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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

$ k$-hyponormality of weighted shifts


Authors: Scott McCullough and Vern Paulsen
Journal: Proc. Amer. Math. Soc. 116 (1992), 165-169
MSC: Primary 47B20; Secondary 47B37
MathSciNet review: 1102858
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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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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1102858-5
PII: S 0002-9939(1992)1102858-5
Article copyright: © Copyright 1992 American Mathematical Society