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

Xia, Jingbo

doi:10.1090/S0002-9939-07-09113-7

On the compactness of the product of Hankel operators on the sphere
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by Jingbo Xia

Proc. Amer. Math. Soc. 136 (2008), 1375-1384

DOI: https://doi.org/10.1090/S0002-9939-07-09113-7

Published electronically: November 23, 2007

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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.

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