We introduce and study a class of Toeplitz operators on the Dirichlet space \({\mathcal {D}_0}\) induced by the symbol class \(\mathcal T({\mathcal {D}_0})\) defined by \({\mathcal {T}}({\mathcal {D}_0}) = \Bigg \{\psi \in h^{\infty }({\mathbb {D}}) : \left|\frac{\partial \psi }{ \partial z} \right|^{2} dA \text{ is } \text{ a } \text{ Carleson } \text{ measure } \text{ for }\,\, {\mathcal {D}}_0\Bigg \}\) , where \(h^{\infty }({\mathbb {D}})\) denotes the set of all bounded harmonic functions on \({\mathbb {D}}.\) We find that this class of Toeplitz operators corresponds to the set of all bounded operators on the Dirichlet space \({\mathcal {D}_0}\) represented by the Toeplitz matrices. We characterize the Toeplitz operators on the Dirichlet space \({\mathcal {D}_0}\) by the Brown-Halmos type operator identity \(T_{\bar{z}}AT_z=A,\) where \(T_z, T_{\bar{z}}\) are the Toeplitz operators induced by the function z and \(\bar{z}\) on the unit circle \({\mathbb {T}}\) , respectively.
