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Ufa Mathematical Journal, 2019, Volume 11, Issue 3, Pages 88–98
DOI: https://doi.org/10.13108/2019-11-3-88
(Mi ufa482)
 

Weak positive matrices and hyponormal weighted shifts

H. El-Azhar, K. Idrissi, 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
References:
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.
Keywords: subnormal operators, $k$-hyponormal operators, $k$-positive matrices, weighted shifts, perturbation, moment problem.
Funding agency Grant number
Centre National pour la Recherche Scientifique et Technique 24UM5R2015
URAC 03
Hassan II Academy of Sciences and Technologies
The first and the last author are supported by the Project URAC 03 of the National center of research and by the Hassan II Academy of Sciences and Technologies. The first author is partially supported by the CNRST grant 24UM5R2015, Morocco.
Received: 29.12.2018
Bibliographic databases:
Document Type: Article
UDC: 517.958, 517.984, 519.21
MSC: 47B37, 44A60
Language: English
Original paper language: English
Citation: H. El-Azhar, K. Idrissi, E. H. Zerouali, “Weak positive matrices and hyponormal weighted shifts”, Ufa Math. J., 11:3 (2019), 88–98
Citation in format AMSBIB
\Bibitem{El-IdrZer19}
\by H.~El-Azhar, K.~Idrissi, E.~H.~Zerouali
\paper Weak positive matrices and hyponormal weighted shifts
\jour Ufa Math. J.
\yr 2019
\vol 11
\issue 3
\pages 88--98
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\crossref{https://doi.org/10.13108/2019-11-3-88}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85081073938}
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    English version PDF:36
    References:38
     
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