A Coburn type theorem for Toeplitz operators on the Dirichlet space

A celebrated theorem of Coburn asserts that, on the setting of the Hardy space, if a Toeplitz operator is nonzero, then either it is one-to-one or its adjoint operator is one-to-one. In this paper, we show that an analogous result holds for Toeplitz operators acting on the Dirichlet space.