Toeplitz operators with monomial symbols on the Dirichlet spaces

In this paper, we are concerned with the various properties of the Toeplitz operators acting on the Dirichlet spaces. First, we consider the matrix representation of Toeplitz operators with harmonic and monomial symbols. Second, we establish the expansivity and contractivity of the Toeplitz operators \(T_{\varphi }\) $T_{\phi }$ with monomial symbols \(\varphi \) $\phi$ . Third, we give a necessary and sufficient conditions for the normality and hyponormality of the Toeplitz operators \(T_{\varphi }\) $T_{\phi }$ with such symbols on the Dirichlet spaces.