Let \(H^{2}(\mathbb {D}^{2})\) be the Hardy module over the bidisc, and \(M^{2}_{\psi ,\phi }\) the submodule generated by \((\psi (z)-\phi (w))^{2}\) , where \(\psi \) and \(\phi \) are two inner functions. Let \(N^{2}_{\psi ,\phi }=H^2(\mathbb {D}^2)\ominus M^{2}_{\psi ,\phi }\) be the corresponding quotient module. The submodules and quotient modules are important objects in multivariable operator theory; Wu and Yu have shown that \(N^{2}_{\psi ,\phi }\) is essential normal. In this paper, the core operator of the submodule \(M^{2}_{\psi ,\phi }=[(\psi (z)-\phi (w))^{2}]\) is proved to be Hilbert–Schmidt, and its norm is computed. Furthermore, the Hilbert–Schmidt norms of the commutators \([S_{z}^{*},S_{z}]\) , \([S_{z}^{*},S_{w}]\) and \([S_{w}^{*},S_{w}]\) on \(N^{2}_{\psi ,\phi }\) are given.
