• Kyungpook Mathematical Journal
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• v.59 no.2
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• pp.315-324
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• 2019

In this note we introduce a local-cubically hyponormal weighted shift of order ${\theta}$ with $0{\leq}{\theta}{\leq}{\frac{\pi}{2}}$, which is a new notion between cubic hyponormality and quadratic hyponormality of operators. We discuss the property of flatness for local-cubically hyponormal weighted shifts.

• Kyungpook Mathematical Journal
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• v.51 no.2
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• pp.187-194
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• 2011

Examples of a quartically hyponormal weighted shift which is not 3-hyponormal are discussed in this note. In [7] Exner-Jung-Park proved that if ${\alpha}$(x) : $\sqrt{x},\sqrt{\frac{2}{3}},\sqrt{\frac{3}{4}},\sqrt{\frac{4}{5}},{\cdots}$ with 0 < x ${\leq}\;\frac{53252}{100000}$, then $W_{\alpha(x)}$ is quartically hyponormal but not 4-hyponormal. And, Curto-Lee([5]) improved their result such as that if ${\alpha}(x)$ : $\sqrt{x},\sqrt{\frac{2}{3}},\sqrt{\frac{3}{4}},\sqrt{\frac{4}{5}},{\cdots}$ with 0 < x ${\leq}\;\frac{667}{990}$, then $W_{\alpha(x)}$ is quartically hyponormal but not 3-hyponormal. In this note, we improve slightly Curto-Lee's extremal value by using an algorithm and computer software tool.