Separation for kernels of Hankel operators

Abstract:

We prove that for two Hankel operators $H_{a_{1}}$ and $H_{a_{2}}$ on the Hardy space of the unit disk either the kernel of $H_{a_{1}}^{*}H_{a_{2}}$ equals the kernel of $H_{a_{2}}$ or the kernel of $H_{a_{2}}^{*}H_{a_{1}}$ equals the kernel of $H_{a_{1}}$. In fact we prove a version of the above result for products of an arbitrary finite number of Hankel operators. Some immediate corollaries are generalizations of the result of Brown and Halmos on zero products of two Hankel operators and the result of Axler, Chang and Sarason on finite rank products of two Hankel operators. Simple examples show our results are sharp.