  We study the class of Hankel matrices for which the $k\times
k$-block-matrices are positive semi-definite. We prove that a $k\times
k$-block-matrix has non zero determinant if and only if all $k\times k$-block
matrices have non zero determinant. We use this result to extend the notion of
propagation phenomena to $k$-hyponormal weighted shifts. Finally we give a
study on invariance of $k$-hyponormal weighted shifts under one rank
perturbation.
