Weak positive matrices and hyponormal weighted shifts

Authors

  • H. El-Azhar
    Center of mathematical research of Rabat, Department of Mathematics, Faculty of sciences, Mohammed V University in Rabat, 4 Avenue Ibn Batouta, B.P. 1014 Rabat, Morocco
  • K. Idrissi
    Center of mathematical research of Rabat, Department of Mathematics, Faculty of sciences, Mohammed V University in Rabat, 4 Avenue Ibn Batouta, B.P. 1014 Rabat, Morocco
  • E.H. Zerouali
    Center of mathematical research of Rabat, Department of Mathematics, Faculty of sciences, Mohammed V University in Rabat, 4 Avenue Ibn Batouta, B.P. 1014 Rabat, Morocco

DOI:

https://doi.org/10.13108/2019-11-3-88

Keywords:

subnormal operators, $k$-hyponormal operators, $k$-positive matrices, weighted shifts, perturbation, moment problem.

Abstract

In the paper we study $k$-positive matrices, that is, the class of Hankel matrices, for which the $(k+1)\times(k+1)$-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 $k\times k$-block-matrix has non zero determinant if and only if all $k\times k$-block matrices have non zero determinant. We provide several applications of our main result. First, we extend the Curto-Stampfly propagation phenomena for for $2$-hyponormal weighted shift $W_\alpha$ stating that if $\alpha_k=\alpha_{k+1}$ for some $n\ge 1$, then for all $n\geq 1, \alpha_n=\alpha_k$, 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.

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Published

20.09.2019