Summary: For a Hankel operator \(H_\xi\), \(\xi\in L^\infty(\mathbb{T}^2)\), on the Hardy space \(H^2\) over the bidisk, \(\operatorname{ker}H_{\bar{\xi}}\) is an invariant subspace of \(H^2\). It is known that there is an invariant subspace \(M\) such that \(\operatorname{ker}H_{\bar{\xi}}\neq M\) for every \(\xi\in L^\infty\). Let \(\eta\in H^\infty\) be a nonconstant function. It is proved that, if \(\eta\bot\phi(z)H^2+\psi(w)H^2\) for Blaschke products \(\phi(z)\) and \(\psi(w)\), then \(\operatorname{ker}H_{\bar{\eta}}=\theta_1(z)\theta_2(w)H^2\) for some subproducts \(\theta_1(z)\) and \(\theta_2(w)\) of \(\phi(z)\) and \(\psi(w)\), respectively. If \(\eta\) is \(\ast\)-cyclic, then it is easy to see that \(\operatorname{ker}H_{\bar{\eta}}=\{0\}\). We give some examples \(\eta\) satisfying \(\operatorname{ker}H_{\bar{\eta}}=\{0\}\) but \(\eta\) is not \(\ast\)-cyclic.