• Kyungpook Mathematical Journal
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• 제56권3호
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• pp.899-910
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• 2016

It is known that a semi-cubically hyponormal weighted shift need not satisfy the flatness property, in which equality of two weights forces all or almost all weights to be equal. So it is a natural question to describe all semi-cubically hyponormal weighted shifts $W_{\alpha}$ with first two weights equal. Let ${\alpha}$ : 1, 1, ${\sqrt{x}}$(${\sqrt{u}}$, ${\sqrt{v}}$, ${\sqrt{w}}$)^ be a backward 3-step extension of a recursively generated weight sequence with 1 < x < u < v < w and let $W_{\alpha}$ be the associated weighted shift. In this paper we characterize completely the semi-cubical hyponormal $W_{\alpha}$ satisfying the additional assumption of the positive determinant coefficient property, which result is parallel to results for quadratic hyponormality.

• Kyungpook Mathematical Journal
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• 제48권4호
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• pp.721-727
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• 2008

Let $W_{\alpha}$ be a weighted shift with positive weight sequence ${\alpha}=\{\alpha_i\}_{i=0}^{\infty}$. The semi-cubical hyponormality of $W_{\alpha}$ is introduced and some flatness properties of $W_{\alpha}$ are discussed in this note. In particular, it is proved that if ${\alpha}_n={\alpha}_{n+1}$ for some $n{\geq}1$, ${{\alpha}_{n+k}}={\alpha}_n$ for all $k{\geq}1$.

• 대한수학회논문집
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• 제34권2호
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• pp.477-486
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• 2019

Let ${\alpha}:1,(1,{\sqrt{x}},{\sqrt{y}})^{\wedge}$ be a weight sequence with Stampfli's subnormal completion and let $W_{\alpha}$ be its associated weighted shift. In this paper we discuss some properties of the region ${\mathcal{U}}:=\{(x,y):W_{\alpha}$ is semi-cubically hyponormal} and describe the shape of the boundary of ${\mathcal{U}}$. In particular, we improve the results of [19, Theorem 4.2].