A Theorem of Brown–Halmos Type for Bergman Space Toeplitz Operators
A Theorem of Brown–Halmos Type for Bergman Space Toeplitz Operators
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DOI：
10.1006/jfan.2001.3811
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发表时间：
2001-12
影响因子：
1.7
通讯作者：
P. Ahern;Z. C̆uc̆ković
中科院分区：
文献类型：
--
作者：
P. Ahern;Z. C̆uc̆ković
本文研究Bergman空间上Toeplitz算子的Brown-Halmos定理的类似定理。证明了对于f和g的调和函数，只要h是C2类且具有不变Laplacian有界的，则只有在平凡的情况下，T f T g = T h.这里平凡的情况是f或g全纯的。由此我们得出结论，调和符号的零积问题只有平凡解。最后，我们提供的例子表明，布朗Halmos定理失败的一般符号，甚至符号连续的边界。
Abstract We study the analogues of the Brown–Halmos theorem for Toeplitz operators on the Bergman space. We show that for f and g harmonic, T f T g = T h only in the trivial case, provided that h is of class C 2 with the invariant laplacian bounded. Here the trivial cases are f or g holomorphic. From this we conclude that the zero-product problem for harmonic symbols has only the trivial solution. Finally, we provide examples that show that the Brown–Halmos theorem fails for general symbols, even for symbols continuous up to the boundary.