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

Abstract:

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