Kernels of Hankel operators on the Hardy space over the bidisk

Abstract

For a Hankel operator $H_{\xi }$, $\xi \in L^{\infty }\left({\mathbb{T}}^2\right)$, on the Hardy space $H^2$ over the bidisk, $ker{H}_{\overline{\xi }}$ is an invariant subspace of $H^2$. It is known that there is an invariant subspace $M$ such that $ker{H}_{\overline{\xi }}\ne M$ for every $\xi \in L^{\infty }$. Let $\eta \in H^{\infty }$ be a nonconstant function. It is proved that if $\eta \perp \phi \left(z\right)H^2+\psi \left(w\right)H^2$ for Blaschke products $\phi \left(z\right)$ and $\psi \left(w\right)$, then $ker{H}_{\overline{\eta }}={\theta }_1\left(z\right){\theta }_2\left(w\right)H^2$ for some subproducts ${\theta }_1\left(z\right)$ and ${\theta }_2\left(w\right)$ of $\phi \left(z\right)$ and $\psi \left(w\right)$, respectively. If $\eta$ is $\ast$-cyclic, then it is easy to see that $ker{H}_{\overline{\eta }}=\left\{0\right\}$. We give some examples $\eta$ satisfying $ker{H}_{\overline{\eta }}=\left\{0\right\}$ but $\eta$ is not $\ast$-cyclic.