Products of Toeplitz operators on the polydisk. (Q1811176)

Let \(\mathbb{D}^n\) be the polydisk in \(\mathbb{C}^n\) and \(T_f\) denote the Toeplitz operator on \(H^2(\mathbb{D}^n)\) with symbol \(f\) in \(L^\infty\). The author studies some properties of the products \(T_fT_g\) of two Toeplitz operators: (1) when the product \(T_fT_g\) is equal to zero; (2) when the product \(T_fT_g\) is still a Toeplitz operator. These problems are solved in terms of finite rank operators. Besides, it is shown that there are no compact semi-commutators \(T_fT_g-T_{fg}\) with bounded pluriharmonic symbols.