H-Toeplitz operators on the Dirichlet type space

In this paper, we conducted a study of H-Toeplitz operators on the Dirichlet type space $ \mathfrak{D}_{t} $, which included several aspects. To begin, we established the matrix representation of the H-Toeplitz operator $ S_{\varphi} $ with respect to the orthonormal basis of $ \mathfrak{D}_t $. Subsequently, we characterized the compactness of $ S_{\varphi} $ in terms of the symbol $ \varphi $. Furthermore, we developed a new method to investigate the algebraic properties of H-Toeplitz operators, including self-adjointness, diagonality, co-isometry, partial isometry as well as commutativity.