Products of composition and analytic Toeplitz operators
type: Item
instance of: scholarly article
title: Products of composition and analytic Toeplitz operators
zbMATH Open document ID: 1032.47015
DOI: 10.1016/S0022-247X(03)00102-1
author: Jun Soo Choa
author: Shuichi Ohno
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: Let \(X\) be either the (classical) Hardy space \(H^2\) or the Bergman space \(L_a^2\). This is a paper on operators \(X\to X\) of the form \(C_\phi\circ T_\psi\) where, for given analytic functions \(\phi:D\to D\) and \(\psi:D\to{\mathbb C}\), \(C_\phi\) is the composition operator \(f\mapsto f\circ\phi\) and \(T_\psi\) is the multiplication operator \(f\mapsto \psi\cdot f\). It is well-known that \(C_\phi\) always exists and is bounded, but \(T_\psi\) need not exist as a bounded operator. In the present paper, existence, boundedness, and compactness of \(C_\phi\circ T_\psi\) is characterized in terms of Carleson measures and appropriate Nevanlinna counting functions. Moreover, boundedness (compactness) in the Hardy case implies the same for the Bergman case. An example is given to show that \(C_\phi\circ T_\psi\) can be bounded even though \(T_\psi\) is not. Similarly, it may happen that neither \(C_\phi\) nor \(T_\psi\) is compact but \(C_\phi\circ T_\psi\) is compact.
reviewed by: Hans Jarchow
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 1956864
zbMATH Keywords: Hardy space
zbMATH Keywords: Bergman space
zbMATH Keywords: composition operator
zbMATH Keywords: multiplication operator
zbMATH Keywords: existence
zbMATH Keywords: boundedness
zbMATH Keywords: compactness
zbMATH Keywords: Carleson measures
MaRDI profile type: Publication
cites work: Multipliers of the range of composition operators
cites work: Weighted composition operators on Hardy spaces
cites work: An analytic Toeplitz operator that commutes with a compact operator and a related class of Toeplitz operators
cites work: Q4888816
cites work: Q3922907
cites work: Q4490700
cites work: Angular Derivatives and Compact Composition Operators on the Hardy and Bergman Spaces
cites work: Q4373547
cites work: The essential norm of a composition operator
cites work: Composition operators and classical function theory
cites work: Q3999007
full work available at URL: s0022-247x(03)00102-1
OpenAlex ID: W2001294034
description: scientific article; zbMATH DE number 1956864