zbMATH Keywords: compactness
MaRDI profile type: Publication
OpenAlex ID: W1985172989
DOI: 10.15352/BJMA/1240336429
description: scientific article; zbMATH DE number 5379955