Hyponormality for commuting pairs of operators

In this paper we study propagation phenomena for mono-weakly hyponormal 2-variable weighted shifts. Next, we describe the positivity and determinant of an immediate ancestor of the classical Hilbert matrix. Finally, using its positivity, we construct a family of examples which shows and elucidates the gap between k-hyponormality and (k+1)-hyponormality for commuting pairs of operators for each k≥1.