A Theorem of Brown–Halmos Type for Bergman Space Toeplitz Operators

We study the analogues of the Brown–Halmos theorem for Toeplitz operators on the Bergman space. We show that for f and g harmonic, TfTg=Th only in the trivial case, provided that h is of class C2 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.