Summary: In this paper, we first prove that a dual Hankel operator \(R _{\phi }\) on the orthogonal complement of the Dirichlet space is compact for \(\phi \in W ^{1,\infty }(D)\), and then that a semicommutator of two Toeplitz operators on the Dirichlet space or two dual Toeplitz operators on the orthogonal complement of the Dirichlet space in Sobolev space is compact. We also prove that a dual Hankel operator \(R _{\phi }\) with \(\phi \in W ^{1,\infty }(D)\) is of finite rank if and only if \(B \bar{\phi}\) is orthogonal to the Dirichlet space for some finite Blaschke product \(B\), and give a sufficient and necessary condition for the semicommutator of two dual Toeplitz operators to be of finite rank.
Editor’s remark. For Part I, see [T. Yu, J. Math. Anal. Appl. 357, No. 1, 300–306 (2009; Zbl 1180.47023)].