Article
Ufa Mathematical Journal
Volume 11, Number 3, pp. 88-98
Weak positive matrices and hyponormal weighted shifts
El-Azhar H., Idrissi K., Zerouali E.H.
DOI:10.13108/2019-11-3-88
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In the paper we study k-positive matrices, that is, the class of Hankel matrices, for which the (k+1)×(k+1)-block-matrices are positive semi-definite. This notion is intimately related to a k-hyponormal weighted shift and to Stieltjes moment sequences. Using elementary determinant techniques, we prove that for a k-positive matrix, a k×k-block-matrix has non zero determinant if and only if all k×k-block matrices have non zero determinant. We provide several applications of our main result. First, we extend the Curto-Stampfly propagation phenomena for %weighted shift given
for 2-hyponormal weighted shift Wα stating that if αk=αk+1 for some n≥1, then for all n≥1,αn=αk, to k-hyponormal weighted shifts to higher order. Second, we apply this result to characterize a recursively generated weighted shift. Finally, we study the
invariance of k-hyponormal weighted shifts under one rank
perturbation. A special attention is paid to calculating the invariance interval of 2-hyponormal weighted shift; here explicit formulae are provided.