Authors’ abstract: 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 \(n\geq 2\), there is no nontrivial compact product of several Toeplitz operators with pluriharmonics symbols, which is a higher dimensional phenomenon whose one-variable analogue is false.