Commutants of analytic Toeplitz operators on the harmonic Bergman space
type: Item
instance of: scholarly article
title: Commutants of analytic Toeplitz operators on the harmonic Bergman space
zbMATH Open document ID: 1088.47016
author: Boo Rim Choe
author: Young Joo Lee
published in: Integral Equations and Operator Theory
publication date: 2005-02-24T00:00:00Z
review text: The analytic Bergman space \(L^2_a\) is the closed subspace of functions in \(L^2\) of the open unit disk that are analytic, while the harmonic Bergman space \(b^2\) is the closed subspace of functions in \(L^2\) that are harmonic on the disk. The orthogonal projections of \(L^2\) onto \(L^2_a\) and \(b^2\) are denoted by \(P\) and \(Q\), respectively. For \(u\in L^\infty\) of the disk, the Toeplitz operator \(T_u\) on \(b^2\) is defined by \(T_uf= Q(uf)\). In their previous work [Mich. Math. J. 46, No. 1, 163--174 (1999; Zbl 0969.47023)], the authors proved that two analytic Toeplitz operators on \(b^2\) commute only when their symbols and 1 are linearly dependent. The main result of this paper is as follows. Theorem. Let \(f\in H^\infty\) and \(u\in L^\infty\). Assume that \(f\) is nonconstant and \(P\overline u\) is not a cyclic vector for the adjoint of the shift operator on \(L^2_a\). Then \(T_uT_f= T_fT_u\) if and only if \(u\) is a linear combination of \(f\) and 1. Whether the noncyclicity hypothesis can be removed remains open. The authors also obtain the analogous result for semi-commutants.
reviewed by: Željko Čučković
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 2138244
zbMATH Keywords: commutant
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: harmonic Bergman space
MaRDI profile type: Publication
full work available at URL: s00020-004-1338-0
OpenAlex ID: W2020413475
DOI: 10.1007/S00020-004-1338-0
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description: scientific article; zbMATH DE number 2138244