Compact products of Toeplitz operators on the Dirichlet space of the unit ball

We consider, on the Dirichlet space of the unit ball, operators which have the form of a finite sum of products of several Toeplitz operators and then give a characterization for which such an operator is compact. We show, as an application, that for n≥2n≥2, there is no nontrivial compact product of several Toeplitz operators with pluriharmonic symbols, which is a higher dimensional phenomenon whose one-variable analogue is false.