  The characterization of the boundedness of operators induced by Hankel
matrices on analytic function spaces can be traced back to the work of Z.
Nehari and H. Widom on the Hardy space, and has been extensively studied on
many other analytic function spaces recently. However, this question remains
open in the context of the Dirichlet space [20]. By Carleson measures, the
Widom type condition and the reproducing kernel thesis, this paper provides a
comprehensive solution to this question. As a beneficial product,
characterizations of the boundedness and compactness of operators induced by
Ces\`aro type matrices on the Dirichlet space are given. In addition, we also
show that a random Dirichlet function almost surely induces a compact Hankel
type operator on the Dirichlet space.
