# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## $k$-hyponormality of weighted shiftsHTML articles powered by AMS MathViewer

by Scott McCullough and Vern Paulsen
Proc. Amer. Math. Soc. 116 (1992), 165-169 Request permission

## 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.
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