A theorem of Brown-Halmos type for Bergman space Toeplitz operators

We study the analogues of the Brown-Halmos theorem for Toeplitz operators o n the Bergman space. We show that for f and g harmonic, TfTg = T-h only in the trivial case, provided that h is of class C-2 with the invariant laplac ian bounded. Here the trivial cases are I or g holomorphic. From this we co nclude that the zero-product problem for harmonic symbols has only the triv ial solution. Finally, we provide examples that show that the Brown Halmos theorem fails for general symbols, even for symbols continuous up to the bo undary. (C) 2001 Academic Press.