A note on weighted composition operators on Dirichlet space

In this paper, we provide some sufficient conditions for the compactness of weighted composition operators on Dirichlet space. Furthermore, we characterize the numerical range of certain classes of weighted composition operators on Dirichlet space and establish the criteria that guarantee the inclusion of zero within the numerical range. Finally, several classes of weighted composition operators are introduced, whose numerical range contains either a circular disk with a given center and radius or an elliptical disk with specified foci and axis lengths.