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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 zeroproduct 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.

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