Products of Hankel operators

This paper studies products of Hankel operators on the Hardy space. We show thatH _f ^(1)* H _f(2) H _f ^(3)* =0 for all permutation if and only if eitherH _f1 orH _f2 orH _f3 is zero. Using Douglas' localization theorem and Izuchi's theorem on Sarason's three functions problem, we show that is a sufficient condition forH _f ^* H _g H _h ^* ,H _g ^* H _f H _h ^* , andH _f ^* H _h H _g ^* to be compact.This work was partly supported by NSF grants. The second author was also partly supported by the Research Council of Vanderbilt University.