Product equivalence of quasihomogeneous Toeplitz operators on the harmonic Bergman space
type: Item
instance of: scholarly article
title: Product equivalence of quasihomogeneous Toeplitz operators on the harmonic Bergman space
zbMATH Open document ID: 1310.47041
author: Xing-Tang Dong
author: Ze-Hua Zhou
published in: Studia Mathematica
publication date: 2013-12-16T00:00:00Z
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 6238590
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: harmonic Bergman space
zbMATH Keywords: quasihomogeneous symbols
MaRDI profile type: Publication
full work available at URL: sm219-2-6
OpenAlex ID: W2323533625
DOI: 10.4064/SM219-2-6
review text: Let \(dA\) denote the Lebesgue area measure on the unit disk \(D\), normalized so that the measure of \(D\) equals 1, and let \(L^2(D,dA)\) be the Hilbert space of Lebesgue square integrable functions on \(D\). The harmonic Bergman space \(L^2_h\) is the closed subspace of \(L^2(D,dA)\) consisting of the harmonic functions on \(D\). The orthogonal projection from \(L^2(D,dA)\) onto \(L^2_h\) is denoted by \(Q\). Given \(z \in D\), let \(K_z(w) = 1/(1-w\bar{z})^2\) be the well-known reproducing kernel for the analytic Bergman space \(L^2_a\) consisting of all \(L^2\)-analytic functions on \(D\). The well-known Bergman projection \(P\) is then the integral operator \(Pf(z)= \int_D f(w)\overline{K_z(w)}\, dA(w)\) for \(f \in L^2(D,dA)\). Thus, \(Q\) can be represented by \(Qf = Pf + \overline{P\bar{f}} - Pf(0)\). For \(u \in L^1(D,dA)\), the Toeplitz operator \(T_u\) with symbol \(u\) is the operator on \(L^2_h\) defined by \(T_u f = Q(uf)\) for \(f \in L^2_h\). This operator is always densely defined on the polynomials and not bounded in general. The authors are interested in the case where it is bounded in the \(L^2_h\) norm, and \(u\) is a \(T\)-function. Then \(T_u\) has the continuous extension. A function \(f\) is said to be quasihomogeneous of degree \(k \in \mathbb{Z}\) if \(f(re^{i\theta})= e^{ik\theta}\varphi(r)\), where \(\varphi\) is a radial function. Let \(f_1\) and \(f_2\) be two quasihomogeneous \(T\)-functions on \(D\). In this case, the authors prove that, if there exists a \(T\)-function \(f\) such that \(T_{f_1}T_{f_2}=T_f\), then \(T_{f_2}T_{f_1}=T_f\) and \(T_{f_1}T_{f_2}=T_{f_2}T_{f_1}\).
reviewed by: Takanori Yamamoto
description: scientific article; zbMATH DE number 6238590