$k$-hyponormality of weighted shifts

Abstract:

An operator $T$ is defined to be $k$-hyponormal if the operator matrix $\left ( {\left [ {{T^{ * j}},{T^i}} \right ]} \right )_{i,j = 1}^k$ is positive, where $\left [ {A,B} \right ] = AB - BA$. In A note on joint hyponormality, Proc. Amer. Math. Soc. 107 (1989), 187-195, we proved that $k$-hyponormality is equivalent to a Bram-type condition, namely, that the operator matrix $\left ( {{T^{ * j}}{T^i}} \right )_{i,j = 0}^k$ is positive. In this note we prove that for weighted shifts, $k$-hyponormality is equivalent to an Embry-type condition, namely, that the operator matrix $\left ( {{T^{ * i + j}}{T^{i + j}}} \right )_{i,j = 0}^k$ is positive. We give an example to show that this latter condition fails even for a rank one perturbation of a weighted shift. For weighted shifts this Embry condition reduces to the positivity of a sequence of $\left ( {k + 1} \right ) \times \left ( {k + 1} \right )$ Hankel matrices and we use this reduction to give a new proof of one of the principal results of Curto.