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Solution of the quadratically hyponormal completion problem

Authors: Raúl E. Curto and Woo Young Lee
Journal: Proc. Amer. Math. Soc. 131 (2003), 2479-2489
MSC (2000): Primary 47B20, 47B35, 47B37; Secondary 47-04, 47A20, 47A57
Published electronically: February 26, 2003
MathSciNet review: 1974646
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Abstract: For $m\ge 1$, let $\alpha : \alpha _{0}<\cdots <\alpha _{m}$ be a collection of ($m+1$) positive weights. The Quadratically Hyponormal Completion Problem seeks necessary and sufficient conditions on $\alpha $ to guarantee the existence of a quadratically hyponormal unilateral weighted shift $W$ with $\alpha $ as the initial segment of weights. We prove that $\alpha $ admits a quadratically hyponormal completion if and only if the self-adjoint $m\times m$matrix

\begin{displaymath}D_{m-1}(s):= \begin{pmatrix}q_{0}&\bar r_{0}&0&\hdots &0&0\\ ... ...m-2}&\bar r_{m-2}\\ 0&0&0&\hdots &r_{m-2}&q_{m-1}\end{pmatrix}\end{displaymath}

is positive and invertible, where $q_{k}:=u_{k}+\vert s\vert^{2} v_{k}$, $r_{k}:=s\sqrt {w_{k}}$, $u_{k}:=\alpha _{k}^{2}-\alpha _{k-1}^{2}$, $v_{k}:=\alpha _{k}^{2}\alpha _{k+1}^{2}-\alpha _{k-1}^{2}\alpha _{k-2}^{2}$, $w_{k}:=\alpha _{k}^{2}(\alpha _{k+1}^{2}-\alpha _{k-1}^{2})^{2}$, and, for notational convenience, $\alpha _{-2}=\alpha _{-1}=0$. As a particular case, this result shows that a collection of four positive numbers $\alpha _{0}<\alpha _{1}<\alpha _{2}<\alpha _{3}$ always admits a quadratically hyponormal completion. This provides a new qualitative criterion to distinguish quadratic hyponormality from 2-hyponormality.

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Additional Information

Raúl E. Curto
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Woo Young Lee
Affiliation: Department of Mathematics, SungKyunKwan University, Suwon 440-746, Korea
Address at time of publication: Department of Mathematics, Seoul National University, Seoul 151-742, Korea

Keywords: Weighted shifts, propagation, subnormal, $k$-hyponormal, quadratically hyponormal, completions
Received by editor(s): March 19, 2002
Published electronically: February 26, 2003
Additional Notes: The work of the first-named author was partially supported by NSF research grants DMS-9800931 and DMS-0099357
The work of the second-named author was partially supported by the Brain Korea 21 Project
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society

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