Zero products of Toeplitz operators with \(n\)-harmonic symbols
type: Item
instance of: scholarly article
title: Zero products of Toeplitz operators with \(n\)-harmonic symbols
zbMATH Open document ID: 1126.47028
author: Boo Rim Choe
author: Hyungwoon Koo
author: Young Joo Lee
published in: Integral Equations and Operator Theory
publication date: 2007-05-03T00:00:00Z
review text: The zero-product problem for Toeplitz operators is one of the most fascinating problems in the theory of Toeplitz operators, both for the Hardy and Bergman spaces. In this paper, the authors study the problem for the Bergman space \(A^2\) of the unit polydisk \(D^n\). If \(V\) is the volume measure on \(D^n\) normalized to have total mass~\(1\), we let \(L^2=L^2(D^n,V)\) denote the usual Lebesgue space and \(A^2\) denote the Bergman space consisting of all holomorphic functions in \(L^2\). As usual, let \(P\) denote the orthogonal projection from \(L^2\) onto \(A^2\). For \(u\in L^{\infty}\), the Toeplitz operator \(T_u\) with symbol \(u\) is defined by \(T_uf=P(uf)\) for \(f\in A^2\). The zero product problem is the following: Does \(T_{u_{1}}\cdots T_{u_{N}}=0\) imply that some \(u_j\) is identically zero? \textit{A.\,Brown} and \textit{P.\,R.\thinspace Halmos} [J.~Reine Angew.\ Math.\ 213, 89--102 (1963; Zbl 0116.32501)] showed that the problem has only a trivial solution for \(N=2\) on the Hardy space. Subsequently, the same was shown to hold for \(N=5\) by \textit{K.-Y.\ Guo} [Proc.\ Am.\ Math.\ Soc.\ 124, No.\,3, 869--871 (1996; Zbl 0841.47015)] and for \(N=6\) by \textit{C.-X.\ Gu} [J.~Funct.\ Anal.\ 171, No.\,2, 483--527 (2000; Zbl 0967.47021)] on the Hardy space on the unit disk. More recently, \textit{X.-H.\ Ding} [Integral Equations Oper.\ Theory 45, No.\,4, 389--403 (2003; Zbl 1036.47015)] solved this problem for products with two factors on the Hardy space of the polydisk. For the \(A^2(\mathbb D)\) case, \textit{P.\,Ahern} and the reviewer solved the zero-product problem for two factors with harmonic symbols [J.~Funct.\ Anal.\ 187, No.\,1, 200--210 (2001; Zbl 0996.47037)] and for two factors with radial symbols [Acta.\ Sci.\ Math.\ 70, No.\,1--2, 373--378 (2004; Zbl 1072.47023)]. Most lately, \textit{B.\,R.\thinspace Choe, Y.\,J.\thinspace Lee, K.\,S.\thinspace Nam} and \textit{D.--C.\thinspace Zheng} [Math.\ Ann.\ 337, No.\,2, 295--316 (2007; Zbl 1122…
reviewed by: Željko Čučković
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 46E20
zbMATH DE Number: 5149189
zbMATH Keywords: zero product
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: Bergman space
zbMATH Keywords: polydisk
MaRDI profile type: Publication
full work available at URL: s00020-006-1444-2
OpenAlex ID: W2082864780
DOI: 10.1007/S00020-006-1444-2
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description: scientific article; zbMATH DE number 5149189