A Coburn type theorem on the Hardy space of the bidisk

A famous theorem of Coburn says that a nonzero Toeplitz operator on the Hardy space of the unit disk is injective or its adjoint operator is injective. In this paper we study the corresponding problem on the Hardy space of the bidisk. We first show the Coburn type theorem fails generally on the bidisk. But, we show that certain pluriharmonic symbols or product symbols of one variable functions induce Toeplitz operators satisfying the Coburn type theorem.