Algebraic properties of Toeplitz operators on the harmonic Dirichlet space
type: Item
instance of: scholarly article
title: Algebraic properties of Toeplitz operators on the harmonic Dirichlet space
zbMATH Open document ID: 1222.47035
author: Yong Chen
author: Young Joo Lee
author: Nguyen Quang Dieu
published in: Integral Equations and Operator Theory
publication date: 2011-03-14T00:00:00Z
review text: By the harmonic Dirichlet space \({\mathcal D}_h\) over the unit disk \(D\) of the complex plane, the authors mean the closed subspace of \({\mathcal L}^{2,1}\) (=\(L^2\)-Sobolev space of order 1 over \(D\)) consisting of harmonic functions. Given \(\varphi\in {\mathcal L}^{2,1}\), the Toeplitz operator \(T_\varphi:{\mathcal D}_h\to {\mathcal D}_h\) with symbol \(\varphi\) is densely defined by \(T_\varphi f= Q(\varphi f)\) whenever \(f\varphi\in{\mathcal L}^{2,1}\), where \(Q\) is the orthogonal projection from \({\mathcal L}^{2,1}\) onto \({\mathcal D}_h\). It is known that every function in \({\mathcal L}^{2,1}\) admits its boundary function defined a.e. via radial limits. First, the authors characterize boundedness/compactness of Toeplitz operators. Boundedness is characterized by means of the boundedness of the Poisson extension of the boundary function of the symbol and certain Carleson conditions associated with the holomorphic Dirichlet space, and compactness is characterized simply by the boundary vanishing property of the symbol. Next, the authors characterize when two Toeplitz operators commute. Their characterization is divided into three subcases, depending on whether or not each symbol satisfies a certain condition. Finally, the authors characterize when the product of two Toeplitz operator is another Toeplitz operator. As an application, they show that an operator of the form \(T_UT_V-T_H\) is compact if and only if it is of finite rank.
reviewed by: Boo Rim Choe
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 30H20
zbMATH DE Number: 5865284
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: harmonic Dirichlet space
zbMATH Keywords: commuting Toeplitz operators
zbMATH Keywords: product problem
MaRDI profile type: Publication
full work available at URL: s00020-010-1822-7
OpenAlex ID: W2014475240
cites work: A theorem of Brown-Halmos type for Bergman space Toeplitz operators
cites work: Commuting Toeplitz operators with harmonic symbols
cites work: Compact operators via the Berezin transform
cites work: Q5725935
cites work: Commuting Toeplitz operators on the Dirichlet space
cites work: Toeplitz and Hankel operators with \(L^{\infty ,1}\) symbols on Dirichlet space
cites work: Commuting Toeplitz operators on the harmonic Bergman space
cites work: Algebraic properties of Toeplitz operators on the Dirichlet space
cites work: Toeplitz operators on the Dirichlet spaces of planar domains
cites work: Multipliers of the Dirichlet space
cites work: Carleson measures and multipliers for Dirichlet spaces
cites work: Hankel and Toeplitz operators on Dirichlet spaces
cites work: Q3840238
cites work: Commutativity of Toeplitz operators on the harmonic Dirichlet space
cites work: Q3447431
DOI: 10.1007/S00020-010-1822-7
description: scientific article; zbMATH DE number 5865284