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Zero products of Toeplitz operators with harmonic symbols. (English) Zbl 1099.47028

The authors study the so-called zero-product problem for Toeplitz operators with bounded harmonic symbols acting on the Bergman space over the unit ball \(B\) in \(\mathbb C^n\). The main results of the article are as follows. Theorem. Suppose that \(u_1,u_2 \in h^{\infty}\) are continuous on \(B \cup W\) for some bounded open set \(W\). If \(T_{u_1}T_{u_2}=0\), then either \(u_1=0\) or \(u_2=0\). Theorem. Let \(u_1,\dots,u_{n+3} \in \text{Lip}_{\varepsilon}(\overline{B}) \cap h^{\infty}\) for some \(\varepsilon > 0\). If \(T_{u_1}\cdots T_{u_{n+3}}=0\), then \(u_j=0\) for some \(j\). Theorem. Let \(u_1,\dots,u_{n+2} \in \text{Lip}_{\varepsilon}(\zeta) \cap h^{\infty}\) for some \(\varepsilon > 0\) and \(\zeta \in \partial B\). If \(T_{u_1}\cdots T_{u_{n+2}}=0\), then \(u_j=0\) for some \(j\).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A36 Bergman spaces of functions in several complex variables
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