Essentially normal multiplication operators on the Dirichlet space
type: Item
instance of: scholarly article
title: Essentially normal multiplication operators on the Dirichlet space
zbMATH Open document ID: 0839.47023
author: Jarosław Lech
published in: Michigan Mathematical Journal
publication date: 1996-06-26T00:00:00Z
review text: Let \(D\) be the Dirichlet space of analytic functions \(f(z)= \sum^\infty_{n= 1} a_n z^n\) in the unit disc \(U\) such that \(f(0)= 0\) and \(\sum^\infty_{n= 1} n|a_n|^2< \infty\). An analytic function \(\varphi\) on \(U\) is called a multiplier of \(D\) if \(\varphi D\subset D\). Each multiplier generates a bounded multiplication operator \(M_\varphi\) on \(D\), \(M_\varphi f= \varphi f\) for \(f\in D\). About ten years ago Axler and Shields asked whether \(M^*_\varphi M_\varphi- M_\varphi M^*_\varphi\) is compact for every multiplier \(\varphi\). The paper gives a negative answer to this question. The author shows that the answer is equivalent to the problem of compactness of \(M_{\varphi'}\).
reviewed by: Jan Janas
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 784587
zbMATH Keywords: Dirichlet space of analytic functions
zbMATH Keywords: multiplier
zbMATH Keywords: bounded multiplication operator
zbMATH Keywords: compactness
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005158
OpenAlex ID: W2037498059
DOI: 10.1307/MMJ/1029005158
Recommended article: Essentially hyponormal multiplication operators on the Dirichlet space
Recommended article: Q4533105
Recommended article: Q4647665
Recommended article: Q4327361
Recommended article: The essential norm of multiplication operators on \(L_p(\mu)\)
Recommended article: On the multiplicity function of real normal operators
Recommended article: Q4285307
Recommended article: Q3108986
Recommended article: Q4021426
Recommended article: On the normality of operator products
description: scientific article; zbMATH DE number 784587