ABSTRACT
The polynomial approach introduced in Fuhrmann [1991] is extended to cover the crucial area of AAK theory, namely the characterization of zero location of the Schmidt vectors of the Hankel operators. This is done using the duality theory developed in that paper but with a twist. First we get the standard, lower bound, estimates on the number of unstable zeroes of the minimal degree Schmidt vectors of the Hankel operator. In the case of the Schmidt vector corresponding to the smallest singular the lower bound is in fact achieved. This leads to a solution of a Bezout equation. We use this Bezout equation to introduce another Hankel operator which has singular values that are the inverse of the singular values of the original Hankel operator. Moreover the singular vectors are closely related to the original singular vectors. The lower bound estimates on the number of antistable zeroes of the new singular vectors lead to an upper bound estimate on the number of antistable zeroes of the original singular vectors. These two estimates turn out to be tight and give the correct number of antistable zeroes. From here the standard results on Hankel norm approximation and Nehari complementation follow easily.
