Zero products of Toeplitz operators with \(n\)-harmonic symbols
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 1122.47022)] extended this result to the polydisk for two factors with pluriharmonic symbols, using the methods of Ahern and the reviewer. Moreover, \textit{B.\,R.\thinspace Choe} and \textit{H.\,W.\thinspace Koo} [J.~Funct.\ Anal.\ 233, No.\,2, 307--334 (2006; Zbl 1099.47028)] solved the zero-product problem for the Bergman space of the unit ball for two factors with harmonic symbols that have local continuous extension property up to the boundary. In the present paper, the authors study the same problem for the polydisk under similar assumptions on the symbols. A function \(u\in C^2(\mathbb D^n)\) is called \(n\)-harmonic if it is harmonic in each variable separately. Let \(h^{\infty}\) denote the class of all bounded \(n\)-harmonic functions on \(\mathbb D^n\). A boundary open set \(W\) is a relatively open subset of the distinguished boundary \(T^n\). The following is the main result of the paper. Theorem. Suppose that \(u_1,u_2\in h^{\infty}\) are continuous on \(\mathbb D^n\cup W\) for some boundary open set \(W\). If \(T_{u_{1}}T_{u_{2}}=0\), then \(u_1=0\) or \(u_2=0\). Next the authors solve the zero-product problem with four factors when symbols are \(n\)-harmonic and have additional global Lipschitz continuity up to the distinguished boundary. For \(n\)-harmonic symbols that have only local Lipschitz continuity up to the distinguished boundary, they prove the same result but with only three factors. The main idea of their method is adapted from the first two authors' last cited previous paper, but going from the ball to the polydisk brings some additional technical difficulties.
scientific article; zbMATH DE number 5149189