On the compactness of the product of Hankel operators on the sphere

Consider Hankel operators $H_\varphi$ and $H_\psi$ on the unit sphere in ${\text{\bf C}}^n$. If $n = 1$, then a necessary condition for $H^\ast _\varphi H_\psi$ to be compact is $\lim _{\vert z\vert\uparrow 1}\Vert H_\varphi k_z\Vert\Vert H_\psi k_z\Vert = 0$. We show that when $n \geq 2$, this condition is no longer necessary for $H^\ast _\varphi H_\psi$ to be compact.