Commuting dual Toeplitz operators on the harmonic Bergman space
type: Item
instance of: scholarly article
title: Commuting dual Toeplitz operators on the harmonic Bergman space
zbMATH Open document ID: 1343.47040
published in: Science China. Mathematics
publication date: 2015-10-27T00:00:00Z
review text: Let \(D\) be the unit disk, and \(dA\) be the normalized area measure on \(D\). The harmonic Bergman space \(L^2_h\) is the closed subspace of \(L^2(D, dA)\) which consists of the harmonic functions on \(D\), and let \(Q\) be the orthogonal projection of \(L^2(D, dA)\) onto \(L^2_h\). The authors define the dual Toeplitz operator \(S_{\varphi}\) with symbol \(\varphi \in L^{\infty}(D)\), acting on \((L^2_h)^{\perp}\), by \(S_{\varphi}(f) = (I-Q)(\varphi f)\). The main result of the paper states: Let \(\varphi\), \(\psi\) be bounded harmonic functions on \(D\), then \(S_{\varphi}S_{\psi} = S_{\psi}S_{\varphi}\) if and only if one of the following conditions hold: (1) Both \(\varphi\) and \(\psi\) are analytic on \(D\). (2) Both \(\varphi\) and \(\psi\) are anti-analytic on \(D\). (3) There exist complex constants \(\alpha\) and \(\beta\), both not 0, such that \(\varphi = \alpha \psi + \beta\). A necessary and sufficient condition for \(S_{\varphi}S_{\psi} = S_{\varphi\psi}\) is also given.
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 6499353
zbMATH Keywords: dual Toeplitz operator
zbMATH Keywords: unit disk
zbMATH Keywords: harmonic Bergman space
author: Jingyu Yang
author: Yufeng Lu
MaRDI profile type: Publication
full work available at URL: s11425-014-4940-x
OpenAlex ID: W1993976499
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DOI: 10.1007/S11425-014-4940-X
description: scientific article; zbMATH DE number 6499353